Abstract

Recently, an intriguing relationship (the double copy) has been discovered between theories like electromagnetism and gravity. This potentially gives us a new way to think about gravity, and there are also practical applications involving the efficient calculation of gravitational observables and how to simulate gravity using optical systems. In this tutorial, we will review what is known about the double copy, and argue that now is the perfect time for researchers in optics and/or condensed matter to become interested in this fascinating correspondence.

© 2021 Optical Society of America

1. INTRODUCTION

Our current understanding of the ultimate building blocks of nature is in terms of matter, acted on by forces. To the best of our knowledge, there are only four fundamental forces: electromagnetism, the weak and strong nuclear forces, and gravity. The first three of these are described by the Standard Model of Particle Physics, which is a quantum field theory (QFT) and thus incorporates the effects of both quantum mechanics and relativity. Gravity, on the other hand, is best described by general relativity (GR), which has so far stubbornly resisted humankind’s attempts to make it quantum. This is unfortunate, given that quantum gravitational effects are believed to be important in extreme regions of the universe, where GR breaks down, e.g., the center of black holes, or at the Big Bang itself. However, even classical GR is at the cutting edge of fundamental physics research: the recent discovery of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) experiment [1] provides an entirely new tool with which to view our universe. Typical signals arise from the inspiral and merger of heavy objects such as black holes and neutron stars, and the precision with which such signals can be understood is limited by our ability to approximate these physical situations in GR calculations.

The above discussion suggests that new ways of thinking about gravity are useful for two main reasons: (i) they could generate new conceptual insights into what a complete (quantum) theory of gravity might look like; (ii) they might provide clever ways of streamlining difficult calculations, even in the incomplete theory of gravity (GR) that we know about. The aim of this paper is to review a particularly intriguing way to think about gravity, namely that it can be seen—in a well-defined sense that we will make precise—as two copies of a gauge theory (the type of theory describing the other forces in nature). This is known as the double copy and was first discovered, to the best of our knowledge, in the so-called scattering amplitudes that are related to probabilities for particles to interact in collider experiments [24] (itself inspired by earlier work [5]). It has since been extended to other types of solutions, and there are even cases where certain quantities in GR can be obtained by “copying” counterparts in conventional electromagnetism. Whilst analogies between electromagnetic (EM) and gravitational physics have been made many times before, the double copy constitutes a very precise relationship that opens up the possibility to simulate and explore gravity using optical or condensed matter systems in a very systematic way. Furthermore, the double copy between gauge and gravity theories turns out to be only one example of a whole host of such copy relationships that have been shown to relate quantities in an increasingly complex web of both physical and non-physical theories (see, e.g., Ref. [6] for a wide-ranging review). This constitutes mounting evidence that our traditional way of thinking about field theories obscures a profound underlying structure, such that the rules of QFT that we take for granted may have to be rewritten. We also do not know if all field theories can be obtained by “copying” other ones, or whether there is a special set of “copiable” theories that are picked out for some sort of special reason.

As the scope and range of applications of the double copy have widened, so has the network of scientists who are interested in its consequences. The field of interested people is an interdisciplinary mix of high energy physicists (trained in either particle physics or string theory), astrophysicists and cosmologists, and both pure and applied mathematicians. There is even an entire annual conference devoted to the topic (QCD Meets Gravity), whose most recent incarnation had over 250 participants from 23 countries! As is common in academia, most of the barriers for knowledge transfer arise from the lack of a common language for what is very often the same physics. Indeed, finding such a language may itself shed light on the origin of the double copy itself. Furthermore, there is clearly scope for researchers from other branches of physics—such as optics and condensed matter—to become involved and to suggest theoretical methods, or experimental set-ups, that can be used to further our knowledge of the double copy. Given that this is a young subject involving unfamiliar territory, there is a high probability of generating novel and impactful research insights by bringing in well-established ideas and techniques from other areas. Ideas from high energy physics and/or gravity have also been successfully applied to optics before, with examples including the use of supersymmetry in describing optical waveguides [7] and the entire field of transformation optics, which originated in Refs. [8,9] (see Ref. [10] for a review). The double copy is different to both of these, and, to define it more precisely, we must first review the various theories that it relates.

2. GAUGE THEORIES

All of the forces and matter particles in the Standard Model are described by fields, whose equations of motion have wave-like solutions. In quantum theory, these waves arrive in discrete quanta or lumps, giving rise to the observed matter and force-carrying particles. For example, the force particles associated with the EM and strong nuclear forces are the photon and gluon, respectively. The former is described by the quantum version of Maxwell’s equations, which in many applications are written in a non-relativistic form involving separate electric and magnetic fields $\vec E$ and $\vec B$. In high energy physics applications, however, it is convenient to use a different language that makes the compatibility of electromagnetism with the theory of special relativity (SR) manifest (for a textbook review of the following, see, e.g., Ref. [11]). In SR, space and time mix with each other under the Lorentz transformations that take us between different inertial frames. It thus makes no sense to separate space and time coordinates, and instead we may combine them to form so-called four-vectors:

$${x^\mu} = (t,\vec x),$$
where we use natural units such that the speed of light $c = 1$ (also Planck’s constant $\hbar = 1$), and the index $\mu\in \{0,1,2,3\}$ labels which component of the four-vector we are talking about. The four-vector in Eq. (1) specifies the position of an event in spacetime, and the vector itself transforms straightforwardly under Lorentz transformations, regarded as $4 \times 4$ matrices. Given Eq. (1), it is also convenient to define a vector with the index downstairs:
$${x_\mu} = (t, - \vec x) = {\eta _{\mu \nu}}{x^\nu}.$$

That is, the downstairs four-vector consists of flipping the spatial components relative to the upstairs one, and the right-hand side shows that this operation can be formalized by multiplying the components of the original four-vector with the metric tensor, expressed in matrix form as

$${\eta _{\mu \nu}} = \left({\begin{array}{*{20}{r}}1&0&0&0\\0&{- 1}&0&0\\0&0&{- 1}&0\\0&0&0&{- 1}\end{array}} \right).$$

We have also used the Einstein summation convention in Eq. (2), where repeated indices are assumed to be summed over. The convenience of the downstairs index notation is that one may define a dot product of two four-vectors,

$$x \cdot y = {x^\mu}{y_\mu} = {\eta _{\mu \nu}}{x^\mu}{y^\nu},$$
which turns out to be invariant under Lorentz transformations. Although we have talked about spacetime positions here, any set of four numbers that mixes appropriately under Lorentz transformations may be combined to make a four-vector. A further example is the four-momentum of a particle:
$${p^\mu} = (E,\vec p),$$
where $E$ and $\vec p$ are the (relativistic) energy and momentum, respectively. By evaluating the dot product of Eq. (5) with itself in a general frame, and, in the remaining frame of the particle, one obtains the relativistic energy–momentum relation,
$${E^2} - {\vec p^2} = {m^2},$$
where $m$ is the particle mass. In electromagnetism, it turns out that one may combine the electrostatic potential $\phi$ and magnetic vector potential $\vec A$ into a single four-vector known as the gauge field:
$${A_\mu} = (\phi ,\vec A),$$
where the electric and magnetic fields are defined in our notation by
$$\vec E = - \nabla \phi - \frac{{\partial \vec A}}{{\partial t}},\quad \vec B = \nabla \times \vec A.$$

With these definitions, the usual non-relativistic Maxwell equations can be shown, after some effort, to arise from the relativistic equations

$${\partial _\mu}{F^{\mu \nu}} = {j^\nu},$$
where ${\partial _\mu} \equiv \partial /\partial {x^\mu}$, and we have defined the field strength tensor
$${F_{\mu \nu}} = {\partial _\mu}{A_\nu} - {\partial _\nu}{A_\mu},$$
as well as the current density four-vector
$${j^\mu} = (\rho ,\vec j),$$
where $\rho$ and $\vec j$ are the charge and (three-vector) current densities, respectively. [Actually, Eq. (9) generates only two of the four usual non-relativistic Maxwell equations. The other two arise from the Bianchi identity ${\partial _{{[\alpha}}}{F_{\beta \gamma]}} = 0$, where square brackets denote antisymmetrization over all indices. This identity is an automatic consequence of the definition of the field strength in Eq. (10).]

The theory of electromagnetism has a remarkably rich symmetry known as gauge invariance: if we subject ${A_\mu}$ to the transformation

$${A_\mu} \to {A^\prime _\mu} = {A_\mu} + {\partial _\mu}\chi ,$$
where $\chi$ is a (spacetime-dependent) scalar field, the field strength tensor of Eq. (10) does not change, and thus the equations of electromagnetism remain invariant. Gauge invariance turns out to be crucial in being able to formulate a sensible quantum theory of electromagnetism, but also means that the gauge field corresponding to a given physical system is not unique, but can be written in infinitely many ways. One may make things precise by imposing additional constraints on ${A_\mu}$, known as fixing a gauge. A common gauge choice is the Lorenz gauge
$${\partial _\mu}{A^\mu} = 0,$$
for which the first equation in Eq. (9) reduces to
$$\left({\frac{{{\partial ^2}}}{{\partial {t^2}}} - {\nabla ^2}} \right){A_\mu} = {j_\mu}.$$

In the vacuum case (${j_\mu} = 0$), this reduces to the wave equation, whose solutions include the well-known plane waves

$${A_\mu} = {\epsilon _\mu}{e^{ik \cdot x}},\quad {k_\mu}{\epsilon ^\mu} = 0,$$
where ${\epsilon _\mu}$ is the polarization vector, and ${k_\mu}$ can be interpreted as the four-momentum. (In natural units, the frequency $\omega$ and energy $E$ are interchangeable, as are the three-momentum $\vec p$ and wave vector $\vec k$.) In QFT, particle states (photons) correspond to quanta of these plane waves.
 figure: Fig. 1.

Fig. 1. Phase of the electron field at each point in spacetime can be thought of as an arrow on the unit circle, where a local gauge transformation may rotate the arrows at different points by different amounts.

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 figure: Fig. 2.

Fig. 2. We can imagine the quark field at each point in spacetime as having an arrow in an internal color space, telling us how much of each color charge it has.

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Above, we saw how gauge transformations act on the photon field ${A_\mu}$. In quantum electrodynamics, there is also an electron field $\psi ({x^\mu})$, which, being complex, has a phase at each point in spacetime. (The electron field is a type of mathematical quantity called a spinor, but the details of this are irrelevant to our main discussion.) A (local) gauge transformation acting on the electron is defined to be a phase shift, which may be defined separately at each point:

$$\psi \to {e^{i\theta ({x^\mu})}}\psi .$$

There is a nice geometric way to think about this: the phase at each spacetime point can be represented as an arrow on the unit circle [Fig. 1(a)], such that performing a local gauge transformation amounts to rotating this arrow differently at different points in spacetime. One may show that requiring that the Dirac equation for the electron be invariant under these local gauge transformations requires the presence of the field ${A_\mu}$ that transforms as in Eq. (12). Thus, electromagnetism can be completely derived as a consequence of local gauge invariance! The question naturally arises of whether the other forces in nature can be phrased in terms of symmetries. Indeed they can, and we may take as an example the strong force that acts on fundamental particles called quarks. Quarks carry a type of charge called color, which can take three different values (conventionally labeled as $r$, $g$, and $b$, or “red,” “green,” and “blue”). We can then think of the quark field as carrying an abstract “color arrow” at each point in spacetime, which tells us how much redness, greenness, and blueness there is: see Fig. 2. A local gauge transformation in this case corresponds to rotations of the color arrow by different amounts at different spacetime points, and demanding that the equation of motion of the quark field be invariant under local rotations of the color arrow leads to the presence of a gauge field for the gluon, albeit more complicated than the EM case above. The group of all possible color rotations of the quark field (i.e., rotations in a three-dimensional complex color space) is denoted by special unitary [SU(3)], and mathematicians call such sets of continuous transformations Lie groups. Acting on the quark field, members of the color rotation group will be a $3 \times 3$ complex matrix, and it turns out that any such member can be written as

$${\textbf{U}} = \exp \left[{i{\theta ^a}({x^\mu}){{\textbf{T}}^a}} \right],$$
where the matrices ${{\textbf{T}}^a}$ are called generators, and the parameters ${\theta ^a}({x^\mu})$ label which member of the group we are talking about. Equation (17) is known as Lie’s third theorem and applies to any simply connected continuous group. (A simply connected group is one where every loop in the space of parameters $\{{\theta ^a}\}$ can be shrunk to a point. All of the groups considered here will have this property.) Note, for example, that the electron phase rotations of Eq. (16) have this form, where there is only one parameter $\theta$ and one generator (the number 1, which is not explicit). For SU(3), it turns out that eight generators are needed, and thus the index $a$ runs from 1 to 8. The explicit form of the generator matrices depends on the choice of a suitable basis in color space, but the generators for any Lie group can be shown to obey a so-called Lie algebra:
$$[{{\textbf{T}}^a},{{\textbf{T}}^b}] = {i{f}^{\textit{abc}}}{{\textbf{T}}^c},\quad [{\textbf{A}},{\textbf{B}}] \equiv {\textbf{AB}} - {\textbf{BA}}.$$

That is, the commutator of any two generators must itself be equal to a superposition of generators, where the relevant coefficients $\{{f^{\textit{abc}}}\}$ are called structure constants, are unique for any given Lie group, and have the property of being completely antisymmetric under interchange of any two indices. (The proof of this result comes from demanding consistency between Eq. (17), and the fact that the product of any two transformations must itself be a transformation in the group.) A familiar example of Eq. (18) from our undergraduate days is for the group SU(2), which arises in the description of angular momentum. Equation (18) is then the statement that a commutator of two angular momentum generators is itself a generator of angular momentum. (The structure constants for SU(2) turn out to be the Levi–Civita symbol ${\epsilon ^{\textit{abc}}}$, so that one has, e.g., $[{J_x},{J_y}] = i{J_z}$ for the angular momentum generators.) Recall that for the phase shifts of electromagnetism there was only one generator (the number 1), which clearly commutes with itself! Thus, the structure constants vanish in electromagnetism.

In electromagnetism, requiring invariance under phase rotations, which had 1 deg of freedom at each point, led to a gauge field ${A_\mu}$. For the strong force, color rotation invariance of the quark field ends up leading to eight separate gauge fields, one for each value of the index $a$ above. We may then write this as a single field $A_\mu ^a$, where $\mu$ is the spacetime index, and $a$ is the color index. Despite the abstract nature of the present discussion, which unavoidably arises from the esoteric nature of the symmetries of the quark field and how these must be expressed mathematically, it is nevertheless possible to understand the extra complications of gluons with respect to photons in highly physical terms. It turns out that a quark may change color by emitting a gluon, such that conservation of color charge requires that the gluon itself carry color charge. The index $a$ in the gluon field then simply tells us which type of gluon we are talking about.

Armed with the above notation, we can finally state the equations describing the gluon. We will focus on the case in which the quark fields are absent, which is known as the Yang–Mills theory. One may then define a field strength tensor

$$F_{\mu \nu}^a = {\partial _\mu}A_\nu ^a - {\partial _\nu}A_\mu ^a + g{f^{\textit{abc}}}A_\mu ^bA_\nu ^c,$$
where $g$ is a number called the coupling constant. It represents the strength of the strong force and is the analogue of the electron charge $e$ in electromagnetism. In terms of this field strength, the vacuum Yang–Mills equations are
$${\partial ^\mu}F_{\mu \nu}^a + g{f^{\textit{abc}}}{A^{\mu b}}F_{\mu \nu}^c = 0.$$

(Similar to the EM case above, we have ignored a Bianchi identity that arises from the definition of the field strength tensor.) We see that Eqs. (19) and (20) are more complicated than their EM counterparts of Eqs. (9) and (10). First, they involve the structure constants, which are absent in the EM case, although their presence here should not surprise us: the symmetry of the theory (as represented by the structure constants) is dictating the equations of motion! Second, the Yang–Mills equations are clearly non-linear, in striking contrast to the Maxwell equations. The physical interpretation of this is that these terms represent interactions between different gluons, which is entirely to be expected given that gluons carry color charge, as noted above. In electromagnetism, the photon carries no charge and thus cannot interact directly with itself. (Photons can mutually interact indirectly, by coupling to an intermediate fermion bubble, but this does not invalidate the above discussion!) We may further check the consistency of the above equations by showing how electromagnetism emerges in the appropriate limit. First, we see that all of the non-linear terms in Eqs. (19) and (20) involve the structure constants ${f^{\textit{abc}}}$. The latter vanish for electromagnetism, so that the equations linearize. Second, the index $a$ can only take a single value in electromagnetism, so that Eqs. (19) and (20) reduce to Eqs. (9) and (10), as required.

Note that, even in Yang–Mills theory, there are situations where the non-linear terms vanish or can be ignored, such that $A_\mu ^a$ (for each $a$) obeys an equation similar to the Maxwell equations. By fixing a gauge, one may obtain the wave equation of Eq. (14). The physical gluon particle then arises as a quantum of a plane wave solution.

3. GRAVITY

Having reviewed what a gauge theory is, let us now turn to gravity. This is described by GR, whose basic idea is that matter and energy curve the spacetime that they are sitting in. Freely falling test particles will then follow paths of the shortest distance in a curved space, and this distorted motion then corresponds to the force of gravity. As an analogy, consider a simple two-dimensional curved space, namely the surface of the sphere in Fig. 3. Let us take two people at points $A$ and $B$ and instruct them both to walk towards the north pole $N$. Each person will feel that they are walking locally in a straight line with no forces acting upon them. However, merely the fact that they are walking on a curved surface will lead to them moving closer together, which looks like an attractive force.

 figure: Fig. 3.

Fig. 3. Trajectories of two people walking in a curved space.

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To make the above idea precise, we first need to use the language of four-vectors to combine space and time. Next, we need to see mathematically how to describe a curved space. Let us consider spacetime without gravity, as in the previous section. Given the spacetime displacement

$$d{x^\mu} = (dt,d\vec x)$$
between two events, one may take its dot product with itself to form an invariant distance
$$d{s^2} = d{t^2} - d\vec x \cdot d\vec x = {\eta _{\mu \nu}}d{x^\mu}d{x^\nu},$$
where we have rewritten things to explicitly involve the metric tensor of Eq. (3). We thus see that ${\eta _{\mu \nu}}$ can be thought of as telling us how to measure distances in spacetime, and it is for this reason that it is called the “metric” tensor, which in this particular case is known as the Minkowski metric.

If we now add gravity, spacetime will be distorted, where this distortion will potentially be different at different spacetime points. We can easily model this by replacing ${\eta _{\mu \nu}} \to {g_{\mu \nu}}({x^\mu})$ in Eq. (21), where the dependence of the new metric tensor ${g_{\mu \nu}}$ on spacetime position indicates that we have warped our previously flat space to make a potentially curved one. Different curved spacetimes will have different metric tensors ${g_{\mu \nu}}$, and, to complete the theory, we need equations that tell us how to determine the metric in all of spacetime that corresponds to a given distribution of matter and energy. These are the Einstein field equations and are extremely complicated to solve in general. Famous solutions include the possibility of black holes (regions of spacetime from which not even light can escape) and universes that expand outwards from a finite time in the past (the Big Bang).

Note that there are many situations in which we are far away from a given matter distribution, in which case we can try to find an approximate solution for the gravitational field. To do this, one may write the following ansatz:

$${g_{\mu \nu}} = {\eta _{\mu \nu}} + \kappa {h_{\mu \nu}},$$
where $\kappa = \sqrt {16\pi {G_N}}$ is a conventional constant factor containing Newton’s constant, and ${h_{\mu \nu}}$ is the graviton field, which represents the deviation from flat space. Upon substituting Eq. (22) into the Einstein equations, we can collect terms involving successive powers of the small number $\kappa$, where keeping each additional power amounts to a better approximation to the exact metric tensor describing the system. This is perturbation theory, and a similar procedure may be used to find classical solutions to the non-linear Yang–Mills theory described in the previous section, where the coupling constant $g$ is the relevant expansion parameter in that case.

In gauge theory, we saw that the field ${A_\mu}$ is not unique for a given physical system, but varies upon making gauge transformations as in Eq. (12). Likewise, one may show that the metric tensor of a given spacetime in GR transforms in the following way under coordinate transformations ${x^\mu} \to {y^\mu}$:

$${g_{\alpha \beta}}({y^\mu}) = \left({\frac{{\partial {x^\mu}}}{{\partial {y^\alpha}}}} \right)\left({\frac{{\partial {x^\nu}}}{{\partial {y^\beta}}}} \right){g_{\mu \nu}}({x^\mu}).$$

There are infinitely many different coordinate systems we can choose and thus infinitely many ways of writing the metric tensor for the same physical system! As in gauge theory, we may fix things by imposing additional constraints on the metric. In perturbation theory, this is usually done at the level of the graviton field itself, and one example is the transverse-traceless (TT) gauge, in which the graviton satisfies

$${\eta ^{\mu \nu}}{h_{\mu \nu}} = 0,\quad {\partial _\mu}{h^{\mu \nu}} = 0.$$

The vacuum Einstein equations then become

$$\left({\frac{{{\partial ^2}}}{{\partial {t^2}}} - {\nabla ^2}} \right){h_{\mu \nu}} = 0,$$
which can immediately be recognized as the wave equation! The solutions constitute small ripples in the fabric of spacetime and are precisely the gravitational waves discovered by LIGO. A particularly straightforward set of solutions are the plane waves
$${h_{\mu \nu}} = {\epsilon _{\mu \nu}}{e^{ik \cdot x}},$$
where ${\epsilon _{\mu \nu}}$ is a polarization tensor. It turns out that gravitational waves have two polarization states, as for photons, and the similarity between Eqs. (15) and (26) is our first glimpse of an intimate relationship between gauge theory and gravity. (The comparison between EM and gravitational waves has also been explored in Ref. [12], in a way that makes the common physics exceptionally clear.) Quanta of these waves are known as gravitons and are the hypothetical particles of quantum gravity.

4. SCATTERING AMPLITUDES AND THE DOUBLE COPY

We have seen in the previous section that the wave equation naturally arises in both gauge theories and gravity, where the force-carrying particles in each case arise as quanta of plane wave solutions, generalizing the origin of the photon in electromagnetism. In applications of QFT, a commonly occurring situation is a scattering experiment, in which two (beams of) particles interact and produce a number of other particles in the final state. In a quantum theory, we cannot predict exactly what will happen, but can instead calculate the probability of given final states. Each scattering process is then associated with a complex number called the scattering amplitude, which, as in non-relativistic quantum mechanics, can be defined in terms of the overlap $\langle f|i\rangle$ between a given initial state $|i\rangle$ and final state $|f\rangle$. The amplitude is a complex number, which makes sense given that the incoming and outgoing particles have a wave-like character, and thus the amplitude must keep track of relative phase differences. If there are different ways of obtaining the same final state, we must add them together to form a total amplitude ${\cal A}$, such that the probability of a given process depends on $|{\cal A}{|^2}$. This is a real number as it should be, and the fact that different possibilities are added before squaring means that quantum interference effects are correctly included.

New methods for calculating amplitudes have blossomed in recent years (see, e.g., Ref. [13] for an up-to-date review), but a more traditional way to calculate them is using Feynman diagrams. These can be viewed as handy spacetime pictures that describe how particles can interact, and an example is shown in Fig. 4(a). This shows two gluons (shown as wavy lines) coming together to form a single intermediate gluon, which then decays to two gluons in the final state. More generally, there are different symbols for different types of particles, and they may be coupled together through interaction vertices in prescribed ways.

 figure: Fig. 4.

Fig. 4. (a) Example Feynman diagram for scattering of two gluons to two gluons; (b) momentum conservation in a loop diagram.

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Feynman diagrams offer much more than a simple way to visualize particle interactions: there are highly precise Feynman rules that convert each diagram into an algebraic contribution to the scattering amplitude! To go into these rules in detail is clearly beyond the scope of this tutorial, but some of the main ideas are as follows:

  • • For a given set of particles in the initial and final states, one must draw all possible diagrams that connect them with a given number of vertices.
  • • Each vertex is associated with a power of the coupling constant of the theory ($e$, $g$, or $\kappa$ for electromagnetism, Yang–Mills theory, or gravity, respectively), plus additional factors involving the momenta of the interacting particles. Note, then, that diagrams with increasing numbers of vertices constitute higher orders in perturbation theory (i.e., an expansion in powers of the coupling).
  • • Each line carries a four-momentum ${p^\mu} = (E,\vec p)$, such that four-momentum is conserved at every vertex. When a diagram involves loops, the external momenta are not sufficient to define all internal momenta. An example is shown in Fig. 4(b), where the lines show the direction of momentum flow. In order to describe the momentum on every line, we have to define the additional loop momentum $k$, whose possible values must then be summed over according to the rule for combining all possibilities to get the total amplitude. Given that each component of the four-momentum is a continuous variable, this is a multidimensional integral,
    $$\int \frac{{{{\rm d}^4}k}}{{{{(2\pi)}^4}}},$$
    where the normalization is conventional.
  • • Internal lines are associated with a factor
    $$\frac{\ldots}{{{p^2}}} \equiv \frac{\ldots}{{{E^2} - {{\vec p}^2}}},$$
    where the numerator is different for each type of particle. (Note that for intermediate particles, $E$ and $\vec p$ are independent degrees of freedom that are not related by the energy–momentum relation of Eq. (6). Such particles are known as virtual particles in QFT language and are not physically measurable directly.)
  • • External particles are associated with polarization vectors or tensors, as appropriate.

Calculating amplitudes in a given theory then amounts to knowing what the exact Feynman rules are and applying them. This is easier said than done: the number of Feynman diagrams increases rapidly with either the number of external particles or loops. Carrying out the integrals over the loop momenta is also highly non-trivial and involves interesting connections with developing branches of pure mathematics (including special function and number theory). Furthermore, this way of calculating amplitudes makes the Yang–Mills theory and gravity look completely different. As an example, the vertex factor for three interacting gluons, written in conventional mathematical notation, has six individual terms in it. The three-graviton vertex instead has over 170 terms! Remarkably, however, a very special structure emerges if we calculate amplitudes in both theories and compare the results.

Diagrams for the total amplitude for an $m$ particle process can be classified according to the number of loops, where each loop order may be calculated separately. One may then consider a gauge theory amplitude ${\cal A}_m^{(L)}$ for $m$ external particles with $L$ loops. The above description of the Feynman rules implies that this will take the general form

$${\cal A}_m^{(L)} = {g^{m - 2 + 2L}}\sum\limits_i \left({\prod\limits_{l = 1}^L \frac{{{d^4}{p_l}}}{{{{(2\pi)}^4}}}} \right)\frac{{{n_i}{c_i}}}{{\prod\limits_{{\alpha _i}} p_{{\alpha _i}}^2}}.$$

Here, the sum is over distinct diagrams $i$, where the overall power of the coupling constant is fixed by the number of external particles and loops. There are integrals over the $L$ loop momenta $\{{p_l}\}$, and each term has a denominator arising from the internal line factors mentioned above. There will be a dependence on the color charges of the gluons for each diagram (represented by ${c_i}$). Finally, there is a so-called kinematic numerator ${n_i}$ for each diagram that collects everything else and depends on particle momenta and polarization vectors. Although the general form of Eq. (28) is correct, the kinematic numerators ${n_i}$ are not unique. For example, gauge transformations (and generalizations of them to include other types of field redefinition) will mix up the numerators associated with different diagrams, such that ${\cal A}_m^{(L)}$ remains invariant. However, it was noticed in 2008 [2] that there is a particular choice for the numerators $\{{n_i}\}$ for a given amplitude, which means that they obey similar mathematical identities to the color factors ${c_i}$, which arise from the fact that the color degrees of freedom are described by a Lie group. This itself implies that there must be some mysterious symmetry underlying the kinematic degrees of freedom of a scattering amplitude and is known as Bern–Carrasco–Johansson (BCJ) duality. It has to this day remained mysterious, although the nature of the symmetry can be glimpsed in certain cases [14]. However, once the numerators $\{{n_i}\}$ have been chosen to have this special BCJ-dual form, it turns out that the formula

$${\cal M}_m^{(L)} = {\left({\frac{\kappa}{2}} \right)^{m - 2 + 2L}}\sum\limits_i \left({\prod\limits_{l = 1}^L \frac{{{d^4}{p_l}}}{{{{(2\pi)}^4}}}} \right)\frac{{{n_i}{{\tilde n}_i}}}{{\prod\limits_{{\alpha _i}} p_{{\alpha _i}}^2}}$$
describes a gravity amplitude, where ${\tilde n_i}$ is a second set of kinematic numerators, and the gauge theory coupling constant $g$ has been replaced by its gravitational counterpart. Even without knowing what any of the symbols mean, it is clear that Eq. (29) is almost identical to Eq. (28)! The only differences are the replacement of coupling constants, and of color information by kinematics. This remarkable relationship is the double copy and was first presented in Refs. [3,4].
 figure: Fig. 5.

Fig. 5. Various theories and the relationships between them. Electromagnetism sits it in the middle, and GR is on the right.

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We have been deliberately imprecise above about which gauge and gravity theories Eqs. (28) and (29) correspond to. There are in fact many different types of gauge theory obtained from Yang–Mills by, e.g., incorporating additional symmetries such as supersymmetry, that relates bosonic and fermionic degrees of freedom. One may then take the kinematic numerators $\{{n_i}\}$ and ${\tilde n_i}$ from different gauge theories and generate amplitudes in various different gravity theories, which are themselves appropriate generalizations of GR. (The simplest case of pure Yang–Mills theory copied with itself does not in fact give GR, but GR plus two additional matter particles. We will see the reason for this below.) It really is amazing that all of these theories should have amplitudes related by the simple replacements above: we stress again that using traditional QFT methods, this structure is entirely invisible.

Above, we started with gauge theory and replaced color information by kinematics. We could instead have done the opposite and obtained the formula

$$\tilde {\cal A}_m^{(L)} = {y^{m - 2 + 2L}}\sum\limits_i \left({\prod\limits_{l = 1}^L \frac{{{d^4}{p_l}}}{{{{(2\pi)}^4}}}} \right)\frac{{{c_i}{{\tilde c}_i}}}{{\prod\limits_{{\alpha _i}} p_{{\alpha _i}}^2}},$$
where we have relabeled the coupling constant to $y$ and introduced a second set of color factors $\tilde c$ that may correspond to a different color symmetry group in general. This is called the zeroth copy and also turns out to yield a scattering amplitude in the so-called biadjoint scalar theory, whose vacuum field equation is
$${\partial ^2}{\Phi ^{{aa^\prime}}} - y{f^{\textit{abc}}}{\tilde f^{a^\prime b^\prime c^\prime}}{\Phi ^{{bb^\prime}}}{\Phi ^{{cc^\prime}}} = 0.$$

In this case, the field ${\Phi ^{{aa^\prime}}}$ has no spacetime indices (i.e., it is a scalar), but has two different types of color charge, where the quantities ${f^{\textit{abc}}}$ snd ${\tilde f^{a^\prime b^\prime c^\prime}}$ are structure constants in the two different color symmetry groups. There are no indications that biadjoint scalar theory is a physical theory by itself. However, the above correspondences suggest that at least some of the dynamics of gauge and gravity theories are inherited from the theory of Eq. (31).

The above discussion has been very technical, so let us now summarize it in more pedestrian terms: scattering amplitudes in a ladder of theories (biadjoint, gauge, and gravity) are related by copying correspondences, such that color charge information in one theory gets replaced by kinematic information (e.g., momenta and polarizations) in another. For the fields themselves, this involves replacing color charge indices with spacetime indices, and these correspondences are depicted in Fig. 5. The term “double copy” is sometimes used loosely in the literature to refer to any of the types of correspondence in the figure. Furthermore, there is mounting evidence that a whole web of interesting theories is related by double copies, not just the three types of theory considered here [6]. We hope it is not too hyperbolic to suggest that the double copy promises to revolutionize our understanding of field theories, given its indication of a deep underlying connection between very different areas of physics. However, for this to be true, we need to know how generally Fig. 5 should be interpreted. Is it merely an accident for scattering amplitudes, or does it apply to the “complete” theories, whatever this means? One way to investigate this is to think of other types of quantity in each theory and to see whether they can be matched up. A natural starting point is to consider exact classical solutions, and this is the subject of the following section.

5. CLASSICAL DOUBLE COPY

In thinking about classical solutions, perhaps the simplest case to think about is that of plane waves, which we have already encountered in gauge and gravity theories in Eqs. (15) and (26). To double copy the EM plane wave of Eq. (15), we can dress it by an arbitrary constant color vector ${c^a}$ to make a Yang–Mills field:

$$A_\mu ^a = {c^a}{A_\mu} = {c^a}{\epsilon _\mu}{e^{ik \cdot x}}.$$

Upon substituting this into the Yang–Mills equations, one may show that they linearize and thus reduce to the Maxwell-like equations,

$${\partial ^\mu}({\partial _\mu}A_\nu ^a - {\partial _\nu}A_\mu ^a) = {c^a}{\partial ^\mu}({\partial _\mu}{A_\nu} - {\partial _\nu}{A_\mu}) = 0.$$

(Each non-linear term involves a contraction of two color vectors ${c^a},\;{c^b}$ and structure constants ${f^{\textit{abc}}}$. Such contractions vanish due to the antisymmetry of the latter.) We may, thus, happily forget about ${c^a}$ (and hence the Yang–Mills nature of the field) and regard the gauge field as living in electromagnetism only. The photon has two polarization states, which are often chosen to correspond to the two independent circular polarizations

$$\epsilon _\mu ^ \pm = \frac{1}{2}(0, \mp 1,i,0),$$
where the normalization is such that ${\epsilon _ +} \cdot {\epsilon _ -} = 1$. (In particle physics language, these are two helicity states of the photon, corresponding to an eigenvalue ${\pm}1$, respectively, for the projection of the spin along the direction of travel.) The graviton also has two polarization states, and one may verify that the appropriate polarization tensors may be written as
$$\epsilon _{\mu \nu}^ \pm = \epsilon _\mu ^ \pm \epsilon _\nu ^ \pm ,$$
such that the gravitational plane waves of Eq. (26) assume the form (for waves of definite helicity)
$$h_{\mu \nu}^ \pm = \epsilon _\mu ^ \pm \epsilon _\nu ^ \pm {e^{ik \cdot x}}.$$

Comparing with Eq. (32), we see that this has the form of a function that does not change (the phase factor ${e^{ik \cdot x}}$) times two copies of quantities taken from a gauge theory. Furthermore, one strips away the color vector in the gauge theory and replaces it with a kinematic quantity (a polarization vector) to get the gravity solution. Doing the opposite, one would obtain a biadjoint field

$${\Phi ^{{aa^\prime}}} = {c^a}{\tilde c^{{a^\prime}}}{e^{ik \cdot x}},$$
which is indeed a plane wave solution of Eq. (31). We thus have a series of fields
$${\Phi ^{{aa^\prime}}}\quad \leftrightarrow \quad A_\mu ^a\quad \leftrightarrow \quad {h_{\mu \nu}},$$
which provide a realization of Fig. 5 for particular classical solutions. Indeed, this can be related to the double copy for scattering amplitudes above, in that the incoming and outgoing states in a scattering process are themselves obtained from plane waves. There is also a more explicit similarity with the amplitude story: in going from Eq. (30) to Eqs. (28) and (29), one must successively remove the color factors and replace them with kinematic numerators, whilst leaving the denominators of the expressions intact. Likewise, in going from Eq. (37) to Eqs. (32) and (36), one must strip off the color vectors and replace them with polarization vectors, leaving a certain function (${e^{ik \cdot x}}$) alone. The latter therefore seems to play an analogous role to the denominator factors in an amplitude, and indeed this analogy can be made more precise [15].

Note that particular combinations of polarization states have been taken in Eq. (36), but that there are two other choices: in combining the two polarization states of two photons, we expect four different combinations. For the missing two, we can take them to be the (anti)symmetric combinations

$$\frac{1}{2}\left[{\epsilon _\mu ^ + \epsilon _\nu ^ - - \epsilon _\mu ^ - \epsilon _\nu ^ +} \right],\quad \frac{1}{2}\left[{\epsilon _\mu ^ + \epsilon _\nu ^ - + \epsilon _\mu ^ - \epsilon _\nu ^ +} \right].$$

Each of these constitutes a single degree of freedom, and the first and second combinations can be associated with a pseudo-scalar and a scalar field, respectively. (A pseudo-scalar field differs from a scalar in that it picks up a minus sign under a parity transformation, which reverses the spatial coordinate axes.) These are known as the axion and the dilaton, and thus the true double copy of pure Yang–Mills theory is not GR but gravity coupled to these two extra fields. There are additional justifications one can give for this result. For Feynman diagrams with no loops, it turns out that the double copy can be derived from string theory, where it reproduces known results [5] and also makes clear that the axion and dilaton should be present. This argument does not, however, easily generalize to amplitudes with loops. However, taking a “product” of gauge fields, on general grounds, generates a field with two indices, which can be decomposed into its symmetric traceless, antisymmetric, and trace degrees of freedom. These are known to mathematicians as irreducible representations of the Lorentz group and correspond physically to the graviton, axion, and dilaton, respectively.

The question naturally arises as to whether the above procedure can be extended, and indeed it can. The double copy of more general exact classical solutions was first considered in Ref. [15], which considered the infinite family of so-called Kerr–Schild solutions in GR. For each case, the metric can be written as in Eq. (22), with the graviton having the special form

$${h_{\mu \nu}} = \phi {k_\mu}{k_\nu},$$
where $\phi$ is a scalar field, and the vector field ${k_\mu}$ has to satisfy
$${k^2} = 0,\quad k \cdot \partial {k^\mu} = 0.$$

Substituting this ansatz into the Einstein equations of GR, they are found (after a great deal of effort) to linearize. Thus, they become much easier to solve, and furthermore any solutions are then known to be exact. Looking at Eq. (39), it is tempting to guess how to take its single copy and zeroth copies. One could simply remove successive factors of the four-vector ${k_\mu}$ and replace them with color vectors to get

$$A_\mu ^a = {c^a}\phi {k_\mu},\quad {\Phi ^{{aa^\prime}}} = {c^a}{\tilde c^{{a^\prime}}}\phi .$$

Reference [15] proved that, for static solutions at least, the fields of Eq. (41) are solutions of the Yang–Mills and biadjoint equations, respectively, where the field equations are also linearized in each case. Some time-dependent cases are also known, including the plane waves already mentioned above! Interestingly, the Kerr–Schild double copy involves solutions of pure GR, without any contamination from the axion and dilaton. To see this, note that the field of Eq. (39) is symmetric in its indices by construction and also traceless due to the conditions on ${k^\mu}$ of Eq. (40). It thus corresponds to the graviton only, and no product of gauge fields involving ${k_\mu}$ would be able to generate the axion (antisymmetric) or dilaton (trace degree of freedom).

 figure: Fig. 6.

Fig. 6. Magnetic field of the single copy of the Kerr (rotating) black hole, corresponding to a rotating disc of charge. The disc is seen sideways on and runs from $x = - 1$ to $x = 1$.

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The Kerr–Schild double copy may look abstract, but it in fact includes some of the most famous gravitational objects known to physics! A first example is the Schwarzschild black hole, a static, spherically symmetric solution, which can be sourced by a point-like mass $M$ at the origin. Its Kerr–Schild scalar field and vector are found to be

$$\phi = - \frac{M}{{4\pi r}},\quad {k^\mu} = (1,{\vec e_r}),$$
where $r$ is the spherical radius, and ${\vec e_r}$ is a unit three-vector in the radial direction. This is very different to how the Schwarzschild solution is conventionally presented (e.g., in introductory GR courses). For the interested reader, we explain the coordinate transformation that relates the two forms of the solution in Appendix A. Taking the single copy of Eq. (42) results in the gauge field
$${A_\mu} = - \frac{Q}{{4\pi r}}{k^\mu}\quad \to \quad \left({- \frac{Q}{{4\pi r}},0,0,0} \right),$$
where we have performed a gauge transformation on the right-hand side, whose details may be found in Ref. [15]. Recalling that the zeroth component of the gauge field is the electrostatic potential, we recognize Eq. (43) as the gauge field of a point charge. Thus, the double copy relates a simple point charge at the origin to a point mass in gravity, which is directly analogous to the replacement of color information by kinematics in the double copy for amplitudes!

As a more non-trivial example, one may consider the Kerr black hole, an axially symmetric system that can be sourced by a rotating disc of mass in gravity. Similar to above, its single copy turns out to be a rotating disc of charge, whose profile matches that of its gravitational counterpart. In Fig. 6, we show the magnetic field of the single copy of the Kerr black hole. As we zoom out, the field becomes dipole-like, as one expects given that a rotating disc of charge looks like a nested set of current loops!

Other examples of the Kerr–Schild double copy involve magnetic monopoles [16], accelerating particles [17], EM vortices [18], and novel solutions in two spatial dimensions [19]. However, the family of solutions that can be put into a Kerr–Schild form remains very special, so that it would be desirable to extend the scope of the classical double copy yet further. To this end, an alternative procedure known as the Weyl double copy has been defined [20], which relies on the so-called spinorial formalism of field theory (reviewed both excellently and encyclopaedically in Refs. [21,22]). This can indeed be made more general than the Kerr–Schild double copy [2325] and has also been explored in the conventional tensor language [26]. Another deficiency of the Kerr–Schild approach is that it is highly dependent on a particular gauge (coordinate system) being chosen in gauge theory (gravity), which itself is reminiscent of the fact that the double copy for amplitudes only works for a particular generalized gauge such that the kinematic numerators $\{{n_i}\}$ are BCJ dual. Alternative formalisms have been developed that can, in principle, work in any gauge [2732], where the “unphysical” degrees of freedom in each theory (i.e., corresponding to gauge redundancy in the fields) can themselves be matched up.

It is not yet known whether a fully general and exact statement of the double copy can be made that works for any type of solution, be it classical or quantum. Attempts to generalize the idea include finding (and trying to copy) non-linear solutions of the biadjoint theory [3336], showing how exact symmetries in different theories can be related [3739] and looking at how topological properties of solutions can be matched up [40,41]. If one is willing to forego exact statements and work order-by-order in perturbation theory, there are by now many different approaches for calculating classical observables in GR by first calculating in a (simpler) gluon theory and then double copying the results (see, e.g., Ref. [6] and references therein). This is continuing to attract intense worldwide interest due to the applications to gravitational waves. A typical signal observed by the LIGO experiment arises from the coming together of two heavy objects such as black holes or neutron stars in a three stage process: (i) an inspiral phase, in which the objects gradually orbit closer to each other; (ii) the merger itself, in which, e.g., two black holes combine to make a larger one; (iii) the ringdown phase, in which the combined heavy object wobbles and settles down (see Ref. [42] for an excellent review of contemporary methods for describing such processes). The double copy has so far been applied to step (i) and may potentially be used for step (iii). Step (ii) relies on complex numerical simulation work, but the expense of this may be reduced if further improvements can be made to the other two steps.

6. WHAT IS IN IT FOR OPTICS?

One of the key scientific problems of our age is a tendency towards increasing specialization and isolation. Physicists start to significantly diverge from each other even before they have completed their Masters degrees, and the lack of a common language between apparently diverse fields such as astrophysics/cosmology, high energy physics, condensed matter, and optics too often impedes the ability of different types of physicists to work together. In this paper, we have reviewed an intriguing set of correspondences—known collectively as the double copy—that relate solutions in widely different field theories. Unlike many of the developments that have preoccupied theoretical high energy physicists in recent years, the double copy connects actual physical theories that are directly observed in nature! It is, thus, only natural to try to widen the notion of the double copy yet further, and to seek situations that may be tested in the lab. It is difficult for the present author to definitively state what these experiments might be, especially given that the aim of this paper is to try to stimulate the kinds of conversations that might lead to the necessary interdisciplinary work. However, there are some potentially useful connections to optics/condensed matter that have appeared in recent years.

Reference [43] showed that classical gravitational waves could, in principle, be emulated using quantum-entangled photon pairs. It was shown how to obtain the appropriate gravitational wave polarizations by combining definite photon helicity states, but it was also noted that two of the possible combinations were associated with helicity zero and thus had to be thrown away. This is in fact the double copy in all but name, and the two “spurious” photon combinations are precisely the axion and dilaton we discussed above! Rather than being discarded, they could be used to simulate the full double copy of a pure gauge theory. Regarding what to simulate, Ref. [43] had in mind the emulation of gravitational waves moving through a background curved spacetime, where the latter could be modeled by passing the entangled photons through a metamaterial. Unknown to the authors was the fact that whether or not the double copy can be made exact in a curved spacetime is in fact an open problem (see, e.g., Refs. [4448] for preliminary investigations), which in turn may shed light on how the double copy may be applied in astrophysics and/or cosmology. Further studies have involved scattering particles on localized wave backgrounds, which could be simulated using, e.g., strong laser pulses [49]. It is highly exciting that the possibility exists of using table-top experiments to probe one of the most intriguing ideas to emerge in high energy physics in recent years!

Another connection noticed by the present author is that of the study of topologically non-trivial states of light. A family of solutions of the Maxwell equations known as Hopfions have electric or magnetic field lines that are knotted, and generalizations exist such that the field lines form so-called torus knots. Gravitational counterparts also exist and have been studied in Refs. [5055], where the latter reference comments that the EM and gravity solutions can be related by the Weyl double copy mentioned above. The authors use twistor theory [5658]—an elegant set of mathematical ideas relating algebraic geometry and complex analysis—to classify knotted radiation solutions in a compact way. This was taken further in Refs. [24,25], which provided a derivation of the Weyl double copy using twistor ideas (see also Refs. [59,60] for related ideas). It would be interesting to know whether experimental efforts exist to study the EM knotted solutions in their own right. If so, might they be combined with the above ideas to simulate gravitational knotted radiation? What could this teach us?

7. CONCLUSION

Field theories occur in many branches of physics, and notable examples include the gauge theories underlying three of the four fundamental forces of nature (including electromagnetism) and GR. Recent years have seen a flurry of activity regarding a newly discovered correspondence between gauge and gravity theories called the double copy. Originating in the study of scattering amplitudes related to the probability for particles to interact, the double copy has since been extended to classical solutions, including those of interest in astrophysics and optics. It has also been shown to apply to an increasing range of theories, with varying degrees of physical relevance.

As the remit of the double copy has increased, so have the number of interested researchers from astrophysics, cosmology, high energy physics, and pure mathematics. Now seems to be the perfect time to try to interest physicists from optics and/or condensed matter, particularly given recent indications that the scope and applicability of the double copy may be amenable to table-top experiments in the lab.

The author humbly hopes that this paper will stimulate interested conversations—and indeed conversations about how to have the conversations—that are able to surmount the often considerable barriers between different subfields, which are often merely due to a difference in language. Recent years have taught us that our traditional approach to QFT is hiding a deep and profound underlying structure. Will the field of optics shine the light that guides the way?

APPENDIX A: THE SCHWARZSCHILD METRIC IN KERR–SCHILD COORDINATES

In this Appendix, we show how to transform from the conventional form of the Schwarzschild metric to the Kerr–Schild form implied by Eq. (42). Given the specialist nature of this section, we will assume slightly more familiarity with GR than in the rest of the article. Our starting point is to recall the line element [the curved space generalization of Eq. (21)] for the Schwarzschild solution, as it is usually presented in textbooks:

$$\begin{split}d{s^2} &= \left({1 - \frac{{2{G_N}M}}{r}} \right)d{t^2} \\&\quad- {\left({1 - \frac{{2{G_N}M}}{r}} \right)^{- 1}}d{r^2} - {r^2}(d{\theta ^2} + \mathop {\sin}\nolimits^2 \theta d{\phi ^2}).\end{split}$$

We may then define the Kerr–Schild time variable

$${t_{{\rm{KS}}}} = t + 2{G_N}M\ln \left({\frac{r}{{2{G_N}M}} - 1} \right),$$
which in turn implies
$$dt = d{t_{{\rm{KS}}}} - \frac{{2{G_N}M}}{{r - 2{G_N}M}}dr.$$

Substituting this directly into Eq. (A1) and rearranging, one obtains

$$d{s^2} = dt_{{\rm{KS}}}^2 - d{r^2} - {r^2}(d{\theta ^2} + \mathop {\sin}\nolimits^2 \theta d{\phi ^2}) - \frac{{2{G_N}M}}{r}{(dt + dr)^2}.$$

We may recognize the first three terms on the right-hand side as the Minkowski-space line element in spherical polars. Thus, the fourth term contains the graviton that, from Eq. (22), is given by

$${h_{\mu \nu}}d{x^\mu}d{x^\nu} = - \frac{\kappa}{2}\frac{M}{{4\pi r}}{(dt + dr)^2}.$$

This agrees precisely with Eq. (42).

Funding

Science and Technology Facilities Council (ST/P000754/1).

Acknowledgment

The author thanks Johannes Courtial and Kurt Busch for their encouragement in writing this paper.

Disclosures

The author declares no conflicts of interest.

Data Availability

No data were generated or analyzed in the presented research.

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52. A. J. de Klerk, R. I. van der Veen, J. W. Dalhuisen, and D. Bouwmeester, “Knotted optical vortices in exact solutions to Maxwell’s equations,” Phys. Rev. A 95, 053820 (2017). [CrossRef]  

53. A. Thompson, A. Wickes, J. Swearngin, and D. Bouwmeester, “Classification of electromagnetic and gravitational hopfions by algebraic type,” J. Phys. A 48, 205202 (2015). [CrossRef]  

54. A. Thompson, J. Swearngin, and D. Bouwmeester, “Linked and knotted gravitational radiation,” J. Phys. A 47, 355205 (2014). [CrossRef]  

55. S. Sabharwal and J. W. Dalhuisen, “Anti-self-dual spacetimes, gravitational instantons and knotted zeros of the Weyl tensor,” J. High Energy Phys. 2019, 004 (2019). [CrossRef]  

56. R. Penrose, “Twistor algebra,” J. Math. Phys. 8, 345 (1967). [CrossRef]  

57. R. Penrose and M. A. H. MacCallum, “Twistor theory: an approach to the quantization of fields and space-time,” Phys. Rep. 6, 241–316 (1972). [CrossRef]  

58. R. Penrose, “Twistor quantization and curved space-time,” Int. J. Theor. Phys. 1, 61–99 (1968). [CrossRef]  

59. G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman–Penrose map and the classical double copy,” J. High Energy Phys. 2020, 121 (2020). [CrossRef]  

60. K. Farnsworth, M. L. Graesser, and G. Herczeg, “Twistor space origins of the Newman–Penrose map,” arXiv: 2104.09525 (2021).

References

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  34. P.-J. De Smet and C. D. White, “Extended solutions for the biadjoint scalar field,” Phys. Lett. B 775, 163–167 (2017).
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  36. N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Monopoles, shockwaves and the classical double copy,” J. High Energy Phys. 2020, 102 (2020).
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  37. R. Alawadhi, D. S. Berman, B. Spence, and D. P. Veiga, “S-duality and the double copy,” J. High Energy Phys. 2020, 059 (2020).
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  38. A. Banerjee, E. O. Colgáin, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” Phys. Rev. D 102, 126017 (2020).
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  39. Y.-T. Huang, U. Kol, and D. O’Connell, “Double copy of electric-magnetic duality,” Phys. Rev. D 102, 046005 (2020).
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  40. D. S. Berman, E. Chacón, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi–Hanson instanton,” J. High Energy Phys. 2019, 107 (2019).
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  41. L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of the double copy,” J. High Energy Phys. 2020, 091 (2020).
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  42. M. Levi, “Effective field theories of post-Newtonian gravity: a comprehensive review,” Rep. Prog. Phys. 83, 075901 (2020).
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  44. N. Bahjat-Abbas, A. Luna, and C. D. White, “The Kerr–Schild double copy in curved spacetime,” J. High Energy Phys. 2017, 004 (2017).
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  45. M. Carrillo-González, R. Penco, and M. Trodden, “The classical double copy in maximally symmetric spacetimes,” J. High Energy Phys. 2018, 028 (2018).
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    [Crossref]
  49. T. Adamo, E. Casali, L. Mason, and S. Nekovar, “Scattering on plane waves and the double copy,” Classical Quantum Gravity 35, 015004 (2018).
    [Crossref]
  50. J. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
    [Crossref]
  51. J. Swearngin, A. Thompson, A. Wickes, J. W. Dalhuisen, and D. Bouwmeester, “Gravitational hopfions,” arXiv: 1302.1431 (2013).
  52. A. J. de Klerk, R. I. van der Veen, J. W. Dalhuisen, and D. Bouwmeester, “Knotted optical vortices in exact solutions to Maxwell’s equations,” Phys. Rev. A 95, 053820 (2017).
    [Crossref]
  53. A. Thompson, A. Wickes, J. Swearngin, and D. Bouwmeester, “Classification of electromagnetic and gravitational hopfions by algebraic type,” J. Phys. A 48, 205202 (2015).
    [Crossref]
  54. A. Thompson, J. Swearngin, and D. Bouwmeester, “Linked and knotted gravitational radiation,” J. Phys. A 47, 355205 (2014).
    [Crossref]
  55. S. Sabharwal and J. W. Dalhuisen, “Anti-self-dual spacetimes, gravitational instantons and knotted zeros of the Weyl tensor,” J. High Energy Phys. 2019, 004 (2019).
    [Crossref]
  56. R. Penrose, “Twistor algebra,” J. Math. Phys. 8, 345 (1967).
    [Crossref]
  57. R. Penrose and M. A. H. MacCallum, “Twistor theory: an approach to the quantization of fields and space-time,” Phys. Rep. 6, 241–316 (1972).
    [Crossref]
  58. R. Penrose, “Twistor quantization and curved space-time,” Int. J. Theor. Phys. 1, 61–99 (1968).
    [Crossref]
  59. G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman–Penrose map and the classical double copy,” J. High Energy Phys. 2020, 121 (2020).
    [Crossref]
  60. K. Farnsworth, M. L. Graesser, and G. Herczeg, “Twistor space origins of the Newman–Penrose map,” arXiv: 2104.09525 (2021).

2021 (8)

H. Godazgar, M. Godazgar, R. Monteiro, D. P. Veiga, and C. N. Pope, “Weyl double copy for gravitational waves,” Phys. Rev. Lett. 126, 101103 (2021).
[Crossref]

C. D. White, “Twistorial foundation for the classical double copy,” Phys. Rev. Lett. 126, 061602 (2021).
[Crossref]

E. Chacón, S. Nagy, and C. D. White, “The Weyl double copy from twistor space,” J. High Energy Phys. 2021, 2239 (2021).
[Crossref]

L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “BRST-Lagrangian double copy of Yang–Mills theory,” Phys. Rev. Lett. 126, 191601 (2021).
[Crossref]

L. Borsten, H. Kim, B. Jurco, T. Macrelli, C. Saemann, and M. Wolf, “Double copy from homotopy algebras,” Fortschr. Phys. 69, 2100075 (2021).
[Crossref]

M. Campiglia and S. Nagy, “A double copy for asymptotic symmetries in the self-dual sector,” J. High Energy Phys. 2021, 262 (2021).
[Crossref]

G. Alkac, M. K. Gumus, and M. Tek, “The Kerr–Schild double copy in Lifshitz spacetime,” J. High Energy Phys. 2021, 214 (2021).
[Crossref]

L. Borsten, I. Jubb, V. Makwana, and S. Nagy, “Gauge × gauge = gravity on homogeneous spaces using tensor convolutions,” J. High Energy Phys. 2021, 117 (2021).
[Crossref]

2020 (9)

G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman–Penrose map and the classical double copy,” J. High Energy Phys. 2020, 121 (2020).
[Crossref]

L. Borsten and S. Nagy, “The pure BRST Einstein–Hilbert Lagrangian from the double-copy to cubic order,” J. High Energy Phys. 2020, 093 (2020).
[Crossref]

N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Monopoles, shockwaves and the classical double copy,” J. High Energy Phys. 2020, 102 (2020).
[Crossref]

R. Alawadhi, D. S. Berman, B. Spence, and D. P. Veiga, “S-duality and the double copy,” J. High Energy Phys. 2020, 059 (2020).
[Crossref]

A. Banerjee, E. O. Colgáin, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” Phys. Rev. D 102, 126017 (2020).
[Crossref]

Y.-T. Huang, U. Kol, and D. O’Connell, “Double copy of electric-magnetic duality,” Phys. Rev. D 102, 046005 (2020).
[Crossref]

L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of the double copy,” J. High Energy Phys. 2020, 091 (2020).
[Crossref]

M. Levi, “Effective field theories of post-Newtonian gravity: a comprehensive review,” Rep. Prog. Phys. 83, 075901 (2020).
[Crossref]

R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling,” J. High Energy Phys. 2020, 127 (2020).
[Crossref]

2019 (5)

N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Biadjoint wires,” Phys. Lett. B 788, 274–279 (2019).
[Crossref]

D. S. Berman, E. Chacón, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi–Hanson instanton,” J. High Energy Phys. 2019, 107 (2019).
[Crossref]

M. Carrillo González, B. Melcher, K. Ratliff, S. Watson, and C. D. White, “The classical double copy in three spacetime dimensions,” J. High Energy Phys. 2019, 167 (2019).
[Crossref]

A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, “Type D spacetimes and the Weyl double copy,” Classical Quantum Gravity 36, 065003 (2019).
[Crossref]

S. Sabharwal and J. W. Dalhuisen, “Anti-self-dual spacetimes, gravitational instantons and knotted zeros of the Weyl tensor,” J. High Energy Phys. 2019, 004 (2019).
[Crossref]

2018 (4)

T. Adamo, E. Casali, L. Mason, and S. Nekovar, “Scattering on plane waves and the double copy,” Classical Quantum Gravity 35, 015004 (2018).
[Crossref]

M. Carrillo-González, R. Penco, and M. Trodden, “The classical double copy in maximally symmetric spacetimes,” J. High Energy Phys. 2018, 028 (2018).
[Crossref]

A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as gauge theory squared: a ghost story,” Phys. Rev. Lett. 121, 211601 (2018).
[Crossref]

A. Ilderton, “Screw-symmetric gravitational waves: a double copy of the vortex,” Phys. Lett. B 782, 22–27 (2018).
[Crossref]

2017 (3)

P.-J. De Smet and C. D. White, “Extended solutions for the biadjoint scalar field,” Phys. Lett. B 775, 163–167 (2017).
[Crossref]

N. Bahjat-Abbas, A. Luna, and C. D. White, “The Kerr–Schild double copy in curved spacetime,” J. High Energy Phys. 2017, 004 (2017).
[Crossref]

A. J. de Klerk, R. I. van der Veen, J. W. Dalhuisen, and D. Bouwmeester, “Knotted optical vortices in exact solutions to Maxwell’s equations,” Phys. Rev. A 95, 053820 (2017).
[Crossref]

2016 (3)

C. D. White, “Exact solutions for the biadjoint scalar field,” Phys. Lett. B 763, 365–369 (2016).
[Crossref]

A. Luna, R. Monteiro, I. Nicholson, D. O’Connell, and C. D. White, “The double copy: Bremsstrahlung and accelerating black holes,” J. High Energy Phys. 2016, 023 (2016).
[Crossref]

B. P. Abbott, LIGO Scientific Collaboration, and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

2015 (3)

A. Luna, R. Monteiro, D. O’Connell, and C. D. White, “The classical double copy for Taub-NUT spacetime,” Phys. Lett. B 750, 272–277 (2015).
[Crossref]

I. Fernandez-Corbaton, M. Cirio, A. Büse, L. Lamata, E. Solano, and G. Molina-Terriza, “Quantum emulation of gravitational waves,” Sci. Rep. 5, 11538 (2015).
[Crossref]

A. Thompson, A. Wickes, J. Swearngin, and D. Bouwmeester, “Classification of electromagnetic and gravitational hopfions by algebraic type,” J. Phys. A 48, 205202 (2015).
[Crossref]

2014 (4)

A. Thompson, J. Swearngin, and D. Bouwmeester, “Linked and knotted gravitational radiation,” J. Phys. A 47, 355205 (2014).
[Crossref]

R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy,” J. High Energy Phys. 2014, 056 (2014).
[Crossref]

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang–Mills origin of gravitational symmetries,” Phys. Rev. Lett. 113, 231606 (2014).
[Crossref]

S. M. Barnett, “Maxwellian theory of gravitational waves and their mechanical properties,” New J. Phys. 16, 023027 (2014).
[Crossref]

2013 (1)

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

2012 (1)

J. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
[Crossref]

2011 (1)

R. Monteiro and D. O’Connell, “The kinematic algebra from the self-dual sector,” J. High Energy Phys. 2011, 007 (2011).
[Crossref]

2010 (2)

Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative quantum gravity as a double copy of gauge theory,” Phys. Rev. Lett. 105, 061602 (2010).
[Crossref]

Z. Bern, T. Dennen, Y.-T. Huang, and M. Kiermaier, “Gravity as the square of gauge theory,” Phys. Rev. D 82, 065003 (2010).
[Crossref]

2008 (1)

Z. Bern, J. Carrasco, and H. Johansson, “New relations for gauge-theory amplitudes,” Phys. Rev. D 78, 085011 (2008).
[Crossref]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

1986 (1)

H. Kawai, D. Lewellen, and S. Tye, “A relation between tree amplitudes of closed and open strings,” Nucl. Phys. B 269, 1–23 (1986).
[Crossref]

1972 (1)

R. Penrose and M. A. H. MacCallum, “Twistor theory: an approach to the quantization of fields and space-time,” Phys. Rep. 6, 241–316 (1972).
[Crossref]

1968 (1)

R. Penrose, “Twistor quantization and curved space-time,” Int. J. Theor. Phys. 1, 61–99 (1968).
[Crossref]

1967 (1)

R. Penrose, “Twistor algebra,” J. Math. Phys. 8, 345 (1967).
[Crossref]

Abbott, B. P.

B. P. Abbott, LIGO Scientific Collaboration, and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

Adamo, T.

T. Adamo, E. Casali, L. Mason, and S. Nekovar, “Scattering on plane waves and the double copy,” Classical Quantum Gravity 35, 015004 (2018).
[Crossref]

Alawadhi, R.

R. Alawadhi, D. S. Berman, B. Spence, and D. P. Veiga, “S-duality and the double copy,” J. High Energy Phys. 2020, 059 (2020).
[Crossref]

R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling,” J. High Energy Phys. 2020, 127 (2020).
[Crossref]

Alfonsi, L.

L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of the double copy,” J. High Energy Phys. 2020, 091 (2020).
[Crossref]

Alkac, G.

G. Alkac, M. K. Gumus, and M. Tek, “The Kerr–Schild double copy in Lifshitz spacetime,” J. High Energy Phys. 2021, 214 (2021).
[Crossref]

Anastasiou, A.

A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as gauge theory squared: a ghost story,” Phys. Rev. Lett. 121, 211601 (2018).
[Crossref]

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang–Mills origin of gravitational symmetries,” Phys. Rev. Lett. 113, 231606 (2014).
[Crossref]

Bahjat-Abbas, N.

N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Monopoles, shockwaves and the classical double copy,” J. High Energy Phys. 2020, 102 (2020).
[Crossref]

N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Biadjoint wires,” Phys. Lett. B 788, 274–279 (2019).
[Crossref]

N. Bahjat-Abbas, A. Luna, and C. D. White, “The Kerr–Schild double copy in curved spacetime,” J. High Energy Phys. 2017, 004 (2017).
[Crossref]

Banerjee, A.

A. Banerjee, E. O. Colgáin, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” Phys. Rev. D 102, 126017 (2020).
[Crossref]

Barnett, S. M.

S. M. Barnett, “Maxwellian theory of gravitational waves and their mechanical properties,” New J. Phys. 16, 023027 (2014).
[Crossref]

Berman, D. S.

R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling,” J. High Energy Phys. 2020, 127 (2020).
[Crossref]

R. Alawadhi, D. S. Berman, B. Spence, and D. P. Veiga, “S-duality and the double copy,” J. High Energy Phys. 2020, 059 (2020).
[Crossref]

D. S. Berman, E. Chacón, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi–Hanson instanton,” J. High Energy Phys. 2019, 107 (2019).
[Crossref]

Bern, Z.

Z. Bern, T. Dennen, Y.-T. Huang, and M. Kiermaier, “Gravity as the square of gauge theory,” Phys. Rev. D 82, 065003 (2010).
[Crossref]

Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative quantum gravity as a double copy of gauge theory,” Phys. Rev. Lett. 105, 061602 (2010).
[Crossref]

Z. Bern, J. Carrasco, and H. Johansson, “New relations for gauge-theory amplitudes,” Phys. Rev. D 78, 085011 (2008).
[Crossref]

Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The duality between color and kinematics and its applications,” arXiv:1909.01358 (2019).

Borsten, L.

L. Borsten, I. Jubb, V. Makwana, and S. Nagy, “Gauge × gauge = gravity on homogeneous spaces using tensor convolutions,” J. High Energy Phys. 2021, 117 (2021).
[Crossref]

L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “BRST-Lagrangian double copy of Yang–Mills theory,” Phys. Rev. Lett. 126, 191601 (2021).
[Crossref]

L. Borsten, H. Kim, B. Jurco, T. Macrelli, C. Saemann, and M. Wolf, “Double copy from homotopy algebras,” Fortschr. Phys. 69, 2100075 (2021).
[Crossref]

L. Borsten and S. Nagy, “The pure BRST Einstein–Hilbert Lagrangian from the double-copy to cubic order,” J. High Energy Phys. 2020, 093 (2020).
[Crossref]

A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as gauge theory squared: a ghost story,” Phys. Rev. Lett. 121, 211601 (2018).
[Crossref]

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang–Mills origin of gravitational symmetries,” Phys. Rev. Lett. 113, 231606 (2014).
[Crossref]

Bouwmeester, D.

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A. J. de Klerk, R. I. van der Veen, J. W. Dalhuisen, and D. Bouwmeester, “Knotted optical vortices in exact solutions to Maxwell’s equations,” Phys. Rev. A 95, 053820 (2017).
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H. Godazgar, M. Godazgar, R. Monteiro, D. P. Veiga, and C. N. Pope, “Weyl double copy for gravitational waves,” Phys. Rev. Lett. 126, 101103 (2021).
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T. Adamo, E. Casali, L. Mason, and S. Nekovar, “Scattering on plane waves and the double copy,” Classical Quantum Gravity 35, 015004 (2018).
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A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, “Type D spacetimes and the Weyl double copy,” Classical Quantum Gravity 36, 065003 (2019).
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A. Banerjee, E. O. Colgáin, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” Phys. Rev. D 102, 126017 (2020).
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S. Sabharwal and J. W. Dalhuisen, “Anti-self-dual spacetimes, gravitational instantons and knotted zeros of the Weyl tensor,” J. High Energy Phys. 2019, 004 (2019).
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L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “BRST-Lagrangian double copy of Yang–Mills theory,” Phys. Rev. Lett. 126, 191601 (2021).
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H. Godazgar, M. Godazgar, R. Monteiro, D. P. Veiga, and C. N. Pope, “Weyl double copy for gravitational waves,” Phys. Rev. Lett. 126, 101103 (2021).
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Data Availability

No data were generated or analyzed in the presented research.

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Figures (6)

Fig. 1.
Fig. 1. Phase of the electron field at each point in spacetime can be thought of as an arrow on the unit circle, where a local gauge transformation may rotate the arrows at different points by different amounts.
Fig. 2.
Fig. 2. We can imagine the quark field at each point in spacetime as having an arrow in an internal color space, telling us how much of each color charge it has.
Fig. 3.
Fig. 3. Trajectories of two people walking in a curved space.
Fig. 4.
Fig. 4. (a) Example Feynman diagram for scattering of two gluons to two gluons; (b) momentum conservation in a loop diagram.
Fig. 5.
Fig. 5. Various theories and the relationships between them. Electromagnetism sits it in the middle, and GR is on the right.
Fig. 6.
Fig. 6. Magnetic field of the single copy of the Kerr (rotating) black hole, corresponding to a rotating disc of charge. The disc is seen sideways on and runs from $x = - 1$ to $x = 1$.

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

x μ = ( t , x ) ,
x μ = ( t , x ) = η μ ν x ν .
η μ ν = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
x y = x μ y μ = η μ ν x μ y ν ,
p μ = ( E , p ) ,
E 2 p 2 = m 2 ,
A μ = ( ϕ , A ) ,
E = ϕ A t , B = × A .
μ F μ ν = j ν ,
F μ ν = μ A ν ν A μ ,
j μ = ( ρ , j ) ,
A μ A μ = A μ + μ χ ,
μ A μ = 0 ,
( 2 t 2 2 ) A μ = j μ .
A μ = ϵ μ e i k x , k μ ϵ μ = 0 ,
ψ e i θ ( x μ ) ψ .
U = exp [ i θ a ( x μ ) T a ] ,
[ T a , T b ] = i f abc T c , [ A , B ] AB BA .
F μ ν a = μ A ν a ν A μ a + g f abc A μ b A ν c ,
μ F μ ν a + g f abc A μ b F μ ν c = 0.
d x μ = ( d t , d x )
d s 2 = d t 2 d x d x = η μ ν d x μ d x ν ,
g μ ν = η μ ν + κ h μ ν ,
g α β ( y μ ) = ( x μ y α ) ( x ν y β ) g μ ν ( x μ ) .
η μ ν h μ ν = 0 , μ h μ ν = 0.
( 2 t 2 2 ) h μ ν = 0 ,
h μ ν = ϵ μ ν e i k x ,
d 4 k ( 2 π ) 4 ,
p 2 E 2 p 2 ,
A m ( L ) = g m 2 + 2 L i ( l = 1 L d 4 p l ( 2 π ) 4 ) n i c i α i p α i 2 .
M m ( L ) = ( κ 2 ) m 2 + 2 L i ( l = 1 L d 4 p l ( 2 π ) 4 ) n i n ~ i α i p α i 2
A ~ m ( L ) = y m 2 + 2 L i ( l = 1 L d 4 p l ( 2 π ) 4 ) c i c ~ i α i p α i 2 ,
2 Φ a a y f abc f ~ a b c Φ b b Φ c c = 0.
A μ a = c a A μ = c a ϵ μ e i k x .
μ ( μ A ν a ν A μ a ) = c a μ ( μ A ν ν A μ ) = 0.
ϵ μ ± = 1 2 ( 0 , 1 , i , 0 ) ,
ϵ μ ν ± = ϵ μ ± ϵ ν ± ,
h μ ν ± = ϵ μ ± ϵ ν ± e i k x .
Φ a a = c a c ~ a e i k x ,
Φ a a A μ a h μ ν ,
1 2 [ ϵ μ + ϵ ν ϵ μ ϵ ν + ] , 1 2 [ ϵ μ + ϵ ν + ϵ μ ϵ ν + ] .
h μ ν = ϕ k μ k ν ,
k 2 = 0 , k k μ = 0.
A μ a = c a ϕ k μ , Φ a a = c a c ~ a ϕ .
ϕ = M 4 π r , k μ = ( 1 , e r ) ,
A μ = Q 4 π r k μ ( Q 4 π r , 0 , 0 , 0 ) ,
d s 2 = ( 1 2 G N M r ) d t 2 ( 1 2 G N M r ) 1 d r 2 r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) .
t K S = t + 2 G N M ln ( r 2 G N M 1 ) ,
d t = d t K S 2 G N M r 2 G N M d r .
d s 2 = d t K S 2 d r 2 r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) 2 G N M r ( d t + d r ) 2 .
h μ ν d x μ d x ν = κ 2 M 4 π r ( d t + d r ) 2 .

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