# 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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### Metric tensor times its inverse using Kronecker delta

From tensor calculus, we know that $$g^{\mu\nu}=\delta_{\lambda}^{\mu}\delta_{\phi}^{\nu}g^{\lambda\phi}.\tag{1}$$ Based on (1), is the following true? g^{\...
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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 ...
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### Determine tetrad of local inertial frame numerically

Let $g_{\mu\nu}$ be a metric of a curved spacetime. However, we do not know its analytical form. We only know $g_{\mu\nu}$ and its derivatives from a former numerical simulation and the underlying ...
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### Dirac equation more natural in $O(1,3)$ than $O(3,1)$?

In physics, the Lorentz group $O(1,3)$ is of central importance, being the setting for electromagnetism and special relativity. According to Wikipedia: Some texts use $O(3,1)$ for the Lorentz group; ...
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### Can we modify classical total energy equation to get desired results (Schwarzschild metric results) for planetary orbits in a Lagrangian analysis?

(A) In a three-dimensional flat space, total energy $H = T + U$ in a conservative field is constant. The three-dimensional Lagrangian ($L = T - U$) gives force on an object (see image- eq. (1)). $T$ ...
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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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### Nonlinear extension of Lorentz Group [duplicate]

The Lorentz group is defined to be the set of linear transformations that leave $ds^2 = -dt^2 + |d\vec{x}|^2$ invariant. The Poincaré group contains the Lorentz group, but now we allow transformations ...
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### The metric exterior of a massive object

The only condition apart from perfect spherical symmetry that is required for the retrieval of the Schwarzschild-metric $g_{ik}$ is actually ($R_{ik}$ being the Ricci tensor, the contraction of the ...