# 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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### Renormalisation and the Fisher-Rao metric

The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be thought of as a flux in a space of possible Hamiltonians for a ...
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### Holonomy group of Schwarzschild spacetime, other interesting examples?

I'm teaching myself a little about holonomy groups in the context of general relativity. This paper by Hall and Lonie classifies a lot of the possibilities for simply connected spacetimes in 3+1 ...
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### Solving Maxwell equations on curved spacetime

I have difficulties to understand how to solve the Maxwell equations on curved spacetime. I want to solve the equations in the weak regime $g_{\mu\nu}=\eta_{\mu\nu}+h{\mu\nu},~ h_{\mu\nu}\ll 1$ ...
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### General relativity 2 particle problem with negative mass

I am looking at a problem of 2-particle system of which one has negative mass. I have situation described on Wiki under section "Runaway motion". Particulary, if we assume, that negative ...
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### In the context of condensed matter physics, what does it mean for time to have two dimensions?

In an online article that describes condensed matter physics for laypersons, the author describes various so-called "designer materials" that have exotic properties, including one in which ...
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### Can some components of metric be Finslerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
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### Pseudo-Riemannian 2D manifold (visualize time curvature)

My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
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In the Hamiltonian description of asymptotically flat spacetimes, the metric should deviate at infinity from the flat metric by terms of order 1/r $$g_{ij} = \delta_{ij} + \frac{\overline{h}_{ij}}{r} +... 5 votes 0 answers 130 views ### Is it possible to create a Nil geometry in real spacetime according to general relativity? (What metrics are possible in the real world?) Background I've heard that it is possible to construct a Penrose triangle in the 3D geometry Nil. And I wondered: Can we build a Penrose triangle in the real world if spacetime is appropriately ... 5 votes 0 answers 180 views ### Can one build Wilson lines in general relativity? This question has two parts: Firstly, I am curious if one can build Wilson lines as a 'parallel transport operator' in general relativity in direct analogy with what is done in gauge theory. For a ... • 1,115 5 votes 0 answers 115 views ### Can two points always be joined by timelike curve? I have asked this question on MSE but now I think it is better suited for the Physics Stack Exchange. Suppose p and q are connected by a causal (i.e. its tangent vectors have non-positive norm) ... • 181 5 votes 0 answers 248 views ### What is the physical motivation behind the mathematical definition of an inertial system? In this German Classical Mechanics lecture by Frederic Schuller, it is given that a Newtonian spacetime with an absolute inertial frame is one in which$$ \nabla_{v} G=0$$Where \nabla_v is the ... 5 votes 0 answers 376 views ### Spin-2 operators from metric fluctuations in the AdS/CFT This is a computational question. I am pretty sure that there is a simple explanation, and something obvious that I am missing but I cannot figure it out. I want to add that this is not meant to be a &... 5 votes 0 answers 301 views ### Linearised diffeomorphisms on an arbitrary gravitational background Part 2 This question is a follow on from my recent post here, in the sense that I will use the notation introduced there. In that post, I considered infinitesimal diffeomorphisms of a metric g_{\mu\nu} ... 5 votes 1 answer 257 views ### A question from cosmological perturbation theory We consider the following scalar perturbation on the FRW metric$$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$where \Phi, B, \Psi and E are ... • 3,691 5 votes 1 answer 405 views ### What is the geometry of light cones if space is curved/non-Euclidean? In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as ... • 251 5 votes 0 answers 763 views ### On the embedding of the Schwarzschild metric in six dimensions At every point of the 4-D space-time, it's metric, being a symmetric 2-tensor, has \frac{D(D+1)}{2}=10 independent components. From this we can subtract four degrees of freedom according to the four ... • 1,349 5 votes 0 answers 1k views ### Scalar Curvature of a Conformally Flat Metric Suppose that you have a metric g_{\mu\nu}=\phi^2\eta_{\mu\nu} for some function \phi. There is a standard formula for what the scalar curvature R looks like in terms of \phi, which is given by ... 5 votes 0 answers 304 views ### Question about derivation of tensor in Di Francesco's CFT This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in ... • 3,569 4 votes 0 answers 214 views ### Justifying the transverse-traceless gauge For weak gravitational fields, we can assume the metric is some perturbation of flat space: g_{ab} = \eta_{ab} + h_{ab}. Following Schutz's argument, you can incorporate a small coordinate ... • 151 4 votes 1 answer 112 views ### How to relate Riemannian and Lorentzian tetrad fields on the same manifold/spacetime? Consider Gibbons and Hawkings paper wherein a Riemannian metric \overset{\mathcal{R}}{g}_{\mu\nu} and everywhere well defined normalized line field l_{\mu} on spacetime M may be used to ... • 2,847 4 votes 0 answers 291 views ### How to derive connection Lie algebra valued one-form on the frame bundle if given the pulled back of it on the physical space? I am following this YouTube lecture by Schuller where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see ... • 489 4 votes 0 answers 99 views ### Electromagnetism in spacetime with 2-times split-signature (+,+,-,-) metric Maxwell's equations can easily be generalized to any (m,n)-spacetime. Is there any material analyzing what such a theory will look like? Note that a particle still moves in spacetime forming a line. ... • 487 4 votes 0 answers 179 views ### Why is AdS spacetime like a "saddle"? When the shape of the universe is discussed, the three cases are flat, closed and open. Where AdS spacetime with a negative cosmological constant describes the open spacetime, as in the middle in the ... • 563 4 votes 0 answers 143 views ### Linearity of Lorentz Transformations from Principle of Relativity Many derivations of the Lorentz transformations assume they must be linear maps on \mathbb R^4, where we identify the components of \mathbb R^4 with orthogonal coordinate systems associated to ... • 3,407 4 votes 0 answers 63 views ### Is there a coordinate system for the Schwarzchild spacetime where r is proper length/time? I don't see any when I search online. The way I would try to construct them would be as follows. Starting from the standard metric$$ds^2 = -\left(1 - \frac{r_s}{r'}\right)dt^2 + \left(1-\frac{r_s}{...
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Consider some spacetime $\big(\mathcal{M},g_{\mu\nu}\big)$ parameterised by local coordinates $x^{\mu}$ ($\mathcal{M}$ is a smooth differentiable manifold equipped with a Lorentzian metric $g_{\mu\nu}$...