All Questions
9 questions
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
- 938
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347
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Isomorphism of the tangent space and the space of directional derivatives [closed]
I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
- 713
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103
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Conceptual confusion about the formula for parallel transport
I am examining the covariant derivative of a vector according to the formula $$\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ and also operating under the ...
- 1,397
1
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2
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143
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Gradient of scalar field
On page183 of Rayd'inverno "An introduction to relativity" he says that the right term in parenthesis is a gradient of some scalar field i.e.
When $$\partial_a (\frac{\ X_b}{\ X^2})=\...
0
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1
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2k
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Derivative of a metric tensor
I would like to ask you a question - maybe simple - but bothering me.
We have two four-position vectors product in curvilinear coordinates given by
$(1) \quad X^{\alpha}g_{\alpha \beta}X^{\beta} = \...
- 9
1
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0
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66
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Differentiating the four-velocity contracted with itself
Let us denote $u^\mu$ as the contravarient component of a four velocity at a point in some coordinate system for a pseudo-Riemannian manifold. I want to examine the following equation.
$$\partial_\nu(...
- 551
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1
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197
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
- 5,991
5
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2
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2k
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Why the unit vector is represented as a partial derivative in GR?
Can someone give a good intuitive explanation why we represent the unit vector as a partial derivative in GR and what does it mean?
- 605
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471
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Why exactly in general relativity are tangent vectors defined as maps from functions to $\mathbb{R}$?
I am basing this on the lectures from the hereaus international winter school on gravity and light.
If $M$ is the manifold of physical spacetime, then at any point $p \in M$, we have a tangent space ...
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