# 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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### Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...
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### Application of Kasner metric

Kasner metric is given by: $$ds^2=-dt^2+\frac{t}{t_0}e^{2K_1}dx^2+\frac{t}{t_0}e^{2K_2}dy^2++\frac{t}{t_0}e^{2K_3}dz^2$$ where $K_i$ are constants How is this metric useful in exploring manifolds with ...
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### Killing Vectors from Killing Equations

I need to find the killing vectors of the FLRW metric. However, it seems that they are complicated. Is there a simple/general equation that gives the killing vectors for a given metric? Or do I have ...
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### Would something in the center of a supermassive shell be pulled apart or remain stationary?

Imagine a supermassive hollow shell in space, and also imagine there is an object at the center of this shell. How does the force of gravity affect the body inside the shell? My reasoning is that the ...
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### Gauge freedom when solving Einstein field equation

In Weinberg's Gravitation and Cosmology, Ch7, the Einstein field equations give us 6 independent equations (totally 10 equations but $∇^\mu G_ {\mu \nu} =0$ gives 4 constraints) while $g_{\mu \nu}$ ...
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### Metric tensor times its inverse using Kronecker delta

From tensor calculus, we know that \begin{equation} g^{\mu\nu}=\delta_{\lambda}^{\mu}\delta_{\phi}^{\nu}g^{\lambda\phi}.\tag{1} \end{equation} Based on (1), is the following true? \begin{equation} g^{\...
1answer
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### Are all maximally symmetric spacetimes conformally flat? What about the converse?

If I'm not mistaken, one of the properties of maximally symmetric spacetimes is that the Riemann tensor can be written as $R_{abcd} = \frac{R}{d(d-1)}(g_{ac}g_{bd} - g_{ad}g_{bc})$, which would imply ...
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### What is the determinant of the induced metric $h_{ab}$ of the NG action?

In the introductory section of Polchinski String Theory: An introduction to the bosonic string we are given the induced metric which reads $$h_{ab} = \partial_a X^{\mu}\partial_b X_{\mu}\tag{1.2.8}$$...
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### Is the space and time dimensions in Schwarzschild metric orthogonal?

Schwarzschild metric does not contain any cross-terms such as $dtd\phi$, does that mean that space and time coordinates are orthogonal to each other? Should the dot product of any time vector with ...
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### Are isometries really global symmetries?

On one hand, a spacetime $(M,g)$ with the Killing vector $\xi^\mu$ and $x^\mu(\tau)$ a geodesic, we can construct the quantity $$Q = \xi_\mu \frac{dx^\mu}{d\tau}\tag{4.32}$$ that is constant along the ...
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### Event horizon from a metric in Cartesian coordinates

How can we derive the event horizon from a metric dependant on $(t,x,y,z)$? I've seen the Schwarzschild solution and the Kerr solution, but both of these are given in $(t,r,θ,φ)$. I tried converting ...
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### How are these two different definitions of covariant vector related?

Definition 1 If under a coordinate transformation $x^i\to \bar{x}^i(x^i)$ certain objects $A^i$ transform as $$A^i\to \bar{A}^{i}=\sum_{j}\frac{\partial \bar{x}^{i}}{\partial x^j}A^j,$$ those objects ...
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### What can be derived from the metric tensor? [closed]

I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl ...