# 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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### Why are there no covariant particles?

Why is physics described by scalar, spinor, vector, and tensor bundles, but not convector bundles or tensors of other ranks? In tensor bundle notation, these are respectively $T_0^0, S, T_0^1, T_0^2$. ...
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### Cosmology - Confusion About Visualising the Universe as the Surface of a 3-Sphere

Consider the FRW metric for the Universe in the form found in many standard cosmology textbooks: $$ds^2 = -dt^2 + a(t)^2\left(\frac{dr^2}{1-Kr^2}+r^2(d\theta^2 + \sin^2\theta d\phi^2)\right)$$ I am ...
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### Linearised diffeomorphisms on 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 $g_{\mu\nu}$ ...
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### Linearised diffeomorphisms on arbitrary gravitational background Part 1

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 $g_{\mu\nu}$...
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### Modulus of four acceleration

The four acceleration is defined as $$\alpha^\mu = \gamma_V ^4 \left(\frac{\vec{v} \cdot \vec{a}}{c},\frac{\vec{v} \cdot \vec{a}}{c^2} \vec{v} + \frac{1}{\gamma_V ^2} \vec{a} \right)$$ where $\vec{v}$ ...
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### Killing equation in coordinates

In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$ I have read that the key observation, to write the equation in ...
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### $SU(2)$ infinitesimal transformation of pion triplet lagrangian raising and lowering indices

How can I show that $$\epsilon^{ijk} (\partial^\mu\pi^i)\pi^j\partial_\mu\alpha^k(x)=\epsilon^{ijk}(\partial_\mu\pi^i)\pi^j\partial^\mu\alpha^k(x)~?$$ can I do the following? $\pi$ is the pion triplet ...
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### Does the invariance of interval hold only for events that occur in inertial reference frames?

For example, if we have two frames that have constant relative velocity, would the interval be the same between any two events? Or should we impose the condition that the particle must not accelerate ...
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### Spacetime translation in QFT

I have a question about the field under the spacetime translation. For example, in page 26 of Peskin's textbook, they give the translation properties of the field. So consider the space translation, ...
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### A problem with parallelism of tangent vectors

In 2D-space with metric $ds^2=du^2+2\lambda dudv+ dv^2$, where $\lambda= \lambda (u,v)$, how can we show that the tangent vectors to the curves $u$ = constant form a field of parallel vectors along ...