# Is space in general relativity defined even when devoid of any energy?

General relativity is said to be background independent. Can we nevertheless think of a completely empty space (devoid of any energy) as if it is furnished with an elastic three-dimensional fabric, a cubic grid the cubes of which have the same size everywhere as long as it remains empty and that it is the insertion of matter, of localizable energy which distorts this fabric locally, which causes spacetime to curve in its neighborhood?

(Though one might say that as empty space is filled with a uniform vacuum energy (the calculated density of which is 120 orders of magnitude larger than is inferred from observations), there exists no empty space; as Einstein wasn’t aware of the existence of such energy at the time he formulated his theory -though he proposed its existence -the cosmological constant- he added it as an afterthought, not as something essential to general relativity- the question remains the same -and the Wikipedia lemma ‘hole argument’ also isn’t very clear on this).

• Like vacuum solutions? – Kyle Kanos May 2 '18 at 10:04
• I'm not sure why this has close votes. As the answers show it's a reasonable question with a non-trivial answer. – StephenG May 2 '18 at 16:48

In general relativity we assume that a four dimensional manifold exists, and it is the solution to Einstein's equation that gives us the geometry of the manifold. What we call spacetime is the combination of this manifold and the metric.

(Incidentally there is a very nice discussion of what we mean by a manifold in the answers to What is a manifold?)

If we have some matter/energy distribution described by the stress-energy tensor $\mathbf T$ then we get the metric $\mathbf g$ by solving Einstein's equation:

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

And if no matter or energy is present this simplifies to the vacuum equation:

$$R_{\mu\nu} - \tfrac{1}{2} R g_{\mu\nu} = 0$$

And this has a number of solutions generically described as vacuum solutions. Even though the matter-energy is zero everywhere, these solutions can still have an energy called the ADM energy. For example the Schwarzschild black hole is actually a vacuum solution and the mass term $M$ in the metric is actually the ADM mass. The vacuum solution with zero ADM energy is just flat spacetime i.e. the Minkowski metric.

To summarise: if we require that the stress-energy tensor be zero everywhere, and we require that the ADM energy be zero, we still get a perfectly good spacetime i.e. Minkowski spacetime. The only assumption we have made is that the manifold exists, but this assumption underlies all of GR. So the answer to your question is that yes spacetime exists even in the absence of matter and energy.

• I think the Schwarzschild solution is a misleading example of a vacuum solution exactly because is has nonvanishing mass (and a singularity). Wouldn't gravitational waves be a better example? – ungerade May 3 '18 at 22:28

In addition to what @John Rennie has already explained one can investigate "devoid of any energy" in the context of the FRW-model. Setting the dimensionless densities $\Omega_i=0$ one obtains the solution $a(t)=H_0t$, which means that comoving objects are moving away from each other with constant velocity (it follows already from the second Friedmann equation that the acceleration is zero in this case). Interestingly the "empty FRW-universe" is equivalent by coordinate transformation (see Chapter 4 in the thesis of Tamara Davis) with the Milne universe which is expanding Minkowski spacetime.

• The Milne metric is just the Minkowski metric in accelerating coordinates. If you calculate the Riemann curvature tensor for the Milne metric you'll find it is zero. – John Rennie May 2 '18 at 8:31
• I'm not sure how to calculate it from the metric. It is clear that if there is no stress energy then the Riemann curvature tensor is zero. – timm May 2 '18 at 9:50
• Not true! For the Schwarzschild metric the stress-energy tensor is everywhere zero but the Riemann tensor is not zero. – John Rennie May 2 '18 at 10:08
• Yes, the Schwarzschild spacetime is curved. Mentioning "no stress energy" I should have added that I refer to the empty FRW-universe respectiveley to the flat Milne universe. – timm May 2 '18 at 13:53
• @John, @ timm: this (and your reference to 'What is a manifold?) is helpful, thanks! I don't know yet if this answers my question, but it gives me something to think about. – Anton May 2 '18 at 23:39