# 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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### Interval preserving transformations are linear in special relativity

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
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### How does the Hubble parameter change with the age of the universe?

How does the Hubble parameter change with the age of the universe? This question was posted recently, and I had almost finished writing an answer when the question was deleted. Since it's a shame to ...
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### How do I derive the Lorentz contraction from the invariant interval?

While reviewing some basic special relativity, I stumbled upon this problem: From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using ...
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### Inverse metric in linearised gravity

From what I've read, in the framework of linearised gravity, one perturbs the metric around a Minkowski background, $\eta_{\mu\nu}$, such that $$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x)\tag{1}$$ ...
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### A helpful proof in contracting the Christoffel symbol?

Out of all of my time learning General relativity, this is the one identity that I cannot get around. $$\Gamma_{\alpha \beta}^{\alpha} = \partial_{\beta}\ln\sqrt{-g} \tag{1}$$ where $g$ is the ...
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### Can general relativity be completely described as a field in a flat space?

Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
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### How to prove that orthochronous Lorentz transformations $O^+(1,3)$ form a group?

Orthochronous Lorentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$) $$\Lambda^0{}_0 \geq +1.$$ How to prove they form a subgroup of ...
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### Schwarzschild metric in Isotropic coordinates

As one wants to jump to Isotropic coordinates in order to write the Schwarzschild metric in terms of them, one does this coordinate transformation: $$r=r'\left(1+\frac{M}{2r'}\right)^2$$ So we start ...
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### Finding the metric tensor from the Einstein field equation?

I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 ...
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### Linearized gravity and perturbation theory

This is a question regarding a calculation in perturbative GR. We have : $g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$ where $h_{\mu\nu}$ is a small perturbation around the flat spacetime metric. In ...
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### Can we exit the event horizon of merging black holes?

I have an intuitive scenario. Consider we have a spaceship just below the event horizon of a BH, which is merging with another black hole. Finally, the singularities merge and we have a single black ...
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### Vector product in a 4-dimensional Minkowski spacetime

I'm studying relativity and I lost track of interpretation along the mathematical formalism. What does vector product mean as an event? I mean, how must one interpret the result of the vector product ...
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### Are the non-standard one-way speed of light conventions just transformations of coordinates?

There are a lot of posts and confusion regarding the fact that different standards of simultaneity result in different one-way speeds of light (OWSOL) (that may be non-isotropic). Of course, the ...
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### Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
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### In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?

In Carroll's Appendix B, he says You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with ...
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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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### Covariant vs contravariant vectors

I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. It seems to me that any vector can be decomposed in either of those basis, thus one can have ...
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