# 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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466 views

### 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 ...
383 views

### 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 ... 3k views

### How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
526 views

### 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 $g_{\mu\nu}=\eta_{\mu\nu}+h{\mu\nu},~ h_{\mu\nu}\ll 1$ ... 231 views

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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 ... 301 views

### 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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### 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 ...
143 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 ...
106 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) ...
204 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 ...
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### 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 &... 720 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 ...
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### 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 ... 296 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 ...
257 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 ...
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### 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. ...
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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 ...
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### 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 ...
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### Angle-preserving linear transformations in 2D space for relativity

I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the $ct$ axis and the worldline of an ...
97 views

### Does every curved spacetime have non-commuting generators of translations?

If we define the generators of translations in a general spacetime to be $P_\mu$, is it true that in every curved spacetime we have $[P_\mu,P_\nu]\neq0$? Is it also true that for every spacetime where ...
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### Solving scalar quantum field in 1+1D Milne space

So our line element is \begin{equation} ds^2=dt^2-a^2t^2dx^2 \end{equation} doing following coordinate transformation \begin{equation} y^0=t\hspace{2pt}\cosh ax, \hspace{2pt}y^1=t\hspace{2pt}\sinh ...
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### Quantum corrections to metric on non-linear sigma model target space

I am trying to make sense of what physicists mean when they talk of quantum corrections to the metric on the target spaces of nonlinear sigma models, for example [GHL99]. First some quick notation. ...
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### How does the Equivalence Principle imply that derivatives of the metric vanish in a freely falling frame?

Why do the first derivatives of $g_{\mu\nu}$ vanish in a freely falling coordinate system? I would like to start from the Equivalence Principle that for any point in spacetime there exists a locally ...
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### Robertson-Walker metric and cosmic homogeneity

The Robertson-Walker metric is of the form $$\tag{1} ds^2 = dt^2 - a(t)^2 \Big(\frac{dr^2}{1 - kr^2} + r^2 d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \Big).$$ My question is related to the $a^2(t)$ ...
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### Role of the observer in Gödel's universe

I am here to clarify myself about the role of the observer in Gödel's solution (1949) of Einstein's field equations. The Universe we are dealing with is anisotropic, since the axis of rotation ...
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### Connection between contra-/covariant vectors in SR and complex numbers?

If we take a spacetime with one spatial dimension, we can write a vector as $A^\mu=(t, x)$. This is a contravariant vector, and we can calculate the covariant vector by multiplying it with the ...
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### 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 ...
131 views