# Lorentzian Manifolds in GR

Gold Member
From the things I've studied till now, the thought came into my mind that all of the known solutions of Einstein's Field Equations are Lorentzian. Is it correct?
And if it is correct, is there something in EFEs that implies all solutions to it should be Lorentzian?
And the last question, do more general pseudo-Riemannian geometries allow us to define proper causal structure? Are there examples of such theories?
Thanks

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Ben Niehoff
Gold Member
From the things I've studied till now, the thought came into my mind that all of the known solutions of Einstein's Field Equations are Lorentzian. Is it correct?

LOL, no. Just all the solutions you know are Lorentzian.

In most Lorentzian solutions, you can analytically continue the time coordinate to obtain a Riemannian solution. All of your favorite black holes allow this; in fact, the Hawking temperature can be obtained as the inverse periodicity of Wick-rotated time.

Beyond that, however, there are many more Riemannian solutions than there are Lorentzian ones. The reason for this is that while nearly any Lorentzian solution can be Wick-rotated into a reasonable Riemannian solution, the converse is not always true. For example, the Israel-Wilson family of Einstein-Maxwell solutions do not have physically reasonable interpretations as Lorentzian metrics, because some of the parameters have to be imaginary. But as Riemannian metrics, they are perfectly cromulent.

Riemannian-signature metrics are important to physics when one discusses the path-integral quantization of GR. By analogy with Yang-Mills theory, the dominant saddle points of the path integral are instantons, which are self-dual field configurations that extremize the Euclidean action. In 4 dimensions, "self-dual" only makes sense in Euclidean signature (in Lorentzian signature, one needs a factor of ##i##). A gravitational instanton is a 4-dimensional Riemannian manifold whose curvature 2-form is self-dual.

And if it is correct, is there something in EFEs that implies all solutions to it should be Lorentzian?

The signature of the metric is completely independent from the EFEs. It's just that physicists are interested in Lorentzian metrics because physics in the real world is locally Lorentz-invariant.

The EFE's in Riemannian signature are quite important in pure math. Ricci-flat manifolds tend to have a lot of nice algebraic properties, such as being hyper-Kaehler or having other special holonomies (a Calabi-Yau manifold, for example, is a complex, Kaehler manifold that is Ricci-flat).

An "Einstein manifold" in pure math is a (usually Riemannian) manifold that satisfies the Einstein equations with cosmological constant. Or, more simply stated,

$$R_{\mu\nu} = \lambda \, g_{\mu\nu}$$
for some constant ##\lambda##. Some simple examples of Einstein manifolds are spheres, projective spaces (both real and complex), etc.

And the last question, do more general pseudo-Riemannian geometries allow us to define proper causal structure? Are there examples of such theories?
Thanks

What do you mean by "more general"? Pseudo-Riemannian geometries that fail to satisfy the Einstein equations? I don't think they pose any problem to causality. Or do you mean pseudo-Riemannian geometries with more than one time direction? Those can be a problem.

Gold Member
So even if a manifold is not Lorentzian, we're still able to reduce its metric at a point to a Minkowski metric with zero first derivatives, right? Because otherwise, it seems to me, it won't satisfy equivalence principle. Can you give me some reference where it is actually done for non-Lorentzian metrics?
What do you mean by "more general"? Pseudo-Riemannian geometries that fail to satisfy the Einstein equations? I don't think they pose any problem to causality. Or do you mean pseudo-Riemannian geometries with more than one time direction? Those can be a problem.

EDIT: But wait, if we consider a Riemannian metric, then how can we model the motion of light? Or causal relationships? Because light traverses null geodesics and events with possible causal relationship have time-like separations. But in a Riemannian manifold, we can't have zero or negative length for spacetime intervals. Right? Can you give some reference on this too?

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Ben Niehoff
Gold Member
So even if a manifold is not Lorentzian, we're still able to reduce its metric at a point to a Minkowski metric with zero first derivatives, right?

No. A Riemannian metric has positive signature; in Riemann normal coordinates, it looks like the flat Euclidean metric at a point, with zero first derivatives.

Because otherwise, it seems to me, it won't satisfy equivalence principle.

The equivalence principle is kind of vacuous. It just means a manifold is locally modelled by flat space. It rules out strange things like Finsler geometry. In the Riemannian case, this means flat Euclidean space, not Minkowski space. I'm not sure why you would expect that.

Maybe you can clarify your question, then.

EDIT: But wait, if we consider a Riemannian metric, then how can we model the motion of light? Or causal relationships? Because light traverses null geodesics and events with possible causal relationship have time-like separations. But in a Riemannian manifold, we can't have zero or negative length for spacetime intervals. Right? Can you give some reference on this too?

Riemannian manifolds are not spacetimes, they are just spaces. So they don't model things like causality or physical motion. They're just curved spaces, like a sphere. The metric tensor measures local distances and angles; there are no timelike directions.

Gold Member
Riemannian manifolds are not spacetimes, they are just spaces. So they don't model things like causality or physical motion. They're just curved spaces, like a sphere. The metric tensor measures local distances and angles; there are no timelike directions.
This is really confusing. So there is no time in such manifolds and this means they can't give us what we want. Because our universe has a time(whatever it means!),light propagates in it and there are causal relationship between events. How can we use something that supports none of these, as a model of our universe? Are you saying that EFEs have solutions that physicists don't use and we should get rid off or interpret them somehow? Or you mean there are actually physical situations that a region of spacetime is only space and there is no time in such regions? Or you're just talking about a mathematical trick?

The equivalence principle is kind of vacuous. It just means a manifold is locally modelled by flat space. It rules out strange things like Finsler geometry. In the Riemannian case, this means flat Euclidean space, not Minkowski space. I'm not sure why you would expect that.
But that's...weird! We know that spacetime in the absence of any significant gravitating mass, is a 4-dimensional Minkowskian manifold and the 3-dimensional Euclidean manifold with a time parameter is only an approximation for ## v<<c ##.
So if we say spacetime can be Euclidean locally(even for velocities near that of light), then this itself is a way of finding out we're in a gravitational field because with no gravitational field, we can be only in a Minkowskian spacetime!
Actually, I never thought equivalence principle as saying spacetime should be locally flat, I always think about it as saying no local experiment can distinguish between being stationary in a gravitational field or being in accelerated motion.(So I think the locally-flat description to be the mathematical manifestation of the physical statement I mentioned) But it seems to me that having a Riemannian space as the universe(now whatever its interpretation is) means locally we can do experiments to see whether we're in a Minkowskian spacetime or in a Euclidean space with a time parameter and this will tell us whether we're in a gravitational field or not and this violates equivalence principle.

Maybe you can clarify your question, then.
The above two paragraphs are actually the clarifications of that question.

P.S.
I just want to make clear I'm not arguing with you, I'm explaining what I think!

Ben Niehoff
Gold Member
I'm equally confused at your confusion. You seem to have a very fundamental misunderstanding. Lorentzian differential geometry is a generalization of Riemannian differential geometry, not the other way around. Whoever taught you GR seems to have given you a very wrong impression.

Differential Riemannian geometry is the study of differentiable manifolds with metrics of positive signature. These are curved spaces of arbitrary dimension. They are not intended to model physical reality in any way; they are just a subject of research in pure math.

If you generalize Riemannian geometry by allowing metrics of indefinite signature, then it turns out that you can model physical reality with such metrics. The reason for this is that physical reality is (locally) Lorentz-invariant.

The metric signature is something you put into the EFEs; not something that comes out of them. You can postulate a metric of (++++) signature, or a metric of (-+++) signature, or even a metric of (---++++) signature if you feel like it. The equations don't care.

If you want to talk about physics in our universe, it is appropriate to choose 4 dimensions and (-+++) signature.

What I am telling you, is that pure mathematicians are also interested in applying the EFEs to metrics of (++++...) signature, or of other weird signatures, because it is interesting mathematically.

Gold Member
I know that the Lorentzian manifolds are a generalization of Riemannian manifolds and a subset of pseudo-Riemannian manifolds. My confusion came from the wrong impression that every solution of EFEs should describe physical reality. With removing this assumption, everything is now clear. EFEs can have solutions with any signature, but just the Lorentzian ones describe physical reality. Because Einstein tensor is important from a pure mathematical perspective too and is not something that is singled out only physically.

P.S.
I think one part of the above explanation is wrong. Lorentzian manifolds are special cases of pseudo-Riemannian manifolds but they aren't generalizations of Riemannian manifolds, because you can do nothing to get a positive-definite signature metric from a metric with the signature (1,n-1)!(The n-1 dimensional subspace of a Lorentzian manifold which results from ignoring the dimension with the negative sign in the metric, is a Riemannian manifold, but does it mean Lorentzian manifolds are generalizations of Riemannian manifolds? I'm not sure!)
So I think Riemannian and Lorentzian manifolds are disjoint subsets of pseudo-Riemannian manifolds.

Then is it correct to say that one of the postulates of GR should be that the spacetime is a Lorentzian manifold? Because EFEs alone may give unphysical solutions like Riemannian spaces!

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Ben Niehoff