Hot answers tagged general-relativity
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I prefer to use Killing vectors and conservation laws to solve stuff like this, so let's analyze the problem using Killing vectors, and see if the results agree with your Euler-Lagrange equations.
Notice that the metric is invariant under translations of $v$. The associated killing vector is $\partial_v$ which in turn gives the following conserved ...
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The gravitational field can indeed be assigned an energy. Unfortunately though whereas for, say, the EM field you can define an energy density at a point ($\bf{E}^2+\bf{B}^2$), for the gravitational field you can't do this. - Whichever way you define the energy in terms of the Christoffel symbols, you run into the problem that you can make them, and hence ...
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After the question was originally asked, the OP changed it to exclude the Big Bang. I don't understand the motivation for imagining that there would be a boundary anywhere else. The following answer addresses the question as originally asked.
First, we should recognize that any answer to this question is going to be model-dependent. The Big Bang is the only ...
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Spacetime (probably) does indeed have at least one boundary. Crazy Buddy mentioned three related questions in his comment, and reading these will help you understand why spacetime has a boundary in the past i.e. the Big Bang. This is a singularity and it is a boundary because you cannot follow geodesics back through it to earlier times.
If the universe were ...
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As noted above in comments, I'm not competely sure I understand the question. But anyway, I'll give it a shot.
The answer is model-dependent. The standard cosmological model at the moment is the Lambda-CDM model. This model has various parameters. Depending on these parameters, the spatial curvature can be positive, negative, or zero. Observation puts ...
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To rephrase the question slightly, you are asking for one of the Betti numbers of the (3+1)-dimensional manifold corresponding to one of the solutions of the Einstein field equations that corresponds to charged or rotating black hole.
The Betti numbers of a manifold are topological invariants that intuitively represent the number of non-contractible ...
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A more recent alternative to Deser's work is that of Gull, Doran, and Lasenby. Framed in the mathematical of geometric (real Clifford) algebra and its associated calculus, this framework presents gravity as a gauge field on a Minkowski spacetime. The formulation is clearly inspired by relativistic quantum mechanics and tetrad approaches, but it has some ...
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There's a great discussion of this sort of thing in the first few pages of a paper by Penrose. Basically, to get an integral conservation law, you need the divergence of a vector to be zero. The energy-momentum tensor satisfies a differential conservation law, of course. But there's no associated quantity that you can generally integrate over a volume on ...
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The FLRW metric can be static, this is the solution that Einstein concocted before Hubble observed the expansion of the universe. The only way that Einstein could make his equations static was by introducing the infamous cosmological constant $\Lambda$. The general FLRW metric has the form
$$
\text{d}s^2 = -c^2\text{d}t^2 + a(t)\left[\frac{\text{d}r^2}{1 - ...
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I suppose $f$ is just an arbitrary scalar function on the manifold. I'm not well-versed with the concept of Ricci flow, so I'll try to give a simple operational answer. I also don't understand what exactly you're looking for.
The Ricci scalar $R$ roughly represents the amount of energy stored in spacetime (as curvature). The dilaton is a scalar field which ...
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