Questions tagged [metric-tensor]
The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.
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Interpretation of degenerate metrics
I was studying the metric tensor and saw all about degenerate metrics. I would like what is the physical or geometrical intuition of a degenerate metric.
What is the meaning of $g(v,w) = 0$ for a ...
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Weyl transformation of induced metric
Consider the Weyl/conformal transformation in four dimenions
$$\tilde{g} \enspace = \enspace \Omega^2 g \quad \Longrightarrow \quad \sqrt{-|\tilde{g}|} \enspace = \enspace \Omega^4 \sqrt{-|g|}$$
The ...
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Physical intuition for the Minkowski space? [closed]
As the title suggests, I am looking for physical intuition to better understand the Minkowski metric.
My original motivation is trying to understand the necessity for distinguishing between co-variant ...
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Checking inverse metric and Christoffel symbols for the Kerr metric against references
I am trying to cross-check the Christoffel symbols and other very laborious geometric components in several metrics. In particular the Kerr metric is notoriously complex and results in expressions ...
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Saddle Shaped Universe
The universe, as described by FLRW metric, if $k = -1$ is clearly a 2 sheet 3-hyperboloid described by $x^2+y^2+z^2-w^2=-R^2$. So where does the more common saddle shaped picture of the open universe ...
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Embedding diagram of $\phi=\mathrm{constant}$ surface in cylindrically symmetric spacetime
I'm trying to generate an embedding diagram for the $\phi=\mathrm{constant}$ hypersurface in a cylindrically symmetric spacetime. I think I'm supposed to start with $$A(p,z)dp^2+A(p,z)dz^2=dw^2+dp^2+...
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Photonic black holes
"Can a photon turn into a black hole?" - usually the answer to this question is - it can't, because it has zero rest mass. However, when we derive the Schwarzchild Metric initially the $2M$ ...
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Confusion about contraction and covariant derivatives [closed]
Understanding Contraction and Second Covariant Derivatives in Tensors
I am confused about contraction in tensors and the second covariant derivative in tensors. Consider a tensor $T_{\mu\nu}$ and the ...
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Deriving Metric for Hypertorus Universe [closed]
The metric tensor for a hyperspherical universe (meaning on the surface of a hypersphere) looks like this:
$$ds^2=-dt^2+\frac{dr^2}{1-r^2}+r^2d\Omega^2$$
How can I derive a metric tensor describing a ...
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Homogeneous and Isotropic But not Maximally Symmetric Space
Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
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On the index orderings for Christoffel symbols [duplicate]
Update: The difference with this question is that it is a much narrower question than the much broader referenced question, and no answer was ever provided to this narrow question either here or on ...
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Stress-energy tensor in terms of the Lagrangian [closed]
In Dirac's "General Theory of Relativity" (Chap 30) he gets
$$T^{μν} = -\frac{2}{√} \frac{∂\mathscr{L}}{∂g_{μν}}$$
where $\mathscr{L}$ is the Lagrangian density and $√$ means $\sqrt{-g}$.
$\...
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Constant curvature on a sphere?
$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
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A few doubts regarding the geometry and representations of spacetime diagrams [closed]
I had a couple questions regarding the geometry of space-time diagrams, and I believe that this specific example in Hartle's book will help me understand.
However, I am unable to wrap my head around ...
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
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