Common properties of all spacetime metrics

Question

I understand that spacetime (in GR) has many possible metrics, all of which are calculated via the Einstein Field Equations. The Minkowski metric (from SR) is one solution of the EFEs; it is a solution of the EFEs for a flat spacetime in a vacuum.

Assuming I got the above right, my question is: what unites all possible spacetime metrics and makes them different from other kinds of metrics (e.g., from the Euclidean metric)? Of course, here I have in mind non-trivial properties of the spacetime metrics.

Background: I am self-learning both the special and the general theory of relativity. Apologies if this question has an obvious answer.

Answer 1

From a mathematical point of view, all you need to describe a 3+1D spacetime solution to the Einstein equations is a pseudo-Riemannian metric with signature $(3, 1)$ or $(1, 3)$ (depending on your convention). This is because any Lorentzian manifold defines a spacetime by simply declaring the associated energy-momentum tensor to be the Einstein tensor divided by $\kappa$: $$ T_{\mu\nu}\equiv\underbrace{\frac{1}{\kappa}G_{\mu\nu}}_\text{geometric}\Rightarrow G_{\mu\nu}=\kappa T_{\mu\nu} $$

For a vacuum solution, the Einstein equations simplify greatly: $$ R_{\mu\nu}=\frac{\Lambda}{n-2}g_{\mu\nu}=kg_{\mu\nu} $$ i.e. the Ricci tensor must be proportional to the metric, defining the so-called "Einstein manifold". With no cosmological constant, this reduces to the well-studied Ricci-flat manifold.

For a given energy-momentum tensor $T_{\mu\nu}$, the uniting property of all spacetime metrics is, somewhat tautologically, that they must satisfy the Einstein field equations. $$ G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu} $$ where the left-hand side of the equation can be computed directly from the metric alone. For completely general $T_{\mu\nu}$, this places no constraints on the metric that are simpler than the Einstein field equations themselves - defining the space of metrics as the intersection of the solution space to 10 nonlinear, second-order partial differential equations. However, we can often use energy conditions or exploit the symmetry of the energy-momentum distribution to, for example, generate ansätze.

For physically realisable (and useful) scenarios however, you may want additional "nice" global spacetime features of various strengths, e.g.

• no closed chronal/causal curves, various other notions of causality
• global hyperbolicity to make the Cauchy problem well-posed
• no isometric embeddings
• asymptotic flatness
• geodesic incompleteness to allow singularities - and by extension, the Penrose-Hawking singularity theorems
• a plethora of other conditions that you will likely find in The Large Scale Structure of Space-Time by Hawking and Ellis.

These constraints amount to identifying a subset of the space of metrics satisfying the EFE. Unfortunately the imposition of most of these conditions is notoriously difficult to analyse analytically in terms of the components of the metric without heavily simplifying assumptions.

Answer 2

The central hypothesis of Relativity - both Special and General - is that at each point and time (at each space-time location), in each direction there is a finite non-zero invariant speed. Since this speed is associated with the speed of wave propagation in a vacuum - particularly of light - then it is usually referred to as "light speed".

Were you to depict this, in a space-time diagram, with the vertical direction being time-like and future-pointing, the locus of all trajectories that emanate from a given point and time at that speed would form an upwardly-directed cone having that space-time location as its vertex. That's its Future Light Cone. Similarly, the locus of all trajectories that arrive at that space-time location at that speed would form a downward-directed cone - the Past Light Cone.

Correspondingly, the geometry is enmeshed with a field of light cones, one pair for each place and time.

The field of light cones - by itself - is actually sufficient to specify the metric for the geometry ... up to a conformal transformation. So, it embodies the conformal part of the metric (i.e the class of metrics that is conformally equivalent to the given metric).

The metrics that correspond to this field will have a set of components that may be arrayed as a matrix. At each space-time location, the matrix for that metric may be reduced to a diagonal form either as (+1, +1, +1, -1), if the metric is treated as a metric for proper distances, or as (+1, -1, -1, -1), if it is treated as a metric for proper times.

Such a metric is said to have "Lorentzian signature" and a space-time which has - at every one of its locations - a Lorentzian signature is a Lorentzian manifold and is precisely what we actually mean by the term "space-time". For other dimensional geometries, the term is normally understood to apply to metrics whose signature is (+,-,-,...,-) or (+,+,...,+,-), where all but one of the dimensions is "space-like" and the other is "time-like".

It's worth pointing out that Hawking and Hartle dealt with signature-changing geometries, where the Lorentzian part of the geometry is connected to a locally 4D-Euclidean part, where the signature is (+,+,+,+). There are other people who work with signature-changing geometries, such as Mansouri. Technically, this violates the central axiom of Relativity, because now you only have a sub-space that is Lorentzian. The interface where it connects up with the non-Lorentzian part would be the boundary of the zone where, strictly speaking, the applicability of Relativity is confined to.

If the light cones can be mapped congruently onto themselves by spatial translations, then they are spatially homogeneous. If they can be mapped congruently onto themselves under time translations, then they are homogeneous in time. In that sense, you could say that the speed of light would then be constant throughout space and in time. If they can be mapped congruently onto themselves by reorientation of the spatial axes, then they are isotropic and you could say (in that sense) that light speed is direction-independent. Finally, if the structure of the light cones remains invariant under a set of transformations that uniformly change the moving state of an observer (these transforms are generically called "boosts", with the Lorentz transform and Galilei transform being two cases in point), then one could say that light speed is observer-independent. The boosts would all be Lorentz transforms.

A field of light cones that has all four set of symmetries is specified up to a conformal transformation by the Minkowski metric. That distinguishes Special Relativity from General Relativity.

So, both Special and General Relativity stipulate that the underlying geometry be Lorentzian ... or at the very least that a subspace of it that contains our world does. Special Relativity goes one step further in asserting that the field of light cones is constant and symmetric so that we may also assert (in this sense) that light speed is a constant. A set of global transformations exist that map the light-speed trajectories at any given space-time location to those of another, and which remain invariant under an arbitrary change in orientation and an arbitrary boost.

If you count the symmetries: 3 degrees of spatial translation, 1 degree of time translation, 3 set of axes for rotation, and 3 sets of directions for boosts, that's 10 in all. Since the underlying metric is Lorentzian, then the transformations together gives you a realization of the Poincaré group. So, the metric in Special Relativity has the Poincaré group as a transformation group, as do the corresponding field of light cones.

The Poincaré group, with conformal transforms added in, forms the Conformal Group, which contains 5 additional sets of transforms. I'm a little fuzzy on what the relation of the light cone field to this group is, though I'm pretty sure it has the full Conformal Group as its transform group, given the previous discussion.