Physical meaning of non-trivial solutions of vacuum Einstein's field equations

According to Einstein, the space-time is curved and the origin of the curvature is the presence of matter i.e. the presence of the energy-momentum tensor $T_{ab}$ in Einstein's field equations. If our universe were empty (i.e. $T_{ab}=0$ and the cosmological constant $\Lambda$ is setted to be $0$) then I would expect only the flat solution to the vacuum field equations $$R_{ab}=0$$ Surprisingly there are non flat (or non trivial) solutions to the above equations, for example the Schwarzschild solution. This conflicts with the fact that the matter curves the spacetime, so what is the origin of the curvature for these non trivial solutions? I understand that mathematically $R_{ab}=0$ (Ricci-flatness) doesn't imply that the metric is flat, i.e. non trivial solutions are formally admissible, but I don't understand how this is explained physically.

• Ricci flatness doesn't mean no curvature. The Riemann tensor can still describe a curved space even when the Ricci tensor is zero. Nov 21 '13 at 11:13
• Yes I know this, but I don't understand what is the physical origin of this curvature in absence of matter. I've explained this in my question. Nov 21 '13 at 11:19
• I think that the Schwarzschild solution is analogous to the 1/r potential of a point charge in electrostatics. In both cases, the solutions satisfy empty space versions their respective field equations (i.e., Einstein field eq. and Poisson eq.) except at the origin. Nov 21 '13 at 11:33
• Yes, but the point is that Schwarzschild solution implies a curvature of the spacetime, where does this curvature come from? Nov 21 '13 at 11:37
• @Galoisfan, surely you see that the Schwarzschild solution has a mass parameter and that the spacetime is flat when the that parameter is zero? Does that fact not illuminate the answer? Nov 21 '13 at 12:09

The Newtonian vacuum field equation $\nabla^2 \phi = \rho$ where $\phi$ is the gravitational potential and $\rho$ is proportional to mass density also has non-trivial vacuum solutions, for example $\phi = -1/r$ for $r$ outside some spherical surface. The Maxwell equations also have non-trivial solutions. In electrostatics, precisely the same as classical gravitation, in elctrodynamics, also radiative solutions of various kinds.

It is not strange that a field theory has non-trivial vacuum solutions. From a mathematical point of view, if it did not, it would not be possible to solve boundary value problems otherwise. Physically, a (local) field theory is supposed to provide a way for spatially separated matter to interact without spooky action at a distance. If interactions were unable to propagate through a region of vacuum we would have a very boring field theory!

If we want to be a little more specific to general relativity, let us note that this theory actually consists of two field equations. The most famous one is Einstein's, $$R_{\mu\nu} = 8\pi T_{\mu\nu}$$ which says that matter is the source for the field $\Gamma^{\mu}{}_{\nu\sigma}$ -- the Christoffel symbols. This equation alone does not contain the fundamental characterization of general relativity. It is just an equation for some field. For this field to actually correspond to the curvature it must also satisfy the Bianchi identity $$R_{\mu\nu[\sigma\tau;\rho]} = 0.$$

The Bianchi identity is redundant if the Christoffel symbols are defined the way they are in terms of the metric. This is actually analogous with electrodynamics (and for a very good reason, because ED is also a theory of curvature). The Maxwell equations are $$F^{\mu\nu}{}_{,\nu} = j^\mu$$ $$F_{[\mu\nu,\sigma]} = 0$$ and the first equation is the one that couples the electromagnetic field to matter. The second equation is redundant if $F_{\mu\nu}$ is defined in terms of the vector potential.

Now, the electromagnetic field has 6 components but as you can see only 4 of them really couple to matter directly. The second equation represents the freedom for the electromagnetic field to propagate in vacuum. (In fact if you do Fourier analysis to find radiation solutions to the Maxwell equations the first only tells you that radiation is transverse, and the second is the one that actually determines the radiation.) The components are naturally not independent since matter and radiation interact, but I think that this is a nice way to think about why of the classical Maxwell equations $$\begin{matrix} \nabla \cdot \mathbf{E} = & \sigma\\ \nabla \cdot \mathbf{B} = & 0 \\ \nabla \times \mathbf{E} = & -\frac{\partial \mathbf B}{\partial t} \\ \nabla \times \mathbf{B} = & \mathbf{j} + \frac{\partial \mathbf E}{\partial t} \end{matrix}$$ two involve only the fields and two involve matter.

Similarly for Einstein's general relativity, in the Einstein field equation $$R_{\mu\nu} = 8\pi T_{\mu\nu}$$ matter only couples to 10 components out of the 20 components in the Riemann curvature tensor. (The Riemann tensor is the physically observable quantity in general relativity.) The other 10 components are in the Weyl tensor. They are the part of the gravitational field that is present in vacuum, so they must include at least the Newtonian potential. By analogy with electrodynamics they also include gravitational radiation.

In the specific case of the Schwarschild and Kerr metrics, not only are all the components of the Ricci tensor 0, one can in fact arrange for all the components of the Weyl tensor except one to be 0 also. This is sort of analogous to how in electrostatics you can always choose the gauge so that the vector potential $\mathbf A = 0$. Perhaps you can think of this as saying that these metrics do not radiate, so only the part of the gravitational field whose limit is the Newtonian potential exists. (But there are radiating metrics with the same property, so maybe this isn't a good way to think.).

There are other vacuum metrics where fewer of the Weyl tensor's components can be made 0, or some gauge freedom remains. It is common to classify metrics along this scheme, which is called Petrov type. In a really famous paper Newman and Penrose show that the Petrov type of gravitational radiation has a near field - transition zone - radiation zone behavior, where more components of the Weyl tensor become irrelevant the further away from the source you go. (This is analogous with electrodynamics again, since in the radiation zone the EM field is transverse, but in the near field it is not.)

• Robin Ekman: You answer starts out with: "The Newtonian vacuum field equation $\nabla^2 \phi = \rho$ where $\phi$ is the gravitational potential and $\rho$ is proportional to mass density [...]" -- Is this equation as such not more correctly called "the Newtonian field equation"? (p.s. I noticed a typo in the first paragraph, and in the penultimate.) Mar 29 '14 at 7:42
• Yes, that's correct, the 'vacuum' should not be there. Mar 29 '14 at 8:08

I'm amazed that there are all of these answers here and no one has pointed out that a spacetime containing nothing but (caustic-free) gravitational radiation will be singularity-free and will be Ricci flat but not Riemann flat, which is a cleaner example than the Schwarzschild metric.

If you consider first the Schwarzschild metric describing the space outside some sphere on non-zero radius, the Ricci curvature is indeed zero outside the sphere and indeed the stress-energy tensor is also zero there.

The black hole is the limit of taking the radius of the sphere to zero. In that case both the Ricci curvature and stress-energy tensor are zero outside the singularity, but mass can still be present because the Ricci tensor is undefined at the singularity, which is the only place that the stress-energy tensor is non-zero.

• Morally speaking, in the Schwarzschild metric , the stress-energy tensor is a distribution... Nov 21 '13 at 11:40
• @Trimok: if you say so :-) I'm at the limit of my comfort zone here! Nov 21 '13 at 11:41
• Why is it a distribution?
– MBN
Nov 21 '13 at 12:47
• @MBN, see Trimok's comment to his answer. Nov 21 '13 at 13:51
• Nov 21 '13 at 15:48

In addition to the other answers here, I would like to add that your intuition would have been right on target in lower dimensions ($2+1$ and $1+1$). There, $T_{ab}=0$ and $\Lambda=0$ does imply (locally) flat space (although nontrivial topological effects are still possible).

In $3+1$ (and higher) dimensions the geometry is not uniquely defined by Ricci tensor. Here, we can separate from the curvature tensor the fully traceless part: Weyl tensor that can be thought of as describing purely gravitational dynamical degrees of freedom.

The existence of such degrees of freedom is exemplified in the phenomenon of gravitational radiation.

• So, these gravitational waves curve the four dimensional space time independently from the presence of matter in the region that we are considering. As a sort of backgrond curvature.... Nov 21 '13 at 15:02
• Not independently. The field equations are nonlinear, after all. So the 'matter' and 'space-time geometry' interact. Nov 21 '13 at 15:14
• In quantum language this means that a material system can absorb graviton increasing its energy -- we have conversion of purely gravitational entity into matter. Reversely, there is nonzero (although very small) probability that electron-positron pair annihilate into a pair of gravitons: we have conversion of matter into pure gravity. Nov 21 '13 at 15:21

A point is that the stress-energy tensor $T_{ab}$ is not conserved (see Wiki). This is because gravitation has its own stress-energy (pseudo) tensor $t_{ab}$, even if we may always choose a frame such as $t_{ab}=0$ for some particular space-time point. Only the sum of the two, (up to a factor $g$ if we take the Landau–Lifshitz pseudotensor) is conserved : $(-g (T^{ab} + t^{ab}_{LL})),_b=0$

If we look at the complicated expression of $t^{ab}_{LL}$, one sees that it involves quadratic quantities of first derivatives of the metric tensor $g_{ab}$ (and the metric tensor itself and its inverse)

The Riemann curvature $R_{abcd}$, by definition, depends on second derivatives, and quadratic quantities of first derivatives, of the metric tensor (and the metric tensor itself and its inverse).

So, if we consider a "vacuum" solution $R_{ab}=0$, so $T_{ab}=0$, but with $t^{ab}_{LL}$ not being zero and not being constant, for instance gravitational waves, this means that the first derivatives of the metric tensor are not zero, and there is no general reason, why the Riemann curvature $R_{abcd}$, made of first and second derivatives of the metric tensor, should be zero (except trivial cases).

[EDIT]

Rigourously speaking, one has to compute the freedom degrees for the curvature tensor $R_{abcd}$ and the Ricci tensor $R_{ab}$. It turns, that, in a space-time of dimension $\geq 4$, the freedom degrees for $R_{abcd}$ are greater than the freedom degrees for $R_{ab}$, so there is enough "place" for Ricci-flat space-time with non-null curvature.

• So the origin of the space-time curvature is not only due to the matter present in the spacetime... There is no matter, i.e. $T_{ab}=0$, (nearly) outside the surface of a planet, but in this region the space-time is curved because of the presence of the gravitational field of the planet. Is this a rough explaination? Nov 21 '13 at 12:18
• This is a different problem (see my comment on the answer of John Rennie). In my answer, I consider $T_{ab}=0$ everywhere. In the Schwarzschild metric, $T_{00}$ should be considered as a distribution (a delta-function) Nov 21 '13 at 12:24
• Good point, and hence the geon, though I don't think anyone has come up with a stable geon so far. Nov 21 '13 at 15:50

I noticed your question from a few months ago and I don't think you were given a very good answer to a very intelligent question.

Perhaps this may help. The Schwarzschild metric is a vacuum solution even though as applied to our solar system, for instance, we regard the sun as the physical source of curvature. The non-vanishing of the stress-energy tensor at points within the sun is not directly relevant to the Schwarzschild solution, since the stress-energy tensor at points within the sun only determines the curvature at those same points within the sun. Since Einstein's vacuum field law (vanishing of the Ricci tensor) is compatible with both flat and non-flat solutions, we need additional assumptions to account for the influence of the sun (that is, since the Ricci tensor vanishes in a vacuum whether or not the sun is out there). Among those assumptions in the Schwarzschild solution, for instance, are the varying of the metric with the variable r and the parameter m (physically designating the mass of the sun). So with a vacuum solution in which matter from elsewhere determines curvature, don't think of the stress-energy tensor from elsewhere as the source of curvature, since the stress-energy tensor only determines curvature locally. There is no "stress-energy tensor of the sun," after all, but only values of the stress-energy tensor as a function of mass-energy density at points within the sun. But there is the total mass of the sun, which is what is reflected in the Schwarzschild solution.

I hope that helps!

To add to the many good points already made, I think I agree with #Steve that what you are concerned about is the conundrum that an empty space seems to reflect the presence of matter. The key point is that we are dealing with a field theory, and the solution only describes the region of space outside of the mass, which is modelled as a single point as which the solution develops a singularity.

Remember that there may be many solutions to a homogenous differential equation, and one needs to fix boundary conditions to identify the physically correct one. So here the Schwarzschild solution is a solution to the homogeneous Einstein equations on a space omitting a point, and the mass is modelled as 'concentrated at a point'. The solution must then be chosen to agree with the boundary conditions, and so form a singularity at this point.

Just an illustration to add to the points already made, this is something which is inherent also in other physical models, such as classical gravity or electrostatics. Take the latter. Here a physically charged particle is modelled as a point, say sitting at the origin, $0\in \mathbb{R}^3$. To work out the electrical potential $V$, one loks for function that satisfy Laplaces equation $\Delta V=0$ on the region of space outside of this point, $\mathbb{R}^3 \setminus 0$. So here again although the solution is a vacuum solution, reflecting the fact that we model no charge away from the point, the solution must still conform to the boundary condition of a singularity at the origin (we expect a solution of the order $V \simeq 1/r$), and so the correct physical solution is not the trivial one $V=0$, which clearly also offers a solution to homogeneous Laplace!

A bit late for an answer, but I think I read your question differently to how others have read it. If I understand correctly, you are expecting a totally empty universe (and with lambda = 0) to be flat. I think this is completely accurate (perhaps with the exception of traveling gravitational waves, as mentioned already - if you allow such things in an 'empty' universe!). However, the vacuum solutions can also describe empty spacetime outside of a mass distribution. So the Schwarzschild solution describes spacetime outside of a spherically symmetric mass distribution. That is, it describes the part of the universe outside the mass. Such a universe is not empty, which is why it is not flat. The curvature is in fact due to the presence of the mass distribution, as you rightly expected it should be.

This mass does not appear in the stress energy tensor, because we are only considering the region exterior to the mass. I hope this helps.