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IV.13 General Relativity and the Einstein Equations

Mihalis Dafermos

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Einstein’s formulation of general relativity represents one of the great triumphs of modern physics and provides the currently accepted classical theory that unifies gravitation, inertia, and geometry. The Einstein equations are the mathematical embodiment of this theory.

The definitive form of the equations,

was attained in November 1915; this was the final act of Einstein’s eight-year struggle to generalize his principle of relativity so as to encompass gravitation, which had been described in the earlier “Newtonian” theory by the Poisson equation

for the potential and mass density µ.

An obvious contrast between the Einstein equations (1) and the Poisson equation (2) is that the mysterious notation of the former makes it far less obvious what they even mean. This has given the subject of general relativity a reputation for difficulty and impenetrability. However, this reputation is to some extent unwarranted. Both (1) and (2) represent the culmination of revolutionary theories whose formulations presuppose a complicated conceptual framework. For better or for worse, however, the structure necessary to formulate Poisson’s equation has been incorporated into our traditional mathematical notation and school education. As a result, ³, with its Cartesian coordinate system, and notions such as functions, partial derivatives, masses, forces, and so on, are familiar to people with a general mathematical background, while the conceptual structure of general relativity is much less so, both with respect to its basic physical notions and with respect to the mathematical objects that are needed to model them. However, once one comes to terms with these, the equations turn out to be more natural and, one might even dare say, simpler.

Thus, the first task of this article is to explain in more detail the conceptual structure of general relativity. Our aim will be to make it clear what the equations (1) actually denote, and, moreover, why they are in a certain sense the simplest equations one can write down, given the general framework of the theory. This in turn will require us to review special relativity and its implications for the structure of matter, which will bring us to the unified concept of stress–energy–momentum, described by a tensorial object T. Finally, we will join Einstein in his inspired leap to the notion of a general four-dimensional Lorentzian manifold (M, g) that represents our space-time continuum. We shall see that equation (1) expresses a relationship between the tensor T and the geometry of g as expressed in its so-called curvature.

There is more to truly understanding a theory than merely knowing how to write down its governing equations. General relativity is associated with some of the most spectacular predictions of twentieth-century physics: gravitational collapse, black holes, space-time singularities, the expansion of the universe. These phenomena (which were completely unknown in 1915 and thus played no role in the formulation of the equations (1)) revealed themselves only when the conceptual issues surrounding the problem of global dynamics of solutions were understood. This took a surprisingly long time, though the story is not as well-known as the heroic struggle to attain (1). The article will conclude with a very brief glimpse into the fascinating dynamics of the Einstein equations.

1 Special Relativity

1.1 Einstein, 1905

Einstein’s 1905 formulation of special relativity stipulated that all fundamental laws of physics should be invariant under Lorentz transformations of the frame of reference defined by x, y, z, and t. A Lorentz transformation is any composition of translations, rotations, and the Lorentz boost, which is given by the formulas

where c is a certain constant and |v| < c. Thus, Einstein’s stipulation was that if one changes coordinates by means of a Lorentz transformation, then the form of all fundamental equations will remain the same. This set of transformations had already been identified in the context of the study of the vacuum Maxwell equations for the electric field E and magnetic field B:

Indeed, the Lorentz transformations are precisely the transformations that keep the form of the above equations invariant if we also transform E and B appropriately. Their significance was emphasized by POINCARÉ [VI.61]. However, it was Einstein’s profound insight to elevate this invariance to the status of fundamental physical principle, despite its incompatibility with what we now usually call Galilean relativity, which corresponds to taking c → ∞ in (3). A surprising consequence of Lorentz invariance is that the notion of simultaneity is not absolute but depends on the observer: given two distinct events that occur at (t, x, y, z) and (t, x′, y′, z′), it is easy to find a Lorentz transformation such that the transformed events no longer have the same t-coordinate.

It follows from a celebrated result in partial differential equations known as the strong Huygens principle, applied to (4), that electromagnetic disturbances in vacuum propagate with speed c, which we thus identify as the speed of light. In view of Lorentz invariance, this statement is independent of the frame! A further postulate of the principle of relativity is that physical theories should not allow massive particles to move at speeds (as measured in any frame) greater than or equal to c.

1.2 Minkowski, 1908

Einstein’s understanding of special relativity was “algebraic.” It was MINKOWSKI [VI.64] who first understood its underlying geometric structure, namely, that the content of the principle was contained in the metric element

defined on ⁴ with coordinates (t, x, y, z). We call ⁴ endowed with the metric (5) Minkowski space-time and denote it ³⁺¹. Points of ³⁺¹ are referred to as events. The expression (5) is classical notation for the inner product defined on tangent vectors v = (c^-1v⁰, v¹,v², v³), w = (c^lw⁰, w^l, w², w³) on ⁴ by

The Lorentz transformations constitute precisely the symmetry group of the geometry defined by (5). Einstein’s principle of relativity could now be understood as the principle that the fundamental equations of physics must refer to space-time only through geometric quantities: that is, quantities that can be defined purely in terms of the metric. For example, from this point of view the reason that the notion of absolute simultaneity is not allowed is that it depends on a privileged hyperplane through any given point of M³⁺¹. But there are Lorentz transformations that preserve the metric and send this hyperplane to another one through the given point, so nothing in the metric can pick out one particular hyperplane. Note that if a physical theory makes use of geometric quantities only, then it is automatically invariant under Lorentz transformations: this observation renders many complicated calculations unnecessary.

Let us explore this geometric point of view further. Note that nonzero vectors v are naturally classified by the inner product ·, · 〉 into three types, called timelike, null, and spacelike, according to whether v,v〉 < 0, v, v〉 = 0, or v, v〉 > 0, respectively. Idealized point particles traverse curves through space-time; these are called the world lines of the corresponding particles. The postulate (referred to earlier) that speed in any frame of reference is bounded by the speed of light? can now be formulated as the following statement: if is the world line of a particle, then the vector d/ds must be timelike. (Null lines correspond to light rays in the geometric optics limit of (4).) This statement is independent of the parameter s of but for world lines we shall always assume that dt/ds > 0. To phrase this more geometrically, d/ds, (c^-1, 0, 0, 0)〉 < 0, which we interpret as the statement that is future-directed.

We can now define the “length” of the world line of a particle by

Classically, the above expression would have been written simply as

which explains the notation (5). We refer to the quantity c^-1L () as proper time. This is the time that is relevant in local physical processes; in particular, if you are the particle traversing the world line then c^-1L () is the time that you will feel.

The metric (5) contains three-dimensional Euclidean geometry

dx² + dy² + dz²,

restricted to t = 0, say. More interestingly, it also contains non-Euclidean geometry

when it is restricted to the hypersurface t = c^-1r = c^-1. It is hard to overestimate how revolutionary the notion was that the time of physical processes (including our very sensations) and the length of measuring rods are two interdependent aspects of a geometric structure that naturally lives on a four-dimensional space-time continuum. Indeed, even Einstein initially rejected Minkowski space-time, preferring to retain the independent reality of a definite “space,” albeit a space with a relative notion of simultaneity. Only as a result of his search for general relativity did he realize that this view is fundamentally untenable. We shall return to this in section 3.

2 Relativistic Dynamics and the Unification of Energy, Momentum, and Stress

Besides the space-time concept and its geometrization, the principle of relativity led to a profound rearrangement and unification of the fundamental concepts of dynamics: mass, energy, and momentum. Einstein’s celebrated relation between mass and energy in the rest frame, is the best-known expression of one aspect of this unification. This relation arises naturally when one attempts to generalize Newton’s second law m(dv/dt) = f to a relation between 4-vectors in Minkowski space.

General relativity has to be formulated in terms of fields rather than particles. As a first step toward understanding it, let us look at continuous media. Now, instead of particles we consider matter fields; the unification of dynamical concepts encompasses what is known as stress, and its complete expression is embodied by the so-called stress–energy–momentum tensor T. This tensor is fundamental to general relativity, so we have no choice but to familiarize ourselves with it. It will be the key to the form of the Einstein equations (1) as well as to the object on their right-hand side.

For each point q ³⁺¹, the stress–energy–momentum tensor field T gives us a map

defined by the formula

Here, T_β = T_β for each and β. By we mean the space of vectors at q. (In Minkowski coordinates, we often identify ⁴ with , but it will be important to distinguish between the two when considering arbitrary coordinates in section 3.2.) Bilinear maps of the form (9) are known as covariant 2-tensors.

If the only matter present is described by what is known as a perfect fluid, then the components of T are given by

where u is the 4-velocity, a timelike vector normalized such that u, u 〉 = -c², p is the mass–energy, p is the pressure, and where δ_ij = 1 if i = j, 0 if i j, and i and j range over 1, 2, 3. Greek indices will range over 0, 1, 2, 3. We identify T₀₀ with energy, T_0i with momentum, and T_ij with stress. These notions are clearly frame-dependent. Finally, observe that T (u, u) = pc². This is the field-theoretic version of the famous equation (8).

In general, T is derived from the totality of all the matter fields by constitutive functions that depend on the nature of the matter fields and their interactions. We need not worry here about such things. But, regardless of the nature of the matter fields involved, we always postulate that the following equations are satisfied:

Defining ∇⁰ = -∂₀, ∇ⁱ = ∂_i, and introducing the Einstein summation convention, under which summation is implicit when an index appears both upstairs and downstairs, we may rewrite this as

These equations are Lorentz invariant.

The above relations embody the conservation of stress–energy–momentum at a differential level. Integrating (10) between homologous hypersurfaces and applying the Minkowski-space version of the divergence theorem, one obtains global balance laws. If one assumes that T_αβ is compactly supported, then, integrating between t = t₁ and t = t₂, one obtains

With respect to the chosen Lorentz frame, the zeroth component of the above equation represents the conservation of total energy, while the remaining components represent conservation of total momentum.

In the case of a perfect fluid, if we close the system (10) by adjoining a conservation law for particle number

∇^α(nu_α) = 0

and postulate constitutive relations between ρ, p, particle number density n, and entropy per particle s, compatible with the laws of thermodynamics, then we arrive at the so-called relativistic Euler equations.

3 From Special to General Relativity

With the elements of special relativity at hand, together with their deep implications for the nature of energy, momentum, and stress, we can now pass to the formulation of general relativity.

3.1 The Equivalence Principle

Einstein understood as early as 1907 that the most profound aspect of the gravitational force could not be described within the relativity principle as he had formulated it in 1905. This aspect is what he called the equivalence principle.

The easiest setting in which to understand this principle is that of the “test particle” with velocity v(t) in a fixed gravitational field . In this case, we have that the classical gravitational force is given by f = -m∇, and we may rewrite Newton’s second law m(dv/dt) = f as

Notice that the mass m has dropped out! Thus, the gravitational field accelerates all objects at a given position in the same way. This explains the fact, recorded already in late antiquity by Ioannes Philoponus and popularized in Western Europe by Galileo, that the time it takes objects to fall from a given height is independent of their weight.

It was Einstein who first interpreted this property as a sort of covariance with respect to transformations to noninertial, that is to say accelerated, frames. For instance, in the case of a constant gravitational field, which corresponds to the case (z) = fz, we can pass to the accelerated frame

= z + ft²

and write (12) as

Similarly, one can reverse the argument to “simulate” a gravitational field when none is present by expressing (13) in an accelerated frame.

3.2 Vectors, Tensors, and Equations in General Coordinates

Exactly what the equivalence principle means in general is somewhat obscure and has been the subject of debate ever since Einstein introduced it. Nevertheless, the above considerations suggest that, even in the absence of gravity, it would be useful to know how various objects and equations appear when expressed in arbitrary coordinate systems. That is to say, let us change from our Minkowski coordinates x⁰, x¹, x², x³ to the most general coordinate system, which we shall write as = (x⁰, x¹, x², x³), where ranges over 0, 1, 2, 3.

Expressing scalar functions in arbitrary coordinates poses no problem. But what about vector fields? If v is a vector field expressed in Minkowski coordinates as (v⁰, v¹, v², v³), how do we express v in our new coordinates ?

One has to think a bit about what a vector field actually is. The correct point of view is to consider a vector field v as a first-order differential operator defined (using Einstein’s summation convention) by v(f) = v∂f . So we seek v such that v(f) = v∂f for all functions f . The chain rule then gives us our answer:

What about tensors, such as the stress–energy–momentum tensor T? In view of the definition (9), we seek such that

where the numbers u are the components of u with respect to the coordinates as we have just calculated them above. (Note that these components depend on the point q. This is why it is now essential to distinguish from ⁴.) Again, the chain rule gives us the answer:

Classically, we write

One can interpret the above as a shorthand notation for (15), but it also tells us how to compute from T_µv by formally applying the chain rule to d.

There is another covariant symmetric 2-tensor besides T that is relevant here. This is the Minkowski metric itself. Indeed, the classical form of the Minkowski metric (5) corresponds to the representation

η_µv dx^µ dx^v,

where the η_µv for Minkowski coordinates x^µ are given by η₀₀ = -1, η_0i = 0, η_ij = 1 if i = j, and η_ij = 0 if i ≠ j. To avoid the cumbersome notation 〈· , ·〉, let us refer to the Minkowski metric as η. Following the above, we may express η in general coordinates by

where η is computed by formal application of the chain rule.

It is clear that if one tries to transform an equation such as (10) into general coordinates, then the components of η and their derivatives will appear in the equations. Einstein (always thinking “algebraically”) was seeking laws of motion for both matter and the gravitational field that would have the same form in all coordinate systems. As he understood it, this meant that all objects that appear should transform as tensors and should be considered a priori “unknown.” He referred to this principle as “general covariance.” This suggests that η should be replaced by an unknown symmetric 2-tensor. Let us call this 2-tensor g. One can of course try to write down an equation for the “unknown” g that forces it to be the “known” Minkowski metric η. Thus, “general covariance” per se does not force one to abandon η. But in view of the fact that g and T have the same number of components, it was a natural step to consider g as the embodiment of the gravitational field and to try to look for an equation that related g and T directly. In this way, the framework of general relativity was born.

3.3 Lorentzian Geometry

The profound insight of replacing the fixed Minkowski η with a dynamic g brought Einstein to what we now call Lorentzian geometry. Lorentzian geometry generalizes Minkowski geometry following the blueprint of RIEMANN [VI.49]. That is, we replace the Minkowski metric η by a general map

In other words, we replace η by a symmetric covariant 2-tensor, which is expressed in arbitrary coordinates x^µ by

g_µv dx^µ dx^v.

Moreover, we require that at each point q the bilinear form g(· , ·) can be diagonalized to the Minkowski form (6). Loosely speaking, a Lorentzian metric is one that “looks locally like the Minkowski metric,” just as a RIEMANNIAN METRIC [I.3 §6.10] looks locally like the Euclidean metric.

Just as with the Minkowski metric, the bilinear form g permits us to classify nonzero vectors v_q at a point q as timelike, null, or spacelike and to define proper times of world lines γ(s) = (x⁰(s), x¹(s), x²(s), x³(s)) by the formula (7), but with 〈,〉 replaced by g_µv^µv. It is in this sense that we can speak of the geometry of g.

In view of Minkowski’s formulation of the special relativity principle as the statement that the equations of physics refer to space-time only through geometric quantities associated with the Minkowski metric, it is natural to look for a generalization of this principle, and indeed a suitable version immediately suggests itself. It is the principle that the equations of physics refer to the space-time coordinates only via geometric quantities naturally associated with g.

We saw earlier that the kinematic constraint on “test particles,” as formulated geometrically for the Minkowski metric, was that dγ/ds should be timelike; this makes sense for an arbitrary Lorentzian metric. But how does one formulate differential equations? For instance, how does one formulate an analogue of (10) that refers only to g?

It turned out that in the Riemannian case, a set of natural geometric concepts suitable for the task had already been developed in the nineteenth and early twentieth centuries by Riemann, Bianchi, Christoffel, Ricci, and Levi-Civita. These carry over directly to the Lorentzian case.

One begins by defining the so-called Christoffel symbols by

= g^λρ(∂_µg_ρv + ∂_vg_µρ - ∂_ρg_µv).

Here, the numbers g^µv are the components of the “inverse metric” of g: that is, they are the unique solution to the equation g^µv g_vλ = , where, as usual, = 1 if λ = µ and 0 otherwise. (It turns out that g^µv is very useful for the calculational gymnastics that are typical of tensor analysis when it exploits the Einstein summation convention.)

One can then define a differential operator ∇_µ called a connection, which acts on vector fields by

and on covariant 2-tensors by

The left-hand sides of (16) and (17) define tensors that can be expressed in any coordinate system by a formal application of the chain rule.

With the help of this differential operator, one could now write the analogue of equations (10) for an arbitrary metric g as

where ∇^µ = g^µv ∇_v refers to the connection associated with g.

If we consider a limit as the matter field becomes concentrated at a point, or rather as the stress–energy–momentum tensor T_µv is nonzero only on a world line, then this curve will be a geodesic of g: that is, a curve that locally maximizes the proper time defined by g. These are the analogues of straight timelike lines in Minkowski space. In this limit, the motion of the matter does not depend on the nature of the stress–energy–momentum tensor, but only on the geometry of the metric that defines geodesics. Thus, all objects fall in the same way. These considerations give a concrete realization to the equivalence principle in general relativity.

Finally, it is important to remark that for a general metric g, the identity (18) does not imply global conservation laws (11) for “total energy” and “total momentum.” Such laws hold only if g has symmetries. The fact that the fundamental conservation laws survive in general only at the infinitesimal level is an important insight into the nature of these principles in physics.

3.4 Curvature and the Einstein Equations

It remains, then, to give a set of equations for the metric g that relate it to T. In anticipation of a Newtonian limit, we expect these equations to be second order, and we expect them to implement “general covariance” in the simplest way possible: they should refer to no other structure but g itself and T.

Again, Riemannian geometry provides ready-made tensorial objects that are invariantly associated with g. One can define the Riemann curvature tensor

R_µvλρ dx^µ dx^v dx^λ dx^ρ

with components given by

One can also define the Ricci curvature

R_µv dx^µ dx^v,

a covariant symmetric 2-tensor with components given by

R_µv = g^λρ R_µvλρ,

and the scalar curvature

R = g^µv R_µv.

If g were the induced (Riemannian) metric on a 2-surface in ³, then R would just be twice the Gauss curvature K. The above expressions should be thought of as complicated tensorial generalizations of Gauss curvature to several dimensions.

The final piece of the puzzle for the formulation of the Einstein equations (1) is provided by the following constraint that Einstein demanded: whatever the equation relating the metric and the stress–energy–momentum tensor of matter, (18) (the infinitesimal conservation of stress–energy–momentum) should hold as a consequence. Now, it turns out that for any metric g, the so-called Bianchi identities imply that

It is thus natural to postulate a linear relation between T_µv and the tensor R_µv - g_µv R. The form

is then uniquely determined by the requirement that it should give the correct Newtonian limit when one makes the identifications

The form (1) corresponds to the usual units G = c = 1. Note that (1), when written out explicitly, is nonlinear in the metric components g_µv.

Einstein did not stop at the Newtonian limit. By considering geodesic motion in solutions of the linearized equations (20), Einstein was able to determine the correct value for the anomalous precession of the perihelion of Mercury, an effect that Newtonian theory was unable to explain. Since (20) had no adjustable parameters after determining the Newtonian limit, this was a genuine test of the theory. A few years later the gravitational “bending” of light was observed. This had been calculated theoretically in the context of the geometric optics approximation where light rays follow null geodesics in a fixed space-time background. Post-Newtonian predictions of (1) have now been verified by various solar system tests, confirming general relativity in this regime to a high degree of accuracy.

One special case of (20) is when we postulate that T_µv = 0. The equations then simplify to

These are known as the vacuum equations. The Minkowski metric (5) is a particular solution (but not the only one!).

The vacuum equations can be derived formally as the EULER–LAGRANGE EQUATIONS [III.94] corresponding to the so-called Hilbert Lagrangian:

(The expression dx⁰ dx¹ dx² dx³ denotes the natural volume form associated with g.) HILBERT [VI.63], who was following closely Einstein’s struggle to formulate a theory of gravity with a dynamic metric g, arrived at his Lagrangian (actually a more general version of the above yielding the coupled Einstein–Maxwell system) very shortly before Einstein obtained the general equations (20).

Many of the most interesting phenomena that come from the equations (20) are already present in the vacuum case (21). This is somewhat ironic, because it was the forms of T and (10) that dictated (20). Note, in contrast, that in the Newtonian theory (2), the “vacuum” equations µ = 0 and standard boundary conditions at infinity imply = 0. Thus, the Newtonian theory of the vacuum is trivial.

The part of the curvature tensor R_µvλρ that is not forced to vanish from (21) is known as the Weyl curvature. This curvature measures the “tidal” distortion of families of geodesics. Thus, the “local strength” of gravitational fields in vacuum regions is related in the Newtonian limit to the tidal forces on macroscopic test matter, not the norm of the gravitational force.

3.5 The Manifold Concept

We have been able to get this far without really addressing the question of where the metric g is defined. In passing from the Minkowski metric to a general g, Einstein did not originally have in mind replacing the domain ⁴. But it is clear in the Riemannian case from the theory of surfaces that the natural object for a metric to live on is not necessarily ² but a general surface. For instance, the metric dθ² + sinθ d² naturally lives on the sphere ². In saying this, we are to understand that one requires several coordinate systems of the type (θ, ) to cover all of ². The n-dimensional generalization of the object where Riemannian or Lorentzian metrics naturally live is a manifold [I.3 §6.9]. Manifolds are the structures obtained by consistently smoothly pasting together local coordinate systems.

Thus, general relativity allows the space-time continuum not to be ⁴ but instead to be a general manifold , which may very well be topologically inequivalent to ⁴, just as ² is inequivalent to ². We call the pair (, g) a Lorentzian manifold. Properly put, the unknown in the Einstein equations is not just g but the pair (, g).

It is interesting that this fundamental fact, namely that the topology of space-time is not a priori determined by the equations, arises almost as an afterthought. Moreover, it was a thought that took many years to be clarified.

3.6 Waves, Gauges, and Hyperbolicity

When written out explicitly in arbitrary coordinates (try it!), the Einstein equations do not appear to be of any usual type, such as elliptic (like THE POISSON EQUATION [IV.12 §1]), parabolic (like THE HEAT EQUATION [I.3 §5.4]), or hyperbolic (like THE WAVE EQUATION [I.3 §5.4]; see [IV.12 §2.5] for more about these different classes of PDEs). This is related to the fact that, given a solution, one can form a “new” solution by composing the old solution with a coordinate transformation. We can do this for new coordinate systems whose coordinate transformations differ from the identity only in a ball. This fact, known as the hole argument, confused Einstein and his mathematical collaborator Marcel Grossmann, who were thinking algebraically in terms of the form of the equations in coordinates, and temporarily led them to reject “general covariance.” The resulting backtracking delayed the final correct formulation of (1) by about two years. The geometric interpretation of the theory immediately suggests the resolution to the dilemma: such solutions are to be considered “the same” because they are the same from the point of view of all geometric measurements. In modern language, a solution to the Einstein vacuum equations (say) is an EQUIVALENCE CLASS [I.2 §2.3] of space-times (, g), where two space-times are equivalent if there exists a diffeomorphism between them such that in any open set the metric has the same coordinate form when one identifies local coordinates by .

It turns out that once these conceptual issues are overcome, the Einstein equations can be viewed as hyperbolic. The easiest way to do this is to impose a gauge: that is to say, a certain restriction on the coordinate system. Specifically, one requires the coordinate functions x^α to satisfy the wave equation _gx^α = 0, where the d’Alembertian operator is defined by the formula

Such coordinates always exist locally and they are traditionally called harmonic coordinates, although the term wave coordinates would perhaps be more appropriate. The Einstein equation can then be written as a system

_gg_µv = N_µv ({g_αβ}, {∂_γ g_αβ}),

where N_µv is a nonlinear expression that is quadratic in the ∂_γg_αβ. In view of the Lorentzian signature of the metric, the above system constitutes what is known as a second-order nonlinear (but quasilinear) hyperbolic system.

At this point, it is instructive to make a comparison with the Maxwell equations. Suppose we are given an electric field E and a magnetic field B defined on Minkowski space. A 4-potential is a vector field A such that E_i = -∂_iA₀ - c^-1 ∂_tA_i, and . (Here ₁₂₃ = 1, and _ijk is totally antisymmetric, i.e., it transforms to its negative under permutation of any two indices.) If one wishes to view A as the fundamental physical object, then one notices that if A is replaced by the field Ã, defined by the formula

à = A + (-c^-1∂_tψ, ∂₁ψ, ∂₂ψ, ∂₃ψ),

where ψ is an arbitrary function, then à is also a 4-potential for E and B. One can expect a determined equation for A only if one imposes further conditions on it: that is, if one “fixes the gauge.” (The terminology “gauge” is originally due to WEYL [VI.80].) In the so-called Lorentz gauge

∇^µ A_µ = 0,

the Maxwell equations can be written

from which the wave properties are completely manifest. The gauge-symmetric point of view lived on to later twentieth century glory: the Yang–Mills equations, which are a nonlinear generalization of the Maxwell equations with a similar gauge symmetry, are the central part of the so-called standard model for particle physics.

The hyperbolicity property of the Einstein equations has two important repercussions. The first is that there should exist gravitational waves. This was noted by Einstein at least as early as 1918, essentially as a result of a linearized version of the considerations in the above discussion. The second is that there is a WELL-POSED INITIAL VALUE PROBLEM [IV.12 §2.4] for the Einstein equations (1) with the domain-of-dependence property, when these are coupled with appropriate matter equations. In particular, this is true in the vacuum case (21). The proper conceptual framework to formulate the latter problem took a long time to get right, and was only completely understood through work of Choquet-Bruhat and Geroch in the 1950s and 1960s, based on the fundamental concept of global hyper-bolicity due to Leray. Well-posedness means that one could associate a unique solution (in the vacuum case, a Lorentzian 4-manifold (, g) satisfying (21)) with a suitable notion of initial data. Of course, “initial data” does not mean “data at time t = 0,” since the concept of t = 0 is not geometric. Instead, the data take the form of some Riemannian 3-manifold (Σ, ) with a symmetric covariant 2-tensor K. The triple (Σ, , K) has to satisfy the so-called Einstein constraint equations. But with this notion, the fundamental problem of general relativity, despite its revolutionary conceptual structure, is thoroughly classical: to determine the relation of the solution to initial data, that is to say, to determine the future from knowledge of the “present.” This is the problem of dynamics.

4 The Dynamics of General Relativity

In this final section we give a taste of our current mathematical understanding of the dynamics of the Einstein equations.

4.1 Stability of Minkowski Space and the Nonlinearity of Gravitational Radiation

In any physical theory in which one can formulate the problem of dynamics, the most basic question is the stability of the trivial solution. In other words, if we make a small change to the “initial conditions,” will the resulting change to the solution be small as well? In the case of general relativity, this is the question of stability of the Minkowski space-time ^{3 + 1}. This fundamental result was proven for the vacuum equations (21) in 1993 by Christodoulou and Klainerman.

The proof of the stability of Minkowski space made it possible to formulate the laws of gravitational radiation rigorously. Gravitational radiation is yet to be observed directly, but it has been inferred, originally by Hulse and Taylor, from the energy loss of a binary system. This work gave them the only Nobel prize (1993) directly associated with the Einstein equations! The blueprint for the mathematical formulation of the radiation problem is based on work of Bondi and later Penrose. One associates with the space-time (M,g) an ideal boundary “at infinity,” known as null infinity and denoted . Physically, the points of correspond to observers who are far away from the isolated self-gravitating system but who are receiving its signals. Gravitational radiation can be identified with certain tensors defined on from rescaled boundary limits of various geometric quantities. As Christodoulou was to discover, the laws of gravitational radiation are themselves nonlinear, and the nonlinearity is potentially relevant for observation.

4.2 Black Holes

Perhaps no prediction of general relativity is better known today than that of black holes.

The story of black holes begins with the so-called Schwarzschild metric:

The parameter m here is a positive constant. This is a solution of the vacuum Einstein equations (21) that was found in 1916. The original interpretation of (22) was that it modeled the gravitational field in a vacuum region outside a star. That is to say, (22) was considered only in some coordinate range r > R_0, for an R₀ > 2m, and the metric was matched at r = R₀ to a “static” interior metric satisfying the coupled Einstein-Euler system in the coordinate range r ≤ R₀. (This latter metric is again of the form (22), but with m = m(r) such that m → 0 as r → 0.)

From the theoretical point of view, a natural problem poses itself. Suppose we do away with the star altogether and try to consider (22) for all values of r. What happens then to the metric (22) at r = 2m? In the (r, t) coordinates, the metric element appears to be singular. But this turns out to be an illusion! By a simple change of coordinates, one can easily extend the metric regularly as a solution of (21) beyond r = 2m. That is, there exists a manifold M that contains both a region r > 2m and a region 0 < r < 2m, separated by a regular (null) hypersurface . The metric element (22) is valid everywhere except on , where it must be rewritten in regular coordinates.

It turns out that the hypersurface can be characterized by an exceptional global property: it defines the boundary of the region of space-time that can send signals to null infinity , or, in the physical interpretation, to distant observers. In general, the set of points that cannot send signals to null infinity is known as the black hole region of space-time. Thus, the region 0 < r < 2m is the black hole region of M, and is known as the event horizon.

These issues took a long time to be sorted out, partly because the language of global Lorentzian geometry was developed long after the original formulation of the Einstein equations. The global geometry of the extended space-time M was clarified by Synge in around 1950 and finally by Kruskal in 1960. The name “black hole” is due to the imaginative physicist John Wheeler. From their beginnings as a theoretical curiosity, black holes have become part of the accepted astro-physical explanation for a wide variety of phenomena, and in particular are thought to represent the end-state for the gravitational collapse of many stars.

4.3 Space-Time Singularities

A second natural problem poses itself in relation to the Schwarzschild metric (22), now considered in the region r < 2m of the extended space-time M: what happens at r = 0?

A computation reveals that as r → 0, the Kretchmann scalar R_µνλρR^µνλρ blows up. Since this expression is a geometric invariant, it follows that, unlike the situation at r = 2m, the space-time is not regularly extendable beyond 0. Moreover, timelike geodesics (freely falling observers in the test particle approximation) entering the black hole region reach r = 0 in finite proper time, so they are “incomplete” in the sense that they cannot be continued indefinitely. They thus “observe” the breakdown of the geometry of the space-time metric. Moreover, macroscopic observers approaching r = 0 are torn apart by the gravitational “tidal forces.”

In the early years of the subject, it was thought that this seemingly pathological behavior was connected to the high degree of symmetry of the Schwarzschild metric and that “generic” solutions would not exhibit such phenomena. That this is not the case was shown by Penrose’s celebrated incompleteness theorem of 1965. This states that solutions to the initial value problem for the Einstein equations coupled to appropriate matter will always contain such incomplete timelike or null geodesics if the initial data hypersurface is noncompact and contains what is known as a closed trapped surface. The Schwarzschild case may appear to suggest that such incomplete geodesics are associated with the curvature blowing up. However, the situation can in fact be very different, as is apparent in the celebrated Kerr solutions, a remarkable two-parameter family of solutions to the vacuum equations (21), discovered only in 1963, which are rotating versions of (22). In the Kerr solutions, incomplete timelike geodesics meet a so-called Cauchy horizon, a smooth boundary of the region of space-time that is uniquely determined by initial data.

The theorem of Penrose gives rise to two important conjectures. The first, known as weak cosmic censorship, says roughly that for generic physically plausible initial data for suitable Einstein-matter systems, geodesic incompleteness, if it occurs, is always confined to black hole regions. The second, strong cosmic censorship, says roughly that for generic admissible initial data, incompleteness of the solution is always associated with a local obstruction to extend-ability, such as the blow-up of curvature. The latter conjecture would ensure that the unique solution of the initial value problem is the only classical space-time that can arise from the data. That is to say, it would imply that classical determinism holds for the Einstein equations.

Both conjectures are false if we drop the assumption that the initial data are generic, and this is one reason for their difficulty. Indeed, Christodoulou has constructed spherically symmetric solutions of the coupled Einstein-scalar field system (arising from regular initial data) that are geodesically incomplete but do not contain black hole regions. Such space-times are said to contain naked singularities.

Naked singularities are easy to construct if one does not require that they arise from the collapse of regular initial data. An example is the Schwarzschild metric (22) for m < 0. This metric, however, does not admit a complete asymptotically flat Cauchy hypersurface. This fact is related to the celebrated positive energy theorem of Schoen and Yau.

4.4 Cosmology

The space-times (M ,g) discussed previously are all idealized representations of isolated systems. The “rest of the universe” is excised and replaced by an “asymptotically flat end”; far-away observers are placed at an ideal boundary “at infinity.” But what if we are more ambitious and consider our space-time (M ,g) as representing the whole universe? The study of this latter problem is known as cosmology.

Observations suggest that on very large scales the universe is approximately homogeneous and isotropic. This is sometimes known as the Copernican principle. Interestingly, one cannot solve the Poisson equation (2) with a constant and constant nonzero µ on ⁴. Thus, in Newtonian physics, cosmology never became a rational science.¹ General relativity, on the other hand, does admit homogeneous and isotropic solutions as well as their perturbations. Indeed, cosmological solutions of the Einstein equations were studied by Einstein himself, de Sitter, Friedmann, and Lemaitre in the early years of the subject.

When general relativity was formulated, the prevailing view was that the universe should be static. This led Einstein to add a term _µν to the left-hand side of his equations, fine-tuned so as to allow for such a solution. The constant is known as the cosmological constant. The expansion of the universe is now considered to be an observational fact, beginning with the fundamental discoveries of Hubble. Expanding universes can be modeled to a first approximation by so-called Friedmann-Lemaitre solutions to the Einstein-Euler system, with various values of . In the past direction, these solutions are singular: this singular behavior is often given the suggestive name “the big bang.”

4.5 Future Developments

The plethora of exact solutions of the Einstein equations gives us a taste of what the qualitative behavior of more general solutions may be. But a true qualitative understanding of the nature of general solutions has been achieved only in a neighborhood of the very simplest solutions. The question of the stability of the black hole solutions described above remains unanswered, as do the cosmic censorship conjectures and the nature of the singularities that occur generically in general relativity. Yet these questions are fundamental to the physical interpretation of the theory, and indeed to assessing its very validity.

How likely is it that these questions can ever be answered by rigorous mathematics? Problems concerning the singular behavior of nonlinear hyperbolic partial differential equations are notoriously difficult. The rich geometric structure of the Einstein equations appears at first as a formidable additional complication, but it may also turn out to be a blessing. One can only hope that the Einstein equations will continue to reveal beautiful mathematical structure that answers fundamental questions about our physical world.

Further Reading

Christodoulou, D. 1999. On the global initial value problem and the issue of singularities. Classical Quantum Gravity 16:A23–A35.

Hawking, S. W., and G. F. R. Ellis. 1973. The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics, number 1. Cambridge: Cambridge University Press.

Penrose, R. 1965. Gravitational collapse and space-time singularities. Physical Review Letters 14:57–59.

Rendall, A. 2008. Partial Differential Equations in General Relativity. Oxford: Oxford University Press.

Weyl, H. 1919. Raum, Zeit, Materie. Berlin: Springer. (Also published in English, in 1952, as Space, Time, Matter. New York: Dover.)

 

1. One can study “Newtonian cosmology” by modifying the foundations of the Newtonian theory so as to describe the theory with a nonmetric connection on, say, ³ × . But this step is of course inspired by general relativity (see section 3.5).