Nathanael Noir's Blog

Chasing Quantum Advantage at Light Speed

Sat, 14 Nov 2020 23:00:00 GMT

The beauty and importance of light in everyday life are clear to everyone, literally under our eyes. Not surprisingly, then, the same is true also in science, where light has always played a central role. On the one hand, the theory of vision was already studied by the Greek and Roman philosophers, as well as by the Arabs during the Islamic Golden Age. On the other hand, light itself is an excellent tool to study our universe, and its phenomena can be found in most of the investigations carried out in physics. Following the studies of Newton, Faraday and many other pioneers, our understanding was finally consolidated by Maxwell’s landmark works, which laid the foundation of modern optics. More recently, optical systems have become a ubiquitous component for technology, to the extent that it’s hard to overrate their essential contribution. In the wake of this enthusiasm, today light is about to deliver new and even deeper thrills under the magnifying lens of quantum mechanics!

The relevance of quantum mechanics for science - and beyond - is well known, with its counter-intuitive phenomena to deal with and figure out. These curious phenomena have been associated with the so-called first quantum revolution, where scientists tried to unveil the mechanisms hidden behind the new observations. One century has passed since then and we’re now entering the second quantum revolution, where scientists are no longer only spectators of weird phenomena but also use them for practical applications! This long-term vision sets a clear departure from the former, more descriptive approach. Only a few decades later, quantum technologies are now believed to dramatically improve classical approaches in several fields, ranging from the simulation and exploration of complex systems to computer science and communication. All these research areas, which come under the name of quantum information, have recently witnessed great achievements both from the theoretical and experimental side.

This notwithstanding, the stage in which quantum technologies outperform classical devices in relevant problems is still beyond our engineering skills. Indeed, for quantum technologies to tackle problems of practical interest (e.g. factorization), we need many more qubits (the quantum equivalent of the classical bit) and of better quality (less noise, longer coherence time, better connectivity etc.). For this reason, to keep everyone happy and motivated - including funding agencies! - scientists have set an intermediate goal whose relevance, albeit symbolic, has driven an enormous effort on a global scale. This goal consists in finding a problem where a quantum hardware outperforms the best classical counterpart with current technology. This means that it’s not important whether 1. the problem is useful (as long as it’s clear and the comparison is fair!) or 2. the classical algorithm is known to be optimal (i.e. in the future, more efficient algorithms could change the outcome of the challenge!). People refer to this goal as the race towards quantum supremacy or (here) towards quantum advantage. The first demonstration of quantum advantage was reported by Google in 2019 using superconducting qubits. This notwithstanding, we will see how light has played, and is still playing, a prominent role in this quest.

In this blog entry, we will briefly retrace the main steps that lead all the way from Maxwell’s equations to the recent, cutting-edge experiments that seek a photonic quantum advantage. First, we will sketch the derivation of single photons in non-relativistic quantum field theory (with integrals and derivatives for the enthusiasts). Then, we will outline the task that aims to unlock a quantum advantage with near-term technology (for the happiness of computer scientists). Finally, we will sketch recent experiments and the problem of validation (for experimentalists and down-to-earth philosophers). Let’s just make a quick remark before we start: even though qubits are the logical units at the core of quantum information, here we will not need them!

From Classical to Quantum

You have all probably seen the t-shirts with the four Maxwell equations. What if I tell you now that you don’t need 4 equations at all?

In the special theory of relativity we described electromagnetism as a field theory which follows from an action principle (or a so-called Lagrangian density). In the following we will derive the 4 Maxwell equations from an a-priori arbitrary Lagrangian. This will legitimize our principle of action a posteriori. We will then quantize this classical field theory into a so-called quantum field theory, where the “quanta” will be identified with photons.

Maxwell’s Theory from Classical Field Theory

Before we start looking for the quantized photon, a brief reminder about Maxwell’s theory from the point of view of classical field theory.

Let’s start with the Lagrangian density for Maxwell’s equations in the absence of any sources. This is simply

\mathcal{L}_{EM}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}

where the field strength is defined by

F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}

As you might know, we get the equations of motion by solving the Euler-Lagrange equations. By doing this we get two of the four famous Maxwell equations

\partial_{\mu}\left(\frac{\partial \mathcal{L}_{EM}}{\partial\left(\partial_{\mu} A_{\nu}\right)}\right)=-\partial_{\mu} F^{\mu \nu}=0

Or in terms of $A^{\mu}$ :

\left[\eta_{\mu \nu}\left(\partial^{\rho} \partial_{\rho}\right)-\partial_{\mu} \partial_{\nu}\right] A^{\nu}=0

Okay, but where are the Maxwell equations we know from highschool? To see the equations we need to go from the 4-vector notation to the 3-vector notation. Let’s define $A^{\mu}=(\phi, \vec{A}),$ then the electric field $\vec{E}$ and magnetic field $\vec{B}$ are defined by

\vec{E}=-\nabla \phi-\frac{\partial \vec{A}}{\partial t} \quad \text { and } \quad \vec{B}=\nabla \times \vec{A}

which, in terms of $F_{\mu \nu},$ becomes

F_{\mu \nu}=\left(\begin{array}{cccc} 0 & E_{x} & E_{y} & E_{z} \\ -E_{x} & 0 & -B_{z} & B_{y} \\ -E_{y} & B_{z} & 0 & -B_{x} \\ -E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)

Using the explicit form of $F_{\mu\nu}$ in the found equations of motion yields two Maxwell equations

\nabla \cdot \vec{E}=0 \quad \text { and } \quad \frac{\partial \vec{E}}{\partial t}=\nabla \times \vec{B}

The other two Maxwell equations do not result from the action principle. They are more of a geometrical consequence. We can observe that $F_{\mu \nu}$ satisfies the so called Bianchi identity.

\partial_{\lambda} F_{\mu \nu}+\partial_{\mu} F_{\nu \lambda}+\partial_{\nu} F_{\lambda \mu}=0

Using again our explicit form of $F_{\mu \nu}$ and the Bianchi identity we get the second pair of Maxwell’s equations,

\nabla \cdot \vec{B}=0 \quad \text { and } \quad \frac{\partial \vec{B}}{\partial t}=-\nabla \times \vec{E}

Radiation Field Quantization

We’ve seen, that our classical field theory reproduces electromagnetism. Now let’s quantize. Quantizing is the moment when you have to hand over the cookbook and get creative. The modern approach would of course be the path-integral formalism. But we’re staying old school and quantize canonically, but first things first:

For the quantization procedure we need to compute the the momentum $\pi^{\mu}$ conjugate to $A_{\mu}$

\begin{aligned} \Pi^{0} &=\frac{\partial \mathcal{L}}{\partial \dot{A}_{0}}=0 \quad \text { and } \quad \Pi^{i} &=\frac{\partial \mathcal{L}}{\partial \dot{A}_{i}}=-F^{0 i} \equiv E^{i} \end{aligned}

so the momentum $\Pi^{0}$ conjugate to $A_{0}$ vanishes. This means $A_{0}$ is not a dynamical field, while $A_i$ is our old friend, the electric field $E^{i}$ .

I’ve been hiding something so far. The Lagrangian has a very large symmetry group, acting on the vector potential as

A_{\mu}(x) \rightarrow A_{\mu}(x)+\partial_{\mu} \chi(x)

for any function $\chi(x) .$ We’ll ask only that $\chi(x)$ dies off suitably quickly at spatial $\vec{x} \rightarrow \infty .$ We call this a gauge symmetry. The field strength is invariant under the gauge symmetry:

F_{\mu \nu} \rightarrow \partial_{\mu}\left(A_{\nu}+\partial_{\nu} \lambda\right)-\partial_{\nu}\left(A_{\mu}+\partial_{\mu} \lambda\right)=F_{\mu \nu}

As always, gauge symmetries spoil that there is a redundancy of the system, therefore we’ll use a gauge. Here we choose the so-called Coulomb or radiation gauge ( $\nabla \cdot \vec{A}=0$ and $\Phi=0$ ). This has the great advantage that the physical degrees of freedom are manifest. However, we’ll lose Lorentz invariance.

In radiation gauge (vector fields satisfying the Coulomb gauge are also called transverse fields) the equations of motions read

\partial_{\mu} \partial^{\mu} \vec{A}=0

which we can solve in the usual way by using plane wave solutions,

\vec{A}=\int \frac{d^{3} p}{(2 \pi)^{3}} \vec{\xi}(\vec{p}) e^{i p \cdot x}

with $p_{0}^{2}=|\vec{p}|^{2}$ . The constraint $\nabla \cdot \vec{A}=0$ tells us that $\vec{\xi}$ must satisfy

\begin{array}{l} \vec{\xi} \cdot \vec{p}=0 \\ \end{array}

which means that $\vec{\xi}$ is perpendicular to the direction of motion $\vec{p}$ . We can pick $\vec{\xi}(\vec{p})$ to be a linear combination of two orthonormal and real vectors $\vec{\epsilon}_{r}$ with $r=1,2,$ each of which satisfies $\vec{\epsilon}_{r}(\vec{p}) \cdot \vec{p}=0$ and

\vec{\epsilon}_{r}(\vec{p}) \cdot \vec{\epsilon}_{s}(\vec{p})=\delta_{r s} \quad r, s=1,2

These two vectors correspond to the two polarization states of the photon.

To quantize we turn the Poisson brackets into commutators. Our first guess would probably be

\begin{array}{l} {\left[\hat{A}_{i}(\vec{x}), \hat{A}_{j}(\vec{y})\right]=\left[\hat{E}^{i}(\vec{x}), \hat{E}^{j}(\vec{y})\right]=0} \quad \text { and } \quad {\left[\hat{A}_{i}(\vec{x}), \hat{E}^{j}(\vec{y})\right]=i \delta_{i}^{j} \delta^{(3)}(\vec{x}-\vec{y})} \end{array}

But there is something wrong, because it’s not consistent with the constraints. We still want to have $\nabla \cdot \hat{\vec{A}}=\nabla \cdot \vec{\hat{E}}=0$ imposed on the operators. But then we would get a weird contradiction

0=[\nabla \cdot \hat{\vec{A}}(\vec{x}), \nabla \cdot \hat{\vec{E}}(\vec{y})]=i \nabla^{2} \delta^{(3)}(\vec{x}-\vec{y}) \neq 0

What’s going on? In imposing the commutator relations we haven’t correctly taken into account the constraints. Clearly, the fix to this problem is to select the following commutator ¹,

\left[\hat{A}_{i}(\vec{x}), \hat{E}_{j}(\vec{y})\right]=i\left(\delta_{i j}-\frac{\partial_{i} \partial_{j}}{\nabla^{2}}\right) \delta^{(3)}(\vec{x}-\vec{y})

To see that this is now consistent with the constraints, we can rewrite the right-hand side of the commutator in momentum space,

\left[\hat{A}_{i}(\vec{x}), \hat{E}_{j}(\vec{y})\right]=i \int \frac{d^{3} p}{(2 \pi)^{3}}\left(\delta_{i j}-\frac{p_{i} p_{j}}{|\vec{p}|^{2}}\right) e^{i \vec{p} \cdot(\vec{x}-\vec{y})}

which is now consistent with the constraints, for example

\left[\partial_{i} \hat{A}_{i}(\vec{x}), \hat{E}_{j}(\vec{y})\right]=i \int \frac{d^{3} p}{(2 \pi)^{3}}\left(\delta_{i j}-\frac{p_{i} p_{j}}{|\vec{p}|^{2}}\right) i p_{i} e^{i \vec{p} \cdot(\vec{x}-\vec{y})}=0

We now write $\hat{\vec{A}}$ in the usual mode expansion,

\begin{aligned} \hat{\vec{A}}(\vec{x}) &=\int \frac{d^{3} p}{(2 \pi)^{3}} \frac{1}{\sqrt{2|\vec{p}|}} \sum_{r=1}^{2} \vec{\epsilon}_{r}(\vec{p})\left[a_{r}(\vec{p}) e^{i \vec{p} \cdot \vec{x}}+a_{r}(\vec{p})^{\dagger} e^{-i \vec{p} \cdot \vec{x}}\right] \\ \hat{\vec{E}}(\vec{x}) &=\int \frac{d^{3} p}{(2 \pi)^{3}}(-i) \sqrt{\frac{|\vec{p}|}{2}} \sum_{r=1}^{2} \vec{\epsilon}_{r}(\vec{p})\left[a_{r}(\vec{p}) e^{i \vec{p} \cdot \vec{x}}-a_{r}(\vec{p})^{ \dagger} e^{-i \vec{p} \cdot \vec{x}}\right] \end{aligned}

where, as before, the polarization vectors satisfy

\vec{\epsilon}_{r}(\vec{p}) \cdot \vec{p}=0 \quad \text { and } \quad \vec{\epsilon}_{r}(\vec{p}) \cdot \vec{\epsilon}_{s}(\vec{p})=\delta_{r s}

Inserting the mode expansion into the commutator relations will give us the commutation relations for the creation and annihilation operators, which are the usual ones for creation and annihilation operators

\left[a_{r}(\vec{p}), a_{s}(\vec{q})\right]=\left[a_{r}(\vec{p})^{ \dagger},a_{s}(\vec{q})^{\dagger}\right]=0 \qquad \qquad \left[a_{r}(\vec{p}), a_{s}(\vec{q})^{\dagger}\right]=(2 \pi)^{3} \delta^{rs} \delta^{(3)}(\vec{p}-\vec{q})

where, in deriving this, we need the completeness relation for the polarization vectors,

\sum_{r=1}^{2} \epsilon_{r}^{i}(\vec{p}) \epsilon_{r}^{j}(\vec{p})=\delta^{i j}-\frac{p^{i} p^{j}}{|\vec{p}|^{2}}

The Hamilton operator is then (after normal ordering, and playing around with $\vec{\epsilon}_{r}$ )

H=\int \frac{d^{3} p}{(2 \pi)^{3}}|\vec{p}| \sum_{r=1}^{2} a_{r}(\vec{p})^{ \dagger} a_{r}(\vec{p})

For every ( $\vec{p}$ , $r$ ) we can then define a basis states $\lvert n \rangle_{\vec{p}, r}$ that are eigenstates of $H$ , with eigenvectors $E_n(\vec{p}) = n \, \hbar \, \omega_{\vec{p}}$ . If we then define the number operator $\hat{n}=\sum_{r} \,\int d\vec{p} \; \hat{a}_r^\dagger (\vec{p}) \, \hat{a}_r (\vec{p})$ , with $\hat{n} \lvert n \rangle_{\vec{p}, r} =n \lvert n \rangle_{\vec{p}, r}$ , we finally arrive to the Fock states $\lvert n \rangle_{\vec{p}, r}$ , which satisfy the well-known relations

\hat{a}_{r}(\vec{p}) \lvert n \rangle_{\vec{p}, r}=\sqrt{n} \lvert n-1 \rangle_{\vec{p}, r} \qquad \qquad \hat{a}_{r}(\vec{p})^\dagger \lvert n \rangle_{\vec{p}, r}=\sqrt{n+1} \lvert n \rangle_{\vec{p}, r}

This suggests that we interpret $\lvert n \rangle_{\vec{p}, r}$ as a state with $n$ particles created by $\hat{a}_{r}^\dagger (\bm{p})$ and destroyed by $\hat{a}_r (\bm{p})$ . Photons are born!

Evolution of Fock States

For closed systems, any Hermitian operator $\hat{O}$ evolves under the action of a unitary operator $U(t)$ given by the complex exponentiation of $\mathcal{H}$ , i.e. $\hat{O}(t)=U(t) \,\hat{O} \,U^\dagger(t)$ . Single-photon wave packet evolution is then described, once again, by the creation and annihilation operators

\hat{a}_\lambda (\bm{k},t)= \hat{a}_\lambda (\bm{k}) \, e^{-i \omega_{\bm{k}} t} \qquad \qquad \lvert 1 \rangle_{j \lambda}= \hat{b}_{j\lambda}^{\dagger} \lvert 0 \rangle = \int d\bm{k} \; \alpha_j (\bm{k}) \, \hat{a}_\lambda^{\dagger} (\bm{k}) \, \lvert 0 \rangle \qquad \qquad \lvert n \rangle_{j\lambda}=\frac{(\hat{b}_{j\lambda}^\dagger)^n}{\sqrt{n!}} \lvert 0 \rangle

where the wave packet mode function $\alpha_j (\bm{k})$ can be Lorentzian or Gaussian. $U(t)$ can be described using the formalism of Bogoliubov transformations of the mode operators $\hat{a}^\dagger$ , whose generators are the interaction Hamiltonians of beam splitters and phase shifters (the most elementary components of linear-optical interferometers) shown in Fig. 1.

Action of the main optical elements in second quantization: phase shifter (a), beam splitter (b) and polarization rotation (c). The indices $j$ and $\lambda$ describe the spatial mode and the polarization, respectively. (Fig. 1)

In this formalism, the most general two-mode interaction that mixes mode operators is given by

\left\{ \begin{array}{ll} \ \hat{c}_i = \sum_j A_{ij} \, \hat{a}_j+B_{ij} \, \hat{a}_j^\dagger \\ \ \hat{c}_i^\dagger = \sum_j B_{ij}^\ast \, \hat{a}_j + A_{ij}^\ast \, \hat{a}_j^\dagger \\ \end{array} \right.

with mixing generator $\mathcal{H}\propto\sum_i \hat{c}_i^\dagger \, \hat{c}_i$ . Here, for simplicity the indices $(i,j)$ include also the polarization degree of freedom. In the non-mixing case ( $\hat a^\dagger \rightarrow \hat a^\dagger$ ), one finds that

\hat U \lvert n_1,...,n_m \rangle = \prod_i^m \frac{1}{\sqrt{n_i !}} \left( \sum_{k_i}^m U_{{k_i},i} \, \hat{a}_{k_i}^\dagger \right)^{n_i} \, \lvert 0_1,...,0_m \rangle

$m$ being the number of optical modes. While this expression can look frightening, there is a very simple way to visualize the physics behind! Here’s the intuition (see also Fig. 2):

the outer product runs over all optical modes $i$ , each with $n_i$ photons.
the factor in parentheses describes the single-photon evolution: a photon enters mode $i$ and exits in superposition over the $m$ modes. The exponent is there because there are $n_i$ photons in mode $i$ !
the normalization factor $n_i!$ takes care of the $n_i$ -particle permutations for the input state.

Next we can derive the scattering amplitude from an input Fock state $\bm{s}=(s_1,...,s_m)$ ( $s_i$ is the number of photons in mode $i$ ) to the output $\bm{t}$

\langle t_1,...,t_m \lvert \hat U \lvert s_1,...,s_m \rangle = \left( \prod_{i=1}^m s_i! \, t_i! \right)^{-\frac{1}{2}} \textup{Per\,} (U_{S,T})

where $U_{S,T}$ is constructed by taking $s_k$ ( $t_k$ ) times the $k$ th row (column) of $U$ , and

\textup{Per}\,(A) = \sum_{ \sigma } \prod_{i=1}^m a_{i, \sigma_i}

is the permanent of a matrix $(A_{m \times m})_{i j}=a_{i j}$ , where the sum is over all permutations $\{\sigma\}$ of $\left[1,..,m\right]$ .

A comment for the curious reader: you’re right, the permanent looks just like the determinant (without the sign permutation)! This is due to the particle statistics: the former (latter) describes bosonic (fermionic) evolutions, with states that are symmetric (anti-symmetric) under particle permutations.

Sketch of a Boson Sampling experiment in the original formulation. From left to right, $n$ photons enter a linear-optical interferometer implementing an $m \times m$ Haar-random unitary transformation $U$ . Interestingly, any unitary transformation can be implemented using only phase shifters and beam splitters (see Fig. 1 : a-b). $U$ shuffles the creation operators in such a way that, at the output, photons exit in superposition over all modes. When the output state is measured, photons are found in a random combination of optical modes, which corresponds to sampling from the unknown underlying probability distribution. Here $n=3$ , while to show quantum advantage we need $n > 50$ (in the ideal case without losses and partial distinguishability… otherwise things become more complicated!). (Fig. 2)

From single photons to the quest for quantum advantage

So far we discovered that the natural evolution of single photons in a linear optical interferometer is described by the permanent of a certain matrix. Cool… so what? Well, it turns out that this innocent-looking function is extremely hard to evaluate on a computer. Can this fact suggest a way to highlight a separation between quantum and classical systems in terms of, say, performance?

This is precisely the idea behind Boson Sampling, a computational problem introduced by Aaronson and Arkhipov (AA) in 2010 . The task is to sample from the output distribution of an interferometer implementing an $m \times m$ unitary evolution $U$ on an input state with $n$ indistinguishable bosons. While a passive quantum device can efficiently solve this task by evolving single photons (the required physical resources scale polynomially in $m$ and $n$ ), a classical computer would be confronted with the monstrous computational complexity of the permanent. Computing the permanent is indeed $\#P$ -hard (it’s tough even for the simplest 0-1 matrices): the best-known algorithm requires $\mathcal{O}(n\,2^n)$ operations! This hardness was numerically tested in 2016, where it took one hour to the most powerful supercomputer with a matrix of size roughly $50 \times 50$ . For an interesting comparison, despite the same number of terms, symmetries in the sign permutation reduce the determinant’s complexity to $\mathcal{O}(n^3)$ .

Below we will briefly overview the key points behind the proof of hardness of Boson Sampling. It’s a little bit hard to follow, especially since the author of this post finds it obscure. Still, it’s interesting to retrace the main ideas and connections that made this result possible. If today this is not your priority, you can directly jump to the end of this section and I’ll join you in a moment!

If you're brave - click here!

The proof of hardness by AA is based on two precedent results:

Stockmeyer: Estimating within a multiplicative error the probability that an efficient randomized algorithm accepts a certain input is in $BPP^{NP}$ .
Troyansky and Tishby: It is always possible to efficiently write an $n \times n$ complex matrix $V$ as a sub-matrix of an $m \times m$ unitary matrix $U$ ( $m > 2n$ ), modulo a rescaling of $V$ by $\epsilon < 1/||V||$ .

Suppose an efficient classical algorithm $cA$ exists to solve Boson Sampling. Using result (2), one can embed an $n \times n$ matrix $M$ in the upper-left corner of a unitary $U$ describing the evolution of the input Fock state. $cA$ outputs the state (1,…,1,0,…,0) with a probability proportional to $|\textup{Per}(M)|^2$ . At the same time, Stockmeyer’s counting algorithm can estimate this probability to within multiplicative error, which implies that a $\#P$ -complete problem is in $BPP^{NP}$ . From here, by summoning Toda’s and Sipser-Lautermann’s theorems, AA obtain

PH = P^{\#P} = BPP^{NP}= NP^{NP^{NP}}

i.e. the whole polynomial hierarchy is contained in its third level, which is considered… well… quite unlikely.

However, the complexity of the permanent is largely reduced if we just seek an approximation. Incidentally, an approximation for matrices with non-negative real entries to within a multiplicative error $\pm \, \epsilon \,|\textup{Per(A)}|$ or additive error $\pm \, \epsilon \,||\textup{U}||$ can be obtained in $poly(n,\frac{1}{\epsilon})$ with a probabilistic algorithm. The proof for the approximate case makes two widely believed assumptions:

Permanent anti-concentration: $\forall X \sim \mathcal{N}^{m \times m}_{\mathcal{C}}\, \exists \, \textup{polynomial} \,Q | \, \forall \,(m, \delta)>0: \textup{prob}\left( \textup{Per}(X)<\frac{\sqrt{m!}}{Q(m,\frac{1}{\delta})} \right)<\delta$ .
Gaussian permanent estimation ( $\textup{GPE}{\times}$ ): Approximating $\textup{Per}(X)$ for a matrix $X \sim \mathcal{N}^{m \times m}_{\mathcal{C}}$ of i.i.d. Gaussian elements to within a multiplicative ( $\times$ ) error is $\#P$ -hard.

Finally, the core of the proof by AA goes through the following two results:

$\textup{GPE}_{\pm}$ : If an efficient algorithm exists to sample from a distribution $\epsilon$ -close to the Boson Sampling distribution in Total Variation Distance, then $|\textup{GPE}|^2$ with an additive ( $\pm$ ) error is in $BPP^{NP}$ .
Haar-unitary hiding: Any $n \times n$ sub-matrix of an $m\times m$ Haar-random unitary is close in variation distance to a Gaussian matrix if $m > n^5 \log^2 n$ .

Combining the above observations, Aaronson and Arkhipov finally prove that a classical polynomial-time algorithm for the approximate Boson Sampling would imply the collapse of the entire polynomial hierarchy in computational complexity theory… which is quite unlikely to say the least!

What does this mean? As you may guess, it means that there’s no way for a classical approach, however fast and smart, to solve this problem in an efficient way. For computer scientists and philosophers, this also represents an attack to the extended Church-Turing thesis, which states that “a probabilistic Turing machine can efficiently simulate any realistic model of computation”. Since a quantum device only needs to run the experiment (efficient in the number of photons and modes), Boson Sampling provides a clear route towards a photonic demonstration of quantum advantage!

From Theory to Experimental Demonstration

The ideas presented in the previous section may appear obscure at first sight. Unfortunately, on a second thought they’re actually even more complicated. Yet, we have good news: an actual experiment is easy to describe instead!

Boson Sampling experiments…

The simplest implementation of Boson Sampling can be divided in three main stages (see Fig. 2):

Generation of $n$ indistinguishable photons. Why indistinguishable? Because if photons were distinguishable, then an $n$ -photon experiment would have the same statistics of $n$ experiments with single photons, which is polynomial in $n$ to simulate. Much more complicated (and interesting!) is the case of partial distinguishability and photon losses, for which many recent studies have tried to gauge the trade-off (among all relevant physical quantities) that prevents a classical computer from solving the problem.
Evolution of $n$ photons in a linear optical interferometer implementing an $m \times m$ ( $m>n^2$ ) unitary transformation $U$ randomly selected according to the Haar measure. Why do we want $U$ to be Haar-random? Well, loosely speaking to avoid simplifications, more specifically to fulfill the assumptions of the problem! n.b. It is very easy to pick a Haar-random $m \times m$ unitary: just use a random unit vector in $\mathcal{C}_m$ for its first row, then a random unit vector orthogonal to the first one for the second row etc.
Detection of single photons in the output modes. The measured output combinations of Fock states correspond to ampling from the underlying (in general unknown) scattering probability distribution.

Despite its apparent simplicity, requiring no entanglement, adaptive measurements or ancillary systems, its experimental demonstration represents a real challenge for our technological state of the art! The first proof-of-concept experiments, reported in 2013, used $n=3$ photons in up to $m=6$ modes, while a threshold of $n \sim 50$ (or more) indistinguishable photons is now considered to be necessary to claim quantum advantage. Various approaches have been proposed to push its limits, both with photons (with discrete and continuous variables) and other systems such as trapped ions. Photonics probably remains the favorite platform in this race towards quantum advantage via Boson Sampling, especially considering the improvements reported in the last few years. At the time of writing (end of 2020), the world record is set by an experiment with $n=20$ photons (14 detected) in $m=60$ modes … and rumors suggest that the target threshold might already be achieved.

… and the problem of validation

Did you really think that was the end of our quest? Bad news: there’s still one issue to take care of! Is it really that bad, though? In this and the past section we’ve seen the ingredients to achieve quantum advantage via Boson Sampling. What happens, then, when one enters that regime? How can classical people be sure that this is really true?

In the hard-to-simulate regime, where a classical computer cannot (efficiently) simulate a quantum device, the following question becomes crucial: how do we verify its correct operation? The answer to this dilemma involves a new class of classical algorithms that aim to validate the outcomes of a quantum machine. While a full certification is exponentially hard to perform, for complexity reasons similar to the ones presented here, we can make some assumptions to simplify this challenge. A common (non-negligible!) simplification is that experiments are run either with fully distinguishable or fully indistinguishable photons. With this and other assumptions, people have developed a set of validation protocols to accept or reject a Boson Sampling experiment. Ideally, these algorithms should be 1. scalable in the number of photons, 2. sample-efficient (in the lab we only have a finite number of measurements!) and 3. reliable (we would not appreciate a false positive).

The list of validation protocols is rather rich, each with its own strong points and limitations. Here we sketch the ideas behind two of them, which well represent the state of the art in this area:

Bayesian hypothesis testing: this approach uses Bayesian inference to identify the most likely between the scenarios with fully distinguishable (D) or indistinguishable (I) photons. The intuition is that it’s more likely to measure $n$ -photon states from the scenario for which their corresponding probability is higher. This idea is quantified using Bayes’ theorem to update our confidence in each hypothesis. The advantages of this approach are its simplicity and sample-efficiency, the drawback is that it requires the evaluation of permanents, which makes it unfeasible when $n \sim 30$ .
Statistical benchmark: this approach uses statistical features of two-mode correlators in the measured $n$ -photon states. These features form clusters in different regions of a suitable space, where each cluster is associated to one of the two scenarios (D, I). By studying what cluster the measured data falls in, and using analytical predictions from random matrix theory, we can cast the validation task as a problem of classification.

Outro

To present the quest for quantum advantage in a compact way, here we had to make a sacrifice and leave out many interesting results. For instance, it would have been nice to add some context on the evolution of optical quantum computing and quantum information, to better appreciate the impact of Boson Sampling. Also, for the sake of brevity we couldn’t present any Boson Sampling experiments, for which we instead refer to a very nice review. Similarly, the role of imperfections would have deserved much more attention, and we could have mentioned relevant studies on the influence of losses and partial distinguishability in Boson Sampling. For a detailed and comprehensive review on these topics, the interested reader can have a look at the following works.

Finally, one last consideration about the journey we just went through. We’ve seen how elegantly light can be described in the classical and quantum framework, and how this is leading us to a demonstration of quantum advantage. This is all fantastic and eye-opening… but is this the end of the story? Well, of course not! While Boson Sampling was designed only to unlock a quantum advantage - with no practical application in mind - it has spurred an entire community to make progress on an unprecedented scale. Remarkable results have been achieved both from the theory side of multiphoton interference and that of the technological platforms. Moreover, Boson Sampling has inspired novel approaches to enhance classical computers with quantum optics. Gaussian Boson Sampling, for instance, can be used to simulate vibronic spectra in molecular dynamics, or to tackle hard problems based on graphs.

Indeed, this certainly doesn’t look like the end of the story, rather like a luminous beginning.

About the Author

Hi! I’m Fulvio Flamini and I’m happy to see you here! Who am I? That’s fun, thanks for asking! During the day, I’m a postdoc in theoretical quantum machine learning, but I was trained as an experimentalist in integrated photonics. At night, I unleash my snail-like creativity writing unfathomable stories or creating a card game for outreach. By the way, when friends joke that I’m good at everything, I just call it being great at nothing :-) One of my aims in life is helping people develop critical thinking and appreciate the wonders of nature. If you know how, or just feel curious about it, you know how to find me.

You can reach me on any of these platforms.

If you’re a mathematician - please don’t kill me for the next one.↩

Gravitational Waves - An Invitation

Fri, 04 Sep 2020 22:00:00 GMT

Gravitational Waves. An Invitation

Albert Einstein originally predicted the existence of gravitational waves in 1916, on the basis of his theory of general relativity. General relativity interprets gravity as a geometric property of spacetime. Einstein also predicted that events in the cosmos would cause ‘ripples’ in spacetime. These ripples are distortions of spacetime itself, which would spread outward, although they would be so minuscule that they would be nearly impossible to detect by any technology foreseen at that time.

The first direct observation of gravitational waves was made on September 14, 2015 and was announced by LIGO (4-kilometer-long interferometers in the United States) and Virgo (3-kilometer-long detector in Italy) in 2016. Previously, gravitational waves had only been inferred indirectly, via their effect on the timing of pulsars in binary star systems. The waveform, detected by both LIGO observatories, matched the predictions of general relativity for a gravitational wave emanating from the inward spiral and merger of a pair of black holes of around 36 and 29 solar masses and the subsequent ‘ringdown’ of the single resulting black hole.

Recently, researchers have detected a signal from what may be the most massive black hole merger yet observed in gravitational waves. The product of the merger is the first clear detection of an ‘intermediate-mass’ black hole, with a mass between 100 and 1.000 times that of the sun. They detected the signal (labeled as GW190521), on May 21, 2019 at LIGO.

The signal, resembling about four short wiggles, is extremely brief in duration, lasting less than one-tenth of a second. From what the researchers can tell, GW190521 was generated by a source that is roughly 5 gigaparsecs away, when the universe was about half its age, making it one of the most distant gravitational-wave sources detected so far.

Many people, with all kinds of backgrounds, are currently talking and discussing this recent event. Many wonder how exactly this is possible and how exactly Einstein already knew what was going on out there in the seemingly infinite space. This blog serves as a stimulation for the curious minded. I tried to keep it short and simple. If there is anything more fascinating than these unbelievable mysterious objects that shake our space and time, then it is the human ability to think about such things. Predicting them, measuring them and hopefully using them for good.

Introduction

There are three essential parts when it comes to any kind of radiation. Here we will formulate it for gravitational radiation. The parts are:

Find and solve a source free equation.
Detection: How do waves influence matter?
Creation: How does matter create waves?

The first point needs a solution of the Einstein equations without sources. To be more precise we want to find a source free wave equation, because they are capable of transmitting energy through a region where there are no sources. In this part we will use a lot of approximations, which are totally fine, since gravitational waves are being originated at events very very far from us. By the time they reach us, they are propagating as a solution of Einstein equations with zero source.

The second point means solving the geodesic equation in the geometry found within the first point. Here we can again use approximations due to the very small amplitudes of the waves reaching us.

The third point asks for solving Einstein’s equations with source.

Find and solve a source free equation.

For the first part we need to linearize general relativity in a similar fashion how it is usually done to get the Newtonian Limit ¹. In this case we can not assume a static approximation because a static approximation does simply not make sense for waves, since we have intrinsically time dependence. So one starts with a flat geometry (often called the background) with small fluctuations depending on $x^\mu$

g_{\mu \nu}\simeq \eta_{\mu \nu}+h_{\mu \nu}(x) + \mathcal{O}(h_{\mu \nu}(x)^2)

The inverse metric is given as

g^{\mu \nu}\simeq\eta^{\mu \nu}-\eta^{\mu \alpha} \eta^{\nu \beta} h_{\alpha \beta}(x) + \mathcal{O}(h_{\mu \nu}(x)^2)

with $\left\|h_{\mu \nu}\right\| \ll 1$ . Next we insert this metric into the source free Einstein equations $R_{\mu \nu}$ in trace-reverse form to find an expression for $\eta_{\mu \nu}(x)$ ².

Why is the last equation considered as the inverse?

Notice that to ensure $g_{\lambda \mu} g^{\mu \nu}=\delta_{\lambda}^{\nu}$ , for a small perturbation $h_{\mu \nu}$ with $\left\|h_{\mu \nu}\right\| \ll 1,$ follows:

\begin{aligned} g_{\lambda \mu} g^{\mu \nu} &=\left(\eta_{\lambda \mu}+h_{\lambda \mu}\right)\left(\eta^{\mu \nu}-\eta^{\mu \alpha} \eta^{\nu \beta} h_{\alpha \beta}\right)=\eta_{\lambda \mu} \eta^{\mu \nu}-\eta_{\lambda \mu} \eta^{\mu \alpha} \eta^{\nu \beta} h_{\alpha \beta}+\eta^{\mu \nu} h_{\lambda \mu}+O\left(h^{2}\right) \\ &=\delta_{\lambda}^{\nu}-\delta_{\lambda}^{\alpha} \eta^{\nu \beta} h_{\alpha \beta}+\eta^{\mu \nu} h_{\lambda \mu}=\delta_{\lambda}^{\nu}-\eta^{\nu \beta} h_{\lambda \beta}+\eta^{\mu \nu} h_{\lambda \mu}=\delta_{\lambda}^{\nu} \end{aligned}

ignoring $\mathcal{O}(h^2)$ and higher.

Let’s calculate the Christoffel symbols

\Gamma_{\alpha \beta}^{\rho}=\frac{1}{2} \eta^{\rho \sigma}\left(\partial_{\alpha} h_{\beta \sigma}+\partial_{\beta} h_{\sigma \alpha}-\partial_{\sigma} h_{\alpha \beta}\right)

since $\partial_{\lambda} \eta_{\mu \nu}=0$ and we are neglecting again terms of order $\mathcal{O}\left(h^{2}\right) .$

The Riemann curvature tensor is

R_{\mu \sigma \nu}^{\rho}=\partial_{\sigma} \Gamma_{\nu \mu}^{\rho}-\partial_{\nu} \Gamma_{\sigma \mu}^{\rho}+\Gamma_{\sigma \lambda}^{\rho} \Gamma_{\nu \mu}^{\lambda}-\Gamma_{\nu \lambda}^{\rho} \Gamma_{\sigma \mu}^{\lambda}

where the last two terms can be ignored because they are $\mathcal{O}\left(h^{2}\right) .$

The Ricci tensor then becomes

\begin{aligned} R_{\mu \nu}\equiv R_{\mu \rho \nu}^{\rho} &=\partial_{\rho} \Gamma_{\nu \mu}^{\rho}-\partial_{\nu} \Gamma_{\rho \mu}^{\rho} \\ &=\frac{1}{2} \eta^{\rho \sigma}\left[\partial_{\rho} \partial_{\nu} h_{\mu \sigma}+\partial_{\rho} \partial_{\mu} h_{\sigma \nu}-\partial_{\rho} \partial_{\sigma} h_{\nu \mu}-\partial_{\nu} \partial_{\rho} h_{\mu \sigma}-\partial_{\nu} \partial_{\mu} h_{\sigma \rho}+\partial_{\nu} \partial_{\sigma} h_{\rho \mu}\right] \\ &=\frac{1}{2} \eta^{\rho \sigma}\left[\partial_{\rho} \partial_{\mu} h_{\sigma \nu}-\partial_{\rho} \partial_{\sigma} h_{\nu \mu}-\partial_{\nu} \partial_{\mu} h_{\sigma \rho}+\partial_{\nu} \partial_{\sigma} h_{\rho \mu}\right] \end{aligned}

We can now define $V_{\mu}\equiv \partial_{\rho} h_{\mu}^{\rho}-\frac{1}{2} \partial_{\mu} h_{\rho}^{\rho}$ and thus the partial derivative reads $\partial_{\nu} V_{\mu}=\partial_{\nu} \partial_{\rho} h_{\mu}^{\rho}-\frac{1}{2} \partial_{\nu} \partial_{\mu} h_{\rho}^{\rho}$ where $h_{\rho}^{\rho}$ is the trace.

With the definition $\square \equiv \partial_{\rho} \partial^{\rho}$ which is $\partial_{\rho} \partial^{\rho}=-\frac{\partial^{2}}{\partial t^{2}}+\vec{\nabla}^{2}$ one gets

R_{\mu \nu}=\frac{1}{2}\left[-\square h_{\mu \nu}+\partial_{\mu} V_{\nu}+\partial_{\nu} V_{\mu}\right]=0

for the Ricci tensor. This looks like a kind of wave equation with some extra chunk. But we can do better. Gauge freedom allows, despite the fact that we had to choose certain coordinates such that $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$ , to make coordinate changes which preserves $\eta_{\mu \nu}$ , but will generally change the form of $h_{\mu \nu}(x)$ .

If one transforms $x^{\mu} \rightarrow x^{\mu^{\prime}}=x^{\mu}+\delta^{\mu}(x)$ , which preserves our form of the metric as a flat backgroud + fluctuations with a small $\delta^{\mu}$ . Recall that the metric changes under coordinate transformations as usual

g_{\mu \nu} \rightarrow g_{\mu^{\prime} \nu^{\prime}}=\frac{\partial x^{\mu}}{\partial x^{\mu^{\prime}}} \frac{\partial x^{\nu}}{\partial x^{\nu^{\prime}}} g_{\mu \nu}

So we get

g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu} \rightarrow g_{\mu^{\prime} \nu^{\prime}}=\eta_{\mu \nu}+\underbrace{h_{\mu \nu}-\partial_{\mu} \delta_{\nu}-\partial_{\nu} \delta_{\mu}}_{h_{\mu^{\prime} \nu^{\prime}}}

where $\delta_{\mu}=\eta_{\mu \nu} \delta^{\nu}$ .

Now we can compare $h_{\mu \nu} \rightarrow h_{\mu \nu}^{\prime}=h_{\mu \nu}-\partial_{\mu} \delta_{\nu}-\partial_{\nu} \delta_{\mu}$ with $A_{\mu} \rightarrow A_{\mu}^{\prime}=A_{\mu}-\partial_{\mu} \Phi$ which we know from electromagnetism. The form of the first equation is very similar to the second one, but taking care that the ‘potential’ has two indices instead of one. A useful aspect of Gauge freedom is that the physical degrees of freedom do not change. In this case, the physical curvature $R_{\nu \lambda \rho}^{\mu}$ is unchanged so that solutions to $R_{\mu \nu}=0$ remain solutions. Here, one can use a gauge so that $V_{\mu} \rightarrow V_{\mu}^{\prime}=\partial_{\alpha} h_{\mu}^{\alpha}-\frac{1}{2} \partial_{\mu} h_{\alpha}^{\alpha}=0 .$

From $R_{\mu \nu}=-\frac{1}{2} \square h_{\mu \nu}=0$ follows that $\square h_{\mu \nu}=0$ .

So we finally found

\square h_{\mu \nu}=0

which is a wave equation, so we we can immediately write down a plane wave solution.

Let’s make the usual Ansatz

h_{\mu \nu}=a_{\mu \nu} e^{i k_{\lambda} x^{\lambda}}

where one can think of $a_{\mu \nu}$ as the polarization and the amplitude of the wave. Feeding this into $\square h_{\mu \nu}=$ $-k_{\lambda} k^{\lambda} h_{\mu \nu}=0$ gives $k_{\lambda} k^{\lambda}=0$ since $h_{\mu \nu}=0$ is a solution but not a very interesting one. Thus $k^{\lambda}$ is a null vector, and one can write it as $k^{\lambda}=(|\vec{k}|, \vec{k})$ with $k_{\lambda} k^{\lambda}=-|\vec{k}|^{2}+\vec{k} \cdot \vec{k}=0 .$ With the frequency $\omega=|\vec{k}|$ and the wavelength $\lambda=\frac{2 \pi}{\omega}$ the wave travels with phase velocity $v=\lambda \frac{\omega}{2 \pi}=1$ in the direction $\vec{k}$ . Thus, the gravitational wave travels with the speed of light as one would expect because the gravitational fluctuations are massless.

The symmetric $4 \times 4$ matrix (10 independent components) $a_{\mu \nu}$ describes amplitude and polarization of the wave and can be simplified because using any $\delta_{\mu}$ so that $\square \delta_{\mu}=0$ these four functions can be used to make any four components of $h_{\mu \nu}$ vanish identically (Residual gauge freedom). Choosing $h_{t i}=0$ (3 degrees of freedom) and $h_{\mu}^{\mu}=0$ (traceless) (1 degree of freedom) or $a_{t i}=0$ and $a_{\mu}^{\mu}=0$ represents three plus one terms such that the ten degrees of freedom of the symmetric matrix $a_{\mu \nu}$ are reduced to six independent components. With the previous gauge condition $V_{\mu}$ follows

\begin{array}{ll} V_{t}=\partial_{\rho} h^{\rho}{ }_{t}-\frac{1}{2} \partial_{t} h_{\rho}^{\rho}=\partial_{t} h_{t}^{t}=i \omega a_{t t} e^{i k_{\lambda} x^{\lambda}}=0 & \Rightarrow \quad a_{t t}=0\\ V_{i}=\partial_{\rho} h_{i}^{\rho}-\frac{1}{2} \partial_{i} h_{\rho}^{\rho}=\partial_{j} h_{i}^{i}=i k_{j} a_{j i} e^{i k_{\lambda} x^{\lambda}}=0 & \Rightarrow \quad k^ja_{j i}=0 \\ \end{array}

where the equation $k^{j} a_{j i}=0$ means that the waves are transverse or, in other words, that the spatial wave vector $\vec{k}$ is ‘perpendicular’ to the polarization tensor. If the wave vector is chosen to be $k^{\mu}=(\omega, 0,0, \omega)$ with the spatial part $\vec{k}=(0,0, \omega)$ then $a_{z i}=0$ follows from the transversality condition. Thus only two independent components of $a_{\mu \nu}v$ remain which are $a_{x x}=-a_{y y}($ traceless $)$ and $a_{x y}=a_{y x}$ (symmetric). This additional choice is called transverse-traceless gauge, and

h_{\mu \nu}(x)=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & a & b & 0 \\ 0 & b & -a & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) e^{i k_{\lambda} x^{\lambda}}

is the final form of $h_{\mu \nu}(x) .$

In the linearized theory one gets more general solutions by adding solutions of this form. This is, however, not possible in the full-blown version of general relativity which is highly non-linear. So, the part of finding a source free wave equation is done.

Detection: How do waves influence matter?

The next part is the detection of gravitational waves. With the metric solution $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}(x)$ where $\eta_{\mu \nu}$ is flat spacetime and $h_{\mu \nu}(x)$ is defined in our last equation. We can now explore how test particles respond to this time-dependent geometry using the geodesic equation

\frac{d^{2} x^{\mu}}{d \lambda^{2}}+\Gamma_{\alpha \beta}^{\mu} \frac{d x^{\alpha}}{d \lambda} \frac{d x^{\beta}}{d \lambda}=0 \quad \Rightarrow \quad \frac{d U^{\mu}}{d \tau}+\Gamma_{\alpha \beta}^{\mu} U^{\alpha} U^{\beta}=0 \quad \Rightarrow \quad \frac{d U^{\mu}}{d \tau}=-\Gamma_{\alpha \beta}^{\mu} U^{\alpha} U^{\beta}

for timelike and $U^{\mu}=\frac{d x^{\mu}}{d \tau} .$ If the particle is at rest for $\tau=0$ such that $U^{\mu}(0)=(1,0,0,0)$ then

\frac{d U^{\mu}}{d \tau}(0)=-\Gamma_{00}^{\mu}=-\frac{1}{2} \eta^{\rho \sigma}\left(\partial_{0} h_{\sigma 0}+\partial_{0} h_{0 \sigma}-\partial_{\sigma} h_{00}\right)=0

since $h_{t i}=h_{t t}=0$ and this means that if the particle begins at rest, it remains at rest as the wave passes! However, this is just a statement that the coordinate position of the mass is unchanged because the particle is at rest and the acceleration is zero. Thus, maybe two test masses and the distance between them show an effect.

Actually, one could have anticipated the need for at least two particles from the equivalence principle. Observing only one particle does not allow to distinguish whether the lab is in the gravitational field of the earth or being accelerated. Only if one observes two particles one can detect tidal forces because the two particles come closer because they move towards the center of the earth while in the case of acceleration they move on parallel paths and keep the distance they initially had. Detecting curvature is impossible with only one test mass because one can always find coordinates in which the particle is and stays at rest. With two test masses one can see whether the distance between them remains constant or not.

Observing only one particle does not allow to distinguish whether the lab is in a gravitational field or being accelerated (a) vs. observing two particle does allow to distinguish whether the lab is in a gravitational field or being accelerated due to geodesic deviation (b). (Fig. 1)

For one mass at the spatial coordinate (0,0,0) and another mass at spatial coordinate $(\varepsilon, 0,0)$ one finds for the distance

\begin{aligned} \int \sqrt{d s^{2}} &=\int \sqrt{g_{\mu \nu} d x^{\mu} d x^{\nu}}=\int_{0}^{\varepsilon} \sqrt{g_{x x}} d x \\ & \approx \sqrt{g_{x x}(x=0)} \varepsilon=\sqrt{1+h_{x x}(x=0)} \varepsilon \approx\left[1+\frac{1}{2} h_{x x}(x=0)\right] \varepsilon=\left[1+a e^{i k_{\lambda} x^{\lambda}}\right] \varepsilon \end{aligned}

Look! This varies now with time. Despite the fact that one particle remains at $x=0$ and the other at $x=\varepsilon$ , the invariant distance between them changes because this value varies with time. This is the difference between the coordinates and the physical reality. In the chosen coordinates the two masses do not move, but in reality they move with respect to each other.

To get a better idea of what a gravitational wave does, the values $a$ and $b$ in our found expression of $h_{\mu \nu}$ are chosen such that $a$ is small and $b$ is zero, and the wave travels along the $z$ -axis. This gives

h_{\mu \nu}(x)=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & -a & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \sin (k z-\omega t)

for the real part. A system of masses is set up in the $x y$ -plane at $z=0$ such that one mass is at the center and the others build a ring around it. The gravitational wave comes along the $z$ -axis perpendicular to the $x y$ -plane. The metric at $z=0$ is

\left.d s^{2}\right|_{z=0}=-d t^{2}+\big[1-a \sin (\omega t)\big]\, d x^{2}+\big[1+a \sin (\omega t)\big]\, d y^{2}

and with $X=\left(1-\frac{1}{2} a \sin (\omega t)\right) x$ and $Y=\left(1+\frac{1}{2} a \sin (\omega t)\right) y$ the metric becomes

\left.d s^{2}\right|_{z=0}=-d t^{2}+d X^{2}+d Y^{2}

plus terms of order $\mathcal{O}\left(a^{2}\right) .$ This is now flat Minkowski space, and one can visualize the geometry with Euclidean intuition. The scenario for the test particle can now be illustrated as in (Fig. 2).

The ring of test particle lies in the X-Y-plane – The gravitational wave passes through the particles along a line perpendicular to the plane of the particles, i.e. following the observer's line of vision into the screen. (Fig. 2)

The result is called the plus-polarization (+ polarization) presented in (Fig. 3(a)).

If one instead chooses the values $a$ and $b$ such that $b$ is small and $a$ is zero, while the wave still travels along the $z$ -axis, the real part of the perturbation of the metric is

h_{\mu \nu}(x)=\left(\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \sin (k z-\omega t)

and corresponds to

\left.d s^{2}\right|_{z=0}=-d t^{2}+d x^{2}+2 b \sin (\omega t)\; d x\; d y+d y^{2}

for the distance. With $X=x+\frac{1}{2} b \sin (\omega t) \;y$ and $Y=y+\frac{1}{2} b \sin (\omega t) \;x$ the metric becomes

\left.d s^{2}\right|_{z=0}=-d t^{2}+d X^{2}+d Y^{2}

because $d X=d x+\frac{1}{2} b \sin (\omega t) \;d y$ and $d Y=d y+\frac{1}{2} b \sin (\omega t)\; d x$ as well as $d X^{2} \approx d x^{2}+b \sin (\omega t)\; d x\; d y$ and $d Y^{2} \approx d y^{2}+b \sin (\omega t) \;d x\; d y .$ This is again flat Minkowski space, and the result is called the cross-polarization ( $\times$ polarization $)$ shown in (Fig. 3(b)).

Polarization of the gravitational wave – The effect of a plus-polarized gravitational wave on a ring of particles (a). The effect of a cross-polarized gravitational wave on a ring of particles (b). (Fig. 3)

These are the two independent polarization states of the plane gravitational wave. The polarization of the electromagnetic wave can similarly be decomposed into an $x$ -polarization and a $y$ -polarization. The difference though is that the electromagnetic wave is invariant if one flips it by $180^{\circ}$ while the gravitational wave is invariant if one flips it by $90^{\circ} .$ One can tie that to the fact that the graviton has spin two and the photon has spin one.

To really detect gravitational waves one could use several such rings of masses because one does not know the direction in which the wave comes and the ring should be perpendicular to this direction. With a ruler one could measure how the ring changes. The ruler does not expand and contract the same way because the above analysis used the geodesic equation for free test particles, and the atoms building the ruler are not free but also experience electromagnetic binding forces which swamp the gravitational distortion. Tiny displacement in physics are not measured with rulers but with interferometers.

The LIGO (Laser Interferometer Gravitational-Wave Observatory) uses four kilometer long Michelson interferometers with mirrors attached to free test masses which are actually hanging but are free to swing. The accuracy is in the order of $10^{-21}$ and the change of length is in the order of $10^{-18}\; \mathrm{m} .$ So much noise has to be eliminated that quantum fluctuations must be filtered out. There are other projects planned. The LISA (Laser Interferometer Space Antenna) project uses satellites in space with a length scale for the arms of $5 \cdot 10^{6}\; \mathrm{km},$ and the PTR (Pulsar Timing Arrays) will observe irregularities in what should be periodic signals from pulsars.

Creation: How does matter create waves?

The last question related to gravitational waves is how they get created. Thus one has to solve Einstein’s equations in the presence of sources, and one cannot use small approximations because one wants to see a signal big enough to be measured. One can create electromagnetic waves by moving charges as in an antenna to get uniform radiation, but for gravitational wave production there is nothing popping energy into the system to make it steady state. One has a time-dependent system which produces the radiation. If two black holes come close then they merge and become a single black hole. Such an event has been observed by LIGO. Studying realistic gravity wave generation is difficult and will not be shown here, but there are two interesting features. One is the possibility of multipole expansion and the other is the observable signals from the binary mergers of two black holes.

Firstly, just like any other form of radiation, one can take the far field limit and do a multipole expansion of the power distribution. For both electromagnetic and gravitational waves, the monopole contribution vanishes because of conservation of charge and mass. The leading electromagnetic term is dipole. For General Relativity, however, also the dipole term vanishes because of conservation of angular momentum and the ability to coordinatize to zero the center of mass motion. Thus the lowest term is quadropole. Secondly, appreciable signals can arise from binary mergers. In particular, when massive black holes merge they can release gravitational wave energies in the order of the mass of the sun. To analyze a merger, the problem is often broken up into stages. Everything could in principle be done numerically, but Einstein’s equations are hard and one would have to simulate over a large region to get far-field behavior.

Two black holes merge in three phases. The first phase is called inspiral, and one can use post-Newtonian approximations (linear approximations) to address the two-body problem. The second phase is called merger, and one uses numerical calculations to handle it. The third and last phase is called ringdown where the resulting black hole still wobbles before it settles down to a Kerr black hole, and one uses single-body black hole perturbation theory for calculations.

One of the fascinating things about black hole mergers compared to other merger events is the ringdown signature, since ringdown only happens for black holes, its observation is a direct observation of black holes.

The 'chirp' characteristic of a typical black hole merge - Polarization of the gravitational wave – Inspiral phase (1), merge phase (2) and the ringdown (3). (Fig. 4)

Apart from my self-made illustration, there are some pretty neat simulations on YouTube, such as this one:

References

Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll.
General Relativity by Rober M. Wald.
Gravitational Waves: Volume I: Theory and Experiments by Michele Maggiore.
Gravitational Waves: Volume II: Astrophysical Sources by Michele Maggiore.
Elements of General Relativity by Piotr T. Chrusciel.

This is often called post-Newtonian approximation in gravitational physics.↩
If you want to make your life even harder, you can also choose a much more complicated geometry. For example, one could replace $\eta$ with the Schwarzschild Metric or a suitable metric to model a cosmological scenario. But that is left as an exercise to you.↩

The Last Frontier

Sat, 29 Aug 2020 22:00:00 GMT

Introduction

In 1967 Bryce DeWitt published his famous trilogy on quantum gravity. ”Quantum Theory of Gravity. I. The Canonical Theory” was the title of his first paper discussing the canonical quantization approach of Einstein’s general theory of relativity. He performs the well known Dirac quantization procedure, emphasising in details on the pitfalls he encounters, such as the factor ordering problem and the problem of time. Nevertheless, he triumphs to write an equation that incorporates quantum mechanics and gravity, a Schrödinger-Einstein type of equation, better known today as the Wheeler-DeWitt equation. The equation though was far from complete, and DeWitt was fully aware of this issue. Yet, he carries on attempting to make sense of it. The Wheeler-DeWitt equation is an equation of motion of $\Psi$ , the wave functional of the universe. Upon analysing the geodesics of $\Psi$ , i.e. the possible trajectories the universe can follow, DeWitt realizes a peculiar feature. The universe may ultimately hit a frontier of infinite curvature beyond which it ceases to exist. Is this a physical boundary that our universe will one day reach? Perhaps it is a mere artefact of the equation, or perhaps the Wheeler-DeWitt equation is telling us something about the fate of the universe, like the possible scenario of collapsing into a black hole. Or maybe the Wheeler-DeWitt equation makes no sense to begin with. In any case, the nature of this frontier remains unknown.

This article does not intend to answer this question. DeWitt never disclosed it neither will I attempt to. However, we will take a couple of steps back to the beginning of the story, and grant the reader the freedom to put an end to it.

Once Upon A 3-Geometry

How do you study the evolution of a structure as our universe? Regardless of our humble attempts and our minuscule size compared to this universe that we know no boundary of, to study its evolution is to study the evolution of its geometry throughout time. To be precise, its 3-geometry.

What exactly is a 3-geometry? Let’s start with a simple example, a line. It is a 1-dimensional object. The only thing you can draw on that line is a a set of points which, if you connect, will give you another line. Now take a surface, a 2-dimensional plane. One can draw a set of points equidistant from a center such that, upon connecting the points, we get a circle, a unit-sphere or a 1-sphere. It is a 1-dimensional surface enclosing a 2-dimensional disk. Now go up in dimensions to 3 and take a cube. One can draw a set of points equidistant from a center to obtain a 2-sphere, which is the commonly known term we use for a sphere. It is a 2-dimensional surface enclosing a 3-dimensional 3-ball (In Euclidean space of dimension $n$ , an $n$ -ball is the volume bounded by a surface known as an ( $n-1$ )-sphere). But now what happens if we go up in dimensions another time to 4, and again draw points equidistant from a center. Well, you get a hyper-sphere, a 3-sphere! It is a 3-dimensional surface enclosing a 4-ball.

What does this have to do with the universe? The way to look at it, we live on a 2-dimensional surface (2-sphere) of a 3-dimensional object (3-ball) we call Earth. Now we and the Earth altogether live inside a 3-dimensional object, our universe. Whether this universe is a 3-sphere or not is still an open question, but it is some 3-manifold with some 3-geometry, each of which lead to consequences on the way we view the universe and study its geometrical properties (distances, angles…). Furthermore, whether this 3-manifold is enclosing a 4-dimensional space is irrelevant. What we are interested in is the study of how the 3-geometry of this 3-manifold behave and evolve. An expanded discussion on the shape of the universe can be found here.

A 3-geometry can be studied through its metric tensor $\gamma_{ij}$ , a measure of distances on that 3-manifold. In cosmology, we usually put constraints on how this 3-manifold should be, such as homogeneity and isotropy. The line element is then reduced from 10 to 2 degrees of freedom and can be written as:

\text{d}\Sigma^2=\gamma_{ij}dx^idx^j= a^2(t)\bigg[\text{d}\chi^2+f^2_{\kappa}(\chi)\bigg(\text{d}\theta^2+\text{sin}^2(\theta)\text{d}\phi^2\bigg)\bigg]

When we look at the night sky and observe the universe, what we are really looking at is a snapshot of this 3-geometry which evolves through time. Of course one can argue here that there exists relativistic effects and that the deeper we look at the night sky the older in time we are viewing things, thus our snapshot extends to several 3-geometries. However, cosmologists are fully aware of these effects, and can construct 3-geometries in such a way that we are in a positions of an external observer taking snapshots from the outside. Here we can start to feel the need for the spirit of general relativity, to merge space and time into a 4-dimensional manifold, namely spacetime. Yes, this is a 4-manifold with a 4-geometry expressed through the metric $g_{\mu\nu}$ whose line elements is:

\text{d}s^2=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=-dt^2+d\Sigma^2 \;\;.

The metric tensor $g_{\mu\nu}$ is the solution to Einstein’s field equations:

R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8{\pi}G}{c^4}T_{\mu\nu}

which is there to tell us how space (3-geometry) and time are linked in a region of a given energy and matter content expressed through the Energy-Momentum tensor $T_{\mu\nu}$ . But at the end of the day, when we study the universe and simulate orbits of merging blackholes and neutron stars and exploding supernovae, we break down spacetime into just space, evolving through time. This “breaking down” process is known as the $3+1$ decomposition (see Fig. 1), and it serves to explain the dynamics of the 3-geometry. Therefore, the dynamical object under study here is just space, not spacetime. This is similar as in particle dynamics, where the dynamical object is not $x$ and $t$ , but only $x$ . This is by no means abandoning the notion of spacetime! Spacetime is there, time is a dynamical parameter, but not a dynamical object. After all, this is what Einstein’s general theory of relativity is about, it is a theory of geometrodynamics: of 3-geometry, not 4-geometry.

3+1 decomposition of 4-manifold spacetime $\mathcal{M}$ into foliations of spacelike hypersurfaces $\Sigma$ (3-manifolds) (Fig. 1)

The Arena of Geometrodynamics

I hope by now I convinced you that the dynamics of the universe is that of its 3-geometry. This should answer the question posed at the beginning of the last section. This concept will be crucial to understand what follows.

As the universe (the 3-manifold $\Sigma$ ) evolves, so does its 3-geometry which we will label as $^{(3)}\mathfrak{G}$ . It evolves through stretching, bending and curving. We assume that $\Sigma$ does not change its topology. No holes or cuttings are allowed, only diffeomorphisms. This means that at every instant of time, the $\Sigma_t$ is diffeomorphic to $\Sigma_{t-1}$ . One can see that the $^{(3)}\mathfrak{G}$ is an equivalence class of metrics that are transformable, one into another, by diffeomorphisms.

One can track the change of $^{(3)}\mathfrak{G}$ and build some kind of a configuration space where every single point in the space refers to an entire $^{(3)}\mathfrak{G}$ . This space is known as Superspace, and it is our arena of geometrodynamics. The term superspace was coined somewhere in the 1950’s by John A. Wheeler. Superspace can be seen as the “space of all spaces”, specifically Riemannian spaces, and those which are related by a diffeomorphism. Formally the superspace of a 3-manifold $\Sigma$ is the quotient set:

S(\Sigma)= \frac{\text{Riem}\;\Sigma}{\text{Diff}\;\Sigma}\;\;.

Now there are infinitely many spaces $\Sigma$ related by a diffeomorphism, hence the superspace has infinitely many points. Furthermore, there are infinitely many spaces that are not related by a diffeomorphisms, therefore, there are infinitely many superspaces, each of which correspond to a 3-manifold $\Sigma$ . For the time being, we will only consider our universe and its evolution, which makes up a point in one of those superspaces.

As the universe evolves, it traces a path $\mathcal{H}$ of 3-manifolds ( $\Sigma_t,\Sigma_{t+1},\Sigma_{t+2}...$ ) each of a $^{(3)}\mathfrak{G}$ diffeomorphic to the other (see Fig. 2). Or to put in elegantly:

^{(3)}{}{\mathfrak{G}}(\Sigma_i) \cong^{(3)}\mathfrak{G}(\Sigma_{i+1}) \;\;\;i=0,1,2...

where diffeomorphic is denoted by $\cong$ .

Superspace $\mathcal{S}$ containing “points” A, B, C… each of which is a 3-geometry diffeomorphic to the other and tracing the path $\mathcal{H}$ . (Remastered artwork; original: J.A. Wheeler.) (Fig. 2)

The word trace is important here, because it gives a notion of time. You see, the spirit of general relativity and spacetime cannot be avoided as we previously claimed. In fact, the path $\mathcal{H}$ is a very familiar object to us, it consists of all those spacelike 3-geometries that can be obtained as spacelike sections through one particular 4-geometry (the 4-geometry of spacetime), that which satisfies Einstein’s classical field equations!

”Time” conceived in these terms means nothing more or less than the location of the $^{(3)}{}{\mathfrak{G}}$ in the $^{(4)}{}{\mathfrak{G}}$ . In this sense “3-geometry is a carrier of information about time”
— J.A.Wheeler

Quantum Geometrodynamics

Wheeler’s ideas of superspace and geometrodynamics did not spring as an alternative way of understanding the behavior of the universe. After all, Einstein’s theory of general relativity served that particular purpose. Wheeler’s main motivation was to go beyond the classical limit, to understand the quantum nature of the universe, quantum geometrodynamics.

Everyone knew back then that general relativity is incomplete. The presence of physical singularities, such as blackholes or the Big Bang, are barriers beyond which we cannot understand. They are a warning sign telling us that our equations no longer work there, and the only way to pass beyond that singularity is by finding an alternative route to what we presently have. General relativity has passed countless tests and predicted phenomena Einstein wrote down 100 years before their discovery. The answer to the problems posed by general relativity do not lie in an alternative description of gravity, the theory works fine in the classical limit. It is in the quantum limit where one needs to rethink about his concept of spacetime. The answer is in a theory of quantum gravity, that which Wheeler hoped to achieve.

To achieve this, Wheeler collaborated with Bryce S. DeWitt, our second protagonist of this story. Their fruitful discussions resulted in a somewhat “Einstein-Schrödinger” equation, a wave equation of $\Psi[^{(3)}\mathfrak{G}]$ , the wave functional of 3-geometries, or loosely speaking the wave functional of the universe. Today this equation is known as the Wheeler-DeWitt equation:

\mathcal{H} \Psi\left[^{(3)} \mathfrak{G}\right]=\left(G_{i j k l} \frac{\delta}{\delta \gamma_{i j}} \frac{\delta}{\delta \gamma_{k l}}+\gamma^{\frac{1}{2}(3)} R\right) \Psi\left[^{(3)} \mathfrak{G}\right]=0

DeWitt extensively analysed this equation. He pointed out its flaws, and frequently called it ”that damn equation”, for it was strange, and full of nasty features. Nevertheless, he was keen in understanding what it meant. One particular aspect DeWitt wanted to understand is the nature of superspace. He was aware that it was a manifold by itself and wrote:

$\mathcal{S}$ ¹ is the domain manifold for the state functional $\Psi$ , and the $^{(3)}\mathfrak{G}$ are its “points.”

Similar to what Wheeler taught us. But he asks the question:

Can one assign a metric, or a pseudo-Riemannian structure, to $\mathcal{S}$ ?

From the Wheeler-DeWitt equation, DeWitt realizes that the object $G_{ijkl}$ can be seen as a contravariant metric of some 6-dimensional Riemannian manifold $M$ . He emphasizes that such an interpretation is inappropriate because $G_{ijkl}$ is part of the operator $\mathcal{H}$ , but is nevertheless useful to study the properties of the manifold it defines. DeWitt was hoping to connect the properties of $M$ to $\mathcal{S}$ .

Now given any metric tensor, one can follow the rules of differential geometry and compute the connection coefficients and the curvature and other properties of the manifold defined by this metric. For $G_{ijkl}$ , DeWitt computed the curvature, now 6-Ricci (equivalent to the 4-Ricci of spacetime):

^{(6)}{}{R}=-\frac{60}{\zeta^2}

Moreover, upon analysing the geodesics in $M$ , i.e. trajectories of free falling particles, he noticed that all geodesics ultimately hit a frontier at $\zeta=0$ . Looking back at the curvature at $\zeta=0$ , we notice the peculiar feature that the curvature $^{(6)}{}{R}$ is infinity! It appears to be that there is some kind of a blockage at $\zeta=0$ , a frontier, beyond which no point in $M$ can go. Does this frontier exist in the superspace $S$ aswell?

DeWitt rewrites the Wheeler-DeWitt equation in another form:

\left[-\frac{\delta^{2}}{\delta \zeta^{2}}+\frac{(32 / 3)}{\zeta^{2}} \bar{G}^{A B} \frac{\delta^{2}}{\delta \zeta^{A} \delta \zeta^{B}}+(3 / 32) \zeta^{2} {}^{(3)}R\right] \times \Psi\left[^{(3)} \mathfrak{G}\right]=0

Here, it clearly shows the presence of a singularity at $\zeta=0$ . But whether the geodesic incompleteness in $\textit{M}$ leads to a geodesic incompleteness in $\mathcal{S}$ is unclear. Could it be that there is a point in superspace beyond which the 3-geometries such as our universe cannot exist? And if so, what does it mean to tell us about the fate of our universe?

“The question at issue is whether the frontier in $\textit{M}$ generates a corresponding barrier in $\mathcal{S}$ beyond which there is no possibility of extending the state functional. Unfortunately, in the present state of our knowledge no clear-cut answer can be given to this question.” — Bryce DeWitt

It seems that the problem of singularities is following us. Wheeler’s hope of a theory of quantum gravity through quantum geometrodynamics was in vain. DeWitt noticed these problems early on and dedicated most of his research to other approaches to quantum gravity that rely on path integral techniques. But then again, the question remains:

“Is the Wheeler-DeWitt equation and the frontier of any meaning?”

Superspace is a scenario where a frontier, marked with the red cross, is present, blockading the evolution of 3-geometries. (Fig. 3)

References

Quantum Theory of Gravity. I. The Canonical Theory Bryce S. DeWitt Phys. Rev. 160, 1113 - Published 25 August 1967.
Superspace and the Nature of Quantum Geometrodynamics. J.A. Wheeler (Princeton U.) 1968.

About the Author

Hi! My name is Ali Lezeik. I’m a human among a billion other and a mind of many. In this short life span we have, we are constantly fascinated by the beauty of the world, many of which we have explained and even more left unanswered. It is this beauty that gives me a meaning, and probably to many of you. And it is the journey, despite the frustrations, that gives me joy. So who am I? I like to think of myself as those old voyagers who sailed to the unknown, and came back home to tell the world about the treasures they found and the stories they lived. I might not know how to sail the oceans, but I am learning how to sail the abstract world of physics and how it connects to our world. This is my journey, and these are the stories I would like to share with you.

You can reach me on any of these platforms.

In the original paper he uses the letter $\mathfrak{M}$ to denote the superspace.↩

General Relativity as a Field Theory

Mon, 27 Jul 2020 22:00:00 GMT

In this post, I’ll try to explain an alternative route to the famous Einstein Field Equations (EFE) through the principle of least action. All our fundamental theories of physics are described by action principles and gravity is no different. The action principle in general is a relatively simple yet powerful prescription to derive the equations of motion, which in our case will be the Einstein Field Equations. Further it gives an insight to the broad field of variational calculus and some rather technical manipulations which are used in many different areas - therefore, every physicist should know them.

At the end of this – hopefully – rather short blog post you should have gotten an idea of how general relativity can be derived from the Lagrangian formalism and how this gives us a basis to discover new attempts to study general relativity in an alternative way.

Before we get started, huge thanks to my followers on Twitter. I originally tried this on Twitter as a micro blog, but it is rather complicated to explain such an equation-loaded topic within 120 characters. However, I’ve gotten very good and supportive feedback and I’m convinced I can help some people to get a compact and simple but yet unadulterated understanding of this topic. The original post can be found here.

Introduction

The most dangerous black holes, accelerating galaxies and waves of pure gravitation hide behind a plain and simple action. The Einstein-Hilbert action in general relativity is the the gravitational part that yields the Einstein Equations in vacuum through the principle of least action. This is sometimes referred to as the Lagrangian formulation of general relativity. Here I will just quote the EH-action in a aesthetic and coordinate-free version.

It’s very impressive how innocent this equation looks, but it harbors incredible secrets which are still being explored and researched to this day.

\mathcal{S}_{E H}=\frac{1}{2 \kappa} \int_{\mathcal{M}} \star\; \mathcal{R}

The exact form of this action will no longer be interesting here, as we as physicists are often interested in concrete results and therefore have to choose coordinates.

But why exactly does the action have this form?

The Einstein-Hilbert Action

The argument is very simple. Integrating over a manifold $\mathcal{M}$ needs a volume-form. Happily, the metric provides a canonical volume form, which we can then multiply by any scalar function. Our dynamical variable is now the metric $g_{\alpha \beta}$ . Since we know that the metric can be set equal to its canonical form and its first derivatives set to zero at any one point, any nontrivial scalar must involve at least second derivatives of the metric. The Riemann tensor is of course made from second derivatives of the metric, and from general relativity we know that the only independent scalar we could construct from the Riemann tensor is the Ricci scalar $\mathcal{R}$ . Further it is true, that any nontrivial tensor made from products of the metric and its first and second derivatives can be expressed in terms of the metric and the Riemann tensor. Therefore, the only independent scalar constructed from the metric, which is no higher than second order in its derivatives, is the Ricci scalar.

This was first proposed by David Hilbert in 1915.

There is also a nice paper about how Hilbert has found the Einstein Equations before Einstein and some forgeries of Hilbert’s page proofs.

Thus, the Einstein-Hilbert action is given as

\begin{aligned} \mathcal{S}_{EH}=\int_{\mathbb{R}^{D}}\mathcal{L}\;d^{D} x = \int_{\mathbb{R}^{D}} \mathcal{R}\sqrt{-g}\;d^{D} x \end{aligned}

where $D$ is the dimension of our Lorentzian manifold. Further, I used the common notation $g \equiv \operatorname{det}(g_{\alpha \beta})$ for the determinant of the metric tensor. Recall that the Ricci tensor takes the schematic form $R \sim \partial \Gamma + \Gamma^{2}$ while the Levi-Civita connection itself is $\Gamma \sim \partial g$ . This means that the Einstein-Hilbert action is second order in derivatives, just like most other actions we consider in physics.

Deriving the EFE in Vacuum

Let us now vary the action and then require that $\delta \mathcal{S}_{EH}=0$ .

\begin{aligned} \delta \mathcal{S}_{EH}&= \delta \int \mathcal{R}\;\sqrt{-g}\;d^D x \int d^{D} x\; \delta\left(g^{\mu \nu} R_{\mu \nu}\sqrt{-g}\right) \\\\ &=\int d^{D} x \left[\left(\delta g^{\mu \nu}\right)R_{\mu \nu}\sqrt{-g}+g^{\mu \nu}\left(\delta R_{\mu \nu} \right)\sqrt{-g}+g^{\mu \nu}R_{\mu \nu}\;\delta \sqrt{-g}\right] \\\\ &=\int d^{D} x \left[R_{\mu \nu}\sqrt{-g}\;\delta g^{\mu \nu}+g^{\mu \nu}\sqrt{-g}\;\delta R_{\mu \nu} +g^{\mu \nu}R_{\mu \nu}\;\delta \sqrt{-g}\right] \end{aligned}

For the next step we will use a fancy identity, which relates the determinant to the trace of any complex square matrix $\operatorname{M}$ .

\operatorname{det}(e^{\operatorname{M}})= e^{\operatorname{Tr}(\operatorname{M})}

Derivation

First we notice that,

\begin{aligned} \operatorname{M}&=\begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix}\\\\ &\therefore\; \operatorname{Tr}(\operatorname{M})=m_1+m_2 \implies e^{\operatorname{Tr}(\operatorname{M})}=e^{m_1+m_2} \end{aligned}

In a second step we can verify that,

\begin{aligned} e^{\operatorname{M}}&=\begin{pmatrix} e^{m_1} & 0 \\ 0 & e^{m_2} \end{pmatrix}\\\\ &\therefore \; \operatorname{det}(e^{\operatorname{M}})=e^{m_1}e^{m_2}=e^{m_1+m_2} \end{aligned}

Combining these two results we recognize that

\operatorname{det}(e^{M})=e^{\operatorname{Tr}(M)}

With the help of the last equation we can now show, that

\delta \sqrt{-g}=\frac{1}{2} \sqrt{-g}\;g^{\alpha \beta}\delta g_{\alpha \beta}

Derivation

Notice that if we choose $e^{\operatorname{M}}\equiv B$ ( or equivalently $\ln{e^{\operatorname{M}}}=\ln{B}$ ) it implies that $\operatorname{M}=\ln{B}$ . Thus, by using our found identity we get

\operatorname{det}(e^{\operatorname{M}})= \operatorname{det}(e^{\ln{B}})=e^{\operatorname{Tr(\ln{B})}}

So it follows that,

\ln({\operatorname{det}(B)})=\operatorname{Tr}(\ln{B})

If we take now the derivative on both sides we conclude,

\frac{1}{\operatorname{det}(B)}\partial \operatorname{det}(B)= \operatorname{Tr}(B^{-1}\partial B)

This result is remarkable, because setting $B\equiv g_{\mu \nu}\; (\text{and } B^{-1}\equiv g^{\mu \nu})$ finally yields

\delta \sqrt{-g}=\frac{1}{2} \sqrt{-g}\;g^{\alpha \beta}\delta g_{\alpha \beta}

Back to the action we have

\begin{aligned} \delta \mathcal{S}_{EH}&=\int d^{D} x \left[R_{\mu \nu}\sqrt{-g}\;\delta g^{\mu \nu}+g^{\mu \nu}\sqrt{-g}\;\delta R_{\mu \nu} +g^{\mu \nu}R_{\mu \nu}\;\delta \sqrt{-g}\right]\\\\ &=\int d^{D} x \left[R_{\mu \nu}\sqrt{-g}\;\delta g^{\mu \nu}+g^{\mu \nu}\sqrt{-g}\;\delta R_{\mu \nu} +\frac{1}{2}g^{\mu \nu}R_{\mu \nu}\;\;\sqrt{-g}\;g^{\alpha \beta}\delta g_{\alpha \beta}\right]\\\\ &=\underbrace{\int d^{D} x \left\{\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} \mathcal{R}\right) \delta g^{\mu \nu} \sqrt{-g}\right\}}_{\delta \mathcal{S}_{1}}+\underbrace{\int d^{D} x\bigg\{g^{\mu \nu}\left(\delta R_{\mu \nu}\right)\sqrt{-g}\bigg\}}_{\delta \mathcal{S}_{2}} \end{aligned}

Let’s now look at the first term $\delta \mathcal{S}_{1}$ .

Since $\delta g^{\mu \nu}$ is arbitrary, the condition $\delta \mathcal{S}_{1} = 0$ yields that

R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} \mathcal{R}=0

which are the Einstein Field Equations in vacuum.

The Boundary Term

Now we could think that we are finished, since we found the desired equations but we forgot to take care of the $\delta \mathcal{S}_{2}$ term.

For that purpose let’s first look at the Riemann tensor

R^{\alpha}_{\mu \sigma \nu} = \partial_{\sigma}\Gamma^{\alpha}_{\mu \nu}-\partial_{\nu}\Gamma^{\alpha}_{\mu \sigma}+ \Gamma^{\alpha}_{\sigma \gamma} \Gamma^{\gamma}_{\mu \nu}-\Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \sigma}

and the Ricci tensor

R_{\mu \nu}=R^{\alpha}_{\mu \alpha \nu} = \partial_{\sigma}\Gamma^{\alpha}_{\mu \nu}-\partial_{\nu}\Gamma^{\alpha}_{\mu \alpha}+ \Gamma^{\alpha}_{\alpha \gamma} \Gamma^{\gamma}_{\mu \nu}-\Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \alpha}

Varying the Ricci tensor yields

\begin{aligned} \delta R_{\mu \nu} &= \partial_{\sigma}\delta \Gamma^{\alpha}_{\mu \nu}-\partial_{\nu}\delta\Gamma^{\alpha}_{\mu \alpha}\\\\ &+ (\delta\Gamma^{\alpha}_{\alpha \gamma}) \Gamma^{\gamma}_{\mu \nu}-(\delta\Gamma^{\alpha}_{\nu \gamma}) \Gamma^{\gamma}_{\mu \alpha}\\\\ &+ \Gamma^{\alpha}_{\alpha \gamma} \delta\Gamma^{\gamma}_{\mu \nu}-\Gamma^{\alpha}_{\nu \gamma} \delta\Gamma^{\gamma}_{\mu \alpha} \end{aligned}

This looks like the difference between two covariant derivatives.

The covariant derivative is given as

\nabla_{\alpha}(\delta \Gamma^{\alpha}_{\mu \nu})= \partial_{\alpha}(\delta\Gamma^{\alpha}_{\mu \nu}) + \Gamma^{\alpha}_{\alpha \gamma}\delta \Gamma^{\gamma}_{\mu \nu}-\delta \Gamma^{\alpha}_{\nu \gamma}\Gamma^{\gamma}_{\mu \alpha} - \delta\Gamma^{\alpha}_{\mu \gamma}\Gamma^{\gamma}_{\nu \alpha}

\nabla_{\nu}(\delta \Gamma^{\alpha}_{\mu \alpha})= \partial_{\nu}(\delta\Gamma^{\alpha}_{\mu \alpha}) + \Gamma^{\alpha}_{\nu \gamma}\delta \Gamma^{\gamma}_{\mu \alpha}-\delta \Gamma^{\alpha}_{\alpha \gamma}\Gamma^{\gamma}_{\mu \nu} - \delta\Gamma^{\alpha}_{\mu \gamma}\Gamma^{\gamma}_{\nu \alpha}

From that follows

\nabla_{\alpha}(\delta \Gamma^{\alpha}_{\mu \nu})-\nabla_{\nu}(\delta \Gamma^{\alpha}_{\mu \alpha})=\delta R_{\mu \nu}

This equation is also known as the Palatini Equation.

Now, coming back to $\delta \mathcal{S}_{2}$ :

\begin{aligned} \delta \mathcal{S}_{2}&=\int d^{D} x\bigg\{g^{\mu \nu}\left(\delta R_{\mu \nu}\right)\sqrt{-g}\bigg\}=\int d^{D} x\bigg\{g^{\mu \nu}\nabla_{\alpha}(\delta \Gamma^{\alpha}_{\mu \nu})-\nabla_{\nu}(\delta \Gamma^{\alpha}_{\mu \alpha})\sqrt{-g}\bigg\}\\\\ &= \int d^{D} x\; \sqrt{-g} \left(\nabla_{\alpha}\left(g^{\mu \nu} \delta \Gamma_{\mu \nu}^{\alpha}\right)-\nabla_{\alpha}\left(g^{\mu \alpha} \delta \Gamma_{\mu \nu}^{\nu}\right)\right)\\\\ &= \int d^{D} x\; \sqrt{-g} \;\nabla_{\alpha}\underbrace{\left(g^{\mu \nu} \delta \Gamma_{\mu \nu}^{\alpha}-g^{\mu \alpha} \delta \Gamma_{\mu \nu}^{\nu}\right)}_{\equiv A^{\alpha}}\\\\ &= \int d^{D} x\; \sqrt{-g}\;\nabla_{\alpha} A^{\alpha} \end{aligned}

which can be converted to a surface integral by the divergence theorem, which vanishes because the variations are assumed to vanish on the surface $\partial\R^{D}$ .

\mathcal{S}_{EH}=\int d^{D} x \left\{\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} \mathcal{R}\right) \delta g^{\mu \nu} \sqrt{-g}\right\}+\underbrace{ \int d^{D} x\; \sqrt{-g}\;\nabla_{\alpha} A^{\alpha} }_{\text{Boundary term}\; \rightarrow\; 0}

However, there is a small sublety regarding the boundary term. The use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold $\mathcal{M}$ is closed (compact and without boundary). If $\mathcal{M}$ has a boundary, the action should be supplemented by a boundary term so that the variational principle is well-defined. This additional term is the so-called Gibbons–Hawking–York boundary term.

Adding Matter

After deriving the famous Einstein Field Equations from $\mathcal{S}_{EH}$ we are now interested in a full action of gravity (if we now consider a spacetime that is not empty). Thus, we have to couple the EH-action with matter fields. This requires some additional terms.

The action of general relativity containing matter is given as

\mathcal{S}_{\text {Gravity}}[g]=\kappa\;\mathcal{S}_{E H}[g]+\mathcal{S}_{\text {Matter}}[g]

Now we have to vary this action again and set $\delta\mathcal{S}_{Gravity}$ = 0. The constant $\kappa$ will be chosen so that we get the right Newtonian limit.

We recall

\begin{aligned} \delta S_{E H}&=\int\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R\right) \sqrt{-g} \;\delta g^{\mu \nu} d^{D} x=\int \frac{\delta S_{E H}}{\delta g^{\mu \nu}} \delta g^{\mu \nu} d^{D} x \\\\ &\Rightarrow \frac{\delta S_{E H}}{\delta g^{\mu \nu}}=\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R\right) \sqrt{-g} \\\\ &\Rightarrow \frac{1}{\sqrt{-g}} \frac{\delta S_{E H}}{\delta g^{\mu \nu}}=R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R \end{aligned}

so varying the whole action gives:

\frac{1}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}=\frac{1}{\sqrt{-g}} \frac{\kappa \delta S_{E H}}{\delta g^{\mu \nu}}+\frac{1}{\sqrt{-g}} \frac{\delta S_{M}}{\delta g^{\mu \nu}}=\kappa\left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R\right)+\frac{1}{\sqrt{-g}} \frac{\delta S_{M}}{\delta g^{\mu \nu}}=0

Rearranging gives

\begin{aligned} \kappa \left(R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R\right)&=-\frac{1}{\sqrt{-g}} \frac{\delta S_{M}}{\delta g^{\mu \nu}}\\\\ R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R&=-\frac{1}{\kappa \sqrt{-g}} \frac{\delta S_{M}}{\delta g^{\mu \nu}} \end{aligned}

Defining the energy momentum tensor $T_{\mu v}$ as

T_{\mu \nu}=-2 \frac{1}{\sqrt{-g}} \frac{\delta S_{M}}{\delta g^{\mu \nu}}

and doing the substitution with $\kappa=c^{4} / 2(8 \pi G)$ leads to the familiar Einstein Equation which relates the spacetime curvature on the left hand side to the matter energy density on the right hand side

R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R=\frac{8 \pi G}{c^{4}} T_{\mu \nu}

Adding a Famous Constant

To be honest I lied a little bit by saying that the Ricci scalar is the simplest we could choose. There is in fact a simpler term which we could add to the action (resulting in not interesting dynamics). This comes from multiplying the volume form by a constant.

When the cosmological constant $\Lambda$ is included we have:

\mathcal{S}_{\text{Gravity}}[g]=\kappa\;\mathcal{S}_{E H}[g]+\mathcal{S}_{\Lambda}[g]+\mathcal{S}_{\text {Matter}}[g]=\int d^{D} x \left[ \kappa\;(\mathcal{R}-2\Lambda)\right]+\mathcal{S}_M

So we finally found

R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} \mathcal{R} + g_{\mu \nu}\Lambda=\frac{8 \pi G}{c^{4}} T_{\mu \nu}

Higher Derivative Terms

We have seen that the Einstein-Hilbert action (with cosmological constant) is the simplest thing we can write down to find the EFE but it is not the only possibility, at least if we allow for higher derivative terms. To give an example, there are three terms (the so-called Gauss–Bonnet term) that contain four derivatives of the metric. This modification of the Einstein–Hilbert action is sometimes referred to as Einstein–Gauss–Bonnet gravity.

S_{\mathcal{G}}=\int d^{D} x \sqrt{-g}\;\mathcal{G}=\int d^{D} x \sqrt{-g}\left(a R^{2}+b R_{\mu \nu} R^{\mu \nu}+ c R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}\right)

with $a, b$ and $c$ dimensionless constants and $\mathcal{G}$ the Gaus-Bonnet term. General choices of these constants will result in higher order equations of motion which do not have a well-defined initial value problem.

Nonetheless, it turns out that one can find certain combinations of these terms, which conspire to keep the equations of motion second order. This is known as Lovelock’s theorem.

Theorem (Lovelock’s Theorem): The only second-order, local gravitational field equations derivable from an action containing solely the $4D$ metric tensor (plus related tensors) are the Einstein Field Equations with a cosmological constant.

This powerful theorem means that if we try to create any gravitational theory in a four-dimensional Riemannian space from an action principle involving the metric tensor and its derivatives only, then the only field equations that are second order or less are Einstein’s equations and/or a cosmological constant. This does not, however, imply that the Einstein-Hilbert action is the only action constructed from $g_{\mu \nu}$ that results in the Einstein Equations. In fact, in four dimensions or less one finds that the most general of such actions is

\mathcal{L}=\alpha \sqrt{-g} R-2 \Lambda \sqrt{-g}+\beta \epsilon^{\mu \nu \rho \lambda} R_{\mu \nu}^{\alpha \beta} R_{\alpha \beta \rho \lambda}+\gamma \sqrt{-g}\left(R^{2}-4 R_{\nu}^{\mu} R_{\mu}^{\nu}+R_{\rho \lambda}^{\mu \nu} R_{\mu \nu}^{\rho \lambda}\right)

The third term vanishes in all dimensions, while the Gaus-Bonnet term is only nontrivial in $4+1D$ or greater, and as such, only applies to extra dimensional models. In $3+1D$ , it reduces to a topological surface term (In lower dimensions, it identically vanishes). This follows from the generalized Gauss–Bonnet theorem on a $4D$ manifold $D=4$ dimensions, this combination has a rather special topological property: a generalization of the Gauss-Bonnet theorem states that

\frac{1}{8 \pi^{2}} \int_{M} d^{4} x \sqrt{g}\left(R^{2}-4 R_{\mu \nu} R^{\mu \nu}+R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}\right)=\chi(M)

where $\chi(M) \in \mathbb{Z}$ is the Euler character of $M$ .

As in any field theory, higher derivative terms in the action only become relevant for fast varying fields. In General Relativity, they are unimportant for all observed physical phenomena and we will not discuss them further in this course.

Modified Gravity

As an outlook I would like to to give an overview of this space of alternatives to general relativities that has been constructed and injected into the literature over the past $17$ or $18$ years.

So this is the round table of modified gravity theories.

While building a modified gravity theory, Lovelock tells us that if we want a gravity theory that is not general relativity we have to brake one of the clauses implicit in the theorem. It further means that we have only five options if we want to modify the Einstein Field Equations. Lets now run through the five options or categories of theories with a cartoon example of an action.

Theorem (Lovelock’s Theorem): The only second order, local gravitational field equations derivable from an action containing solely the $4D$ metric tensor (plus related tensors) are the Einstein Field Equations with a cosmological constant.

New Field Content

A: Add other fields rather than the metric tensor (e.g. scalar-tensor theories like the Brans-Dicke theory):

S_{\mathrm{Grav}}\sim \int \sqrt{-g} d^{4} x\left[\phi R-\frac{\omega(\phi)}{\phi}(\nabla \phi)^{2}-2 V(\phi)\right]

This first option adds new field content and couples it to the Einstein-Hilbert action and is involved in mediating gravitational forces. One can add scalar fields, vector fields, tensor fields or even mixtures of the previous ones. The action above shows an example of a scalar-tensor theory with an scalar field $\phi$ , with a kinetic term controlled by the function $\omega$ . Further, we have a potential $V$ and we could cook up a potential that gives a universe that accelerate at late times - a dark-energy-like candidate.

Higher Dimensions

B: Use more or less than four spacetime dimensions (Kaluza Klein theories):

S_{\mathrm{Grav}}\sim\int \sqrt{-g} d^{D} x[\mathcal{R}+\alpha \mathcal{G}]

For this option we build a gravity theory in higher dimensional spacetime and work out what the effective $4D$ theory is. That can be very different from general relativity (brane worlds, brane bending modes…)

> 2nd Order Derivatives

C: Add more than second order derivatives of the metric

S_{\mathrm{Grav}}\int d^{n} x \sqrt{-g}\left(R+\alpha_{1} R^{2}+\alpha_{2} R_{\mu \nu} R^{\mu \nu}+\alpha_{3} g^{\mu \nu} \nabla_{\mu} R \nabla_{\nu} R+\ldots\right)

This one is more of a mathematical trick. The field equations contain greater than second order time derivatives. Generally, higher order theories sell trouble. They suffer from Ostrogradski instability instabilities where the Hamiltonians are unbounded from below and spontaneous vacuum decays. However, there are some special cases, in which higher orders are possible and stable.

Non-Locality

D: Non-locality, e.g. for example the inverse d'Alembertian

S_{\mathrm{Grav}}=\frac{M_{P l}^{2}}{2} \int \sqrt{-g} d^{4} x\left[R+f\left(\frac{1}{\square} R\right)\right.

In this option we would get things like a inverse d’Alembertian – a non local operator. Now, when we say non-local it brings to mind some bad things, too. Violation of causality, superluminality and tachyons. But there are good news. We don’t have to suffer from these sicknesses, because there are conditions to include non-local operators and avoid the kind of pathologies just mentioned.

Emergence

E: Emergence – the idea that the field equations don't come from the action.

Finally, I would like to show a graphic that Dr. Tessa Baker created. This shows only a fraction of the various alternative branches and theories. This is also a problem because it has become very difficult to classify all theories and many have not yet been tested against experimental data. But that’s a whole different story..

Modified Gravity – A roadmap. Source: Tessa Baker

If you are interested and want to know more about alternative gravity theory, I recommend this wonderful paper by Timothy Clifton, Pedro G. Ferreira, Antonio Padilla and Constantinos Skordis.

Hello World!

Thu, 23 Jul 2020 22:00:00 GMT

Hi, How Are You?

Welcome to my new blog! Gone is my former WordPress site. This is the first post on a new site I created myself. It’s JavaScript from head to heel using the amazing site generator Gatsby and styled-components for the design and layout. The latter relies heavily on the awesome CSS grid, by the way.

Why Did I Do This?

When writing papers in an academic environment, the goal is to make them as compact as possible, easy and quick to read and understand. In such writings, the inclusion of as many results as possible is often more important than pointing out the path that led to those results.

This is why it is so difficult to understand papers from topics other than one’s own and to find access to new fields. With this blog I want to try to create a more intuitive access for people who feel the same. Maybe we can find better insights into foreign fields while interacting and reading about each others works. That is my motivation and the goal of this project.

How Did I Do This?

The site is fully responsive, built with Gatsby, has fluid typography, relies heavily on React Hooks for stateful function components and CSS grid for layout. It uses the following libraries:

MDX for interactive content.
styled-components for appearance + DarkMode.
KaTeX for typesetting math.
gatsby-remark-vscode for syntax highlighting.
Disqus for blog post comments - I’d appreciate if you left a comment under the blog posts
Algolia for custom search.
react-spring for animations.

In the following I will test a few of these libraries here.

Dark Mode

All the cool kids these days have websites with a dark color scheme. The really cool kids even have a dark and a light mode with a neat little toggle for you to pick your preference. Being regularly annoyed myself when browsing pages that insist on being eye-piercingly bright even late in the evening, I decided that my site needs a dark mode as well. In dark mode, the website adopts a darker color palette for all windows, views, menus, and controls. The system also uses more vibrancy to make foreground content stand out against the darker backgrounds.

The major parts of this implementation were heavily inspired by Joshua Comeau, a Gatsby core member. You can read about his dark mode implementation in this awesome post.

To try out the dark mode on this site, click/tap this toggle:

MDX

MDX is an authorable format that lets you seamlessly write JSX in your Markdown documents. You can import components, such as interactive charts or alerts, and embed them within your content. This makes writing long-form content with components a blast. The possibilities and authoring experience when creating content for the web have advanced by leaps and bounds in recent years. MDX — the latest in a long line of game-changing innovations — allows writers to sprinkle snippets of JSX directly into their markdown content (hence MDX). Think about WordPress shortcodes but fancier. Not only does this open up seemingly endless possibilities in terms of more interactive and engaging content, it also facilitates the clear separation of code and content (i.e. no more need to write text directly into .js files).

But let’s not lose too many words and experience MDX using two examples.

Framework Popularity

Let’s start simple with a 2d plot.

Frontend framework popularity over time measured by Google search frequency. Source: Google Trends

All the MDX this requires is

import fpProps from './frameworkPopularity'

### Frontend Framework Popularity

<LazyPlot {...fpProps} />

frameworkPopularity.js

const colors = [`red`, `green`, `blue`, `orange`]

// prettier-ignore
const months = [`2012/01`, `2012/02`, `2012/03`, `2012/04`, `2012/05`, `2012/06`, `2012/07`, `2012/08`, `2012/09`, `2012/10`, `2012/11`, `2012/12`, `2013/01`, `2013/02`, `2013/03`, `2013/04`, `2013/05`, `2013/06`, `2013/07`, `2013/08`, `2013/09`, `2013/10`, `2013/11`, `2013/12`, `2014/01`, `2014/02`, `2014/03`, `2014/04`, `2014/05`, `2014/06`, `2014/07`, `2014/08`, `2014/09`, `2014/10`, `2014/11`, `2014/12`, `2015/01`, `2015/02`, `2015/03`, `2015/04`, `2015/05`, `2015/06`, `2015/07`, `2015/08`, `2015/09`, `2015/10`, `2015/11`, `2015/12`, `2016/01`, `2016/02`, `2016/03`, `2016/04`, `2016/05`, `2016/06`, `2016/07`, `2016/08`, `2016/09`, `2016/10`, `2016/11`, `2016/12`, `2017/01`, `2017/02`, `2017/03`, `2017/04`, `2017/05`, `2017/06`, `2017/07`, `2017/08`, `2017/09`, `2017/10`, `2017/11`, `2017/12`, `2018/01`, `2018/02`, `2018/03`, `2018/04`, `2018/05`, `2018/06`, `2018/07`, `2018/08`]

// prettier-ignore
const data = {
  React: [2, 2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 3, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 7, 7, 10, 10, 10, 13, 18, 18, 19, 20, 21, 24, 25, 27, 28, 29, 29, 32, 39, 39, 41, 42, 43, 41, 43, 41, 47, 49, 50, 55, 65, 68, 68, 71, 79, 76, 83, 73, 80, 74, 66, 74, 82, 88, 89, 94, 95, 98, 100],
  AngularJS: [1, 1, 1, 2, 3, 3, 4, 4, 4, 6, 8, 9, 11, 13, 17, 17, 20, 22, 25, 24, 23, 30, 33, 35, 39, 41, 45, 49, 53, 53, 58, 51, 48, 52, 58, 61, 60, 61, 69, 74, 67, 67, 65, 58, 57, 53, 61, 62, 59, 59, 64, 56, 59, 51, 53, 51, 51, 54, 57, 62, 55, 55, 55, 51, 52, 47, 46, 41, 34, 37, 41, 38, 37, 38, 38, 35, 35],
  Angular: [2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 6, 6, 8, 7, 9, 9, 12, 12, 11, 13, 15, 15, 16, 17, 20, 20, 21, 20, 24, 22, 20, 24, 27, 26, 29, 30, 33, 35, 33, 34, 31, 30, 29, 29, 32, 36, 33, 37, 40, 35, 36, 38, 36, 38, 36, 42, 48, 52, 49, 49, 55, 48, 55, 49, 52, 51, 44, 48, 53, 53, 54, 56, 54, 60, 56],
  Vue: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 3, 3, 3, 3, 4, 5, 5, 6, 6, 8, 8, 8, 9, 10, 10, 12, 12, 12, 12, 12, 14, 14, 16, 16, 16, 18, 18, 19]
}

export default {
  data: Object.keys(data).map((key, index) => ({
    x: months,
    y: data[key],
    type: `scatter`,
    mode: `lines+markers`,
    name: key,
    marker: { color: colors[index] },
  })),
}

Trimodal Distribution

Let’s look at the following surface. This one is a trimodal distribution consisting of three unnormalized Gaussians.

$\exp[-\frac{1}{20}(x^2 + y^2)] + \exp\{-\frac{1}{10}[(x-10)^2 + y^2]\} + \exp\{-\frac{1}{10}[(x-7)^2 + (y-7)^2]\}$ .

This is generated by

import triModalProps from './triModal'

### Trimodal Distribution

<LazyPlot {...triModalProps} />

triModal.js

const [points, middle] = [51, 25]
const range = Array.from(Array(points), (e, i) => 0.5 * (i - middle))
const z = range.map(x =>
  range.map(
    y =>
      Math.exp(-0.05 * (x ** 2 + y ** 2)) +
      0.7 * Math.exp(-0.1 * ((x - 10) ** 2 + y ** 2)) +
      0.5 * Math.exp(-0.1 * ((x + 7) ** 2 + (y - 7) ** 2))
  )
)

export default {
  data: [
    {
      z,
      x: range,
      y: range,
      type: `surface`,
      contours: {
        z: {
          show: true,
          usecolormap: true,
          highlightcolor: `white`,
          project: { z: true },
        },
      },
      showscale: false,
    },
  ],
  style: { height: `30em` },
}

KaTeX

People who know me are aware of the fact that I don’t like how LaTeX is used in many blogs. A form of implementing LaTeX is often provided, but unfortunately only a picture of it is rendered. Obviously, this is neither particularly practical, nor is it beautiful or contemporary. MathJax was also unable to completely convince me. Finally I came across KaTeX, which is not perfect at all, but for me personally it brings the most advantages. Some major advantages:

Fast: KaTeX renders its math synchronously and doesn’t need to reflow the page.
Print quality: KaTeX’s layout is based on Donald Knuth’s TeX, the gold standard for math typesetting.
Self contained: KaTeX has no dependencies and can easily be bundled with your website resources.
Server side rendering: KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.

Examples

f(x) = \int \hat f(\xi)\,e^{2 \pi i \xi x} \,d\xi \tag{1}

\star\, \alpha=\frac{1}{(n-k) !} a_{i_{k+1} \ldots i_{n}} d x^{i_{k+1}} \wedge \cdots \wedge d x^{i_{n}} \tag{2}

\bigotimes_{i=2}^{25} \bigotimes_{j=1}^{\infty}\left(\alpha_{-j}^{i}\right)^{m^{i} j}|0 ; p\rangle \tag{3}

Procreate

I would also like to thank the people of Procreate. Procreate is really a game changer when it comes to drawing on a tablet. With the help of this app, I was able to design the drawings and graphics on this website. Even if I’m far from being good at it, of course.

Outro

In case you’re considering switching your own site to Gatsby or more generally the JAMstack, I definitely recommend it. It was an incredibly enjoyable learning experience. Gatsby is open-source, has excellent docs and an awesome team of maintainers allowing you to contribute your own code and ideas to the project anytime you like.

At this point I’ve pretty much run out of things to say. And admittedly, this post is mostly for testing purposes anyway.