5
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Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
5
votes
Accepted
Can an observer sitting at rest at infinity, simultaneously measure the proper time and proper distance of a particle travelling in any geodesic?
You probably need to say exactly how the "velocity" is defined by Hoson. This would be the rate of change of some spatial coordinate with respect to the proper time measured by the observer.
...
- 122k
5
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Can an observer sitting at rest at infinity, simultaneously measure the proper time and proper distance of a particle travelling in any geodesic?
Your question is lacking a lot of details and it is thus not clear what exactly you are asking for.
However, you seem to be having a common misunderstanding of what is happening. An observer at ...
- 3,676
3
votes
Partial derivatives vs Covariant derivatives in polar coordinates
As OP correctly points out connections introduce a concept of differentiation of tensor fields or more in general of sections of vector bundles that takes into account how the bases of the fibers ...
- 1,607
3
votes
The center of the Schwarzschild black hole
The metric in isotropic coordinates (1) and (2) has a (coordinate) singularity at $r=a$. Consequently, there is no a priori relationship between the metric for $r>a$ and $r<a$. They are both ...
- 9,705
2
votes
The center of the Schwarzschild black hole
Equation for $r$ in terms of $r'$ (in the question as originally posed) has them the wrong way round. In notation $r$=isotropic, $r'$=Schwarzschild coordinate it should be
$$r' = r (1 + a/r)^2 .$$
...
- 53.1k
2
votes
Accepted
Self-connected Einstein-Rosen wormhole
The wormhole itself is regarded as the common boundaries of one or two manifolds when identified with each other to glue them together. Therefore the (three-dimensional) hypersurface with $r=r_\mathrm{...
- 2,072
1
vote
Accepted
Difference between $R^{a}_{bcd}$ and $R_{abcd}$ Riemann tensor types
There is no deep intuitive geometrical meaning behind a Riemann tensor with some indices moved up/down. You could say that the two variants are "dual" to each other, loosely speaking. The ...
- 3,503
1
vote
Does off-shell graviton in 3+1D still have two degree of freedom?
The off-shell graviton has 6 degrees of freedom, while the on-shell graviton has 2 dynamical degrees of freedom. The off-shell graviton can be decomposed into a spin-2, a spin-1 and a spin-0 component,...
- 306
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Equating 2 sides of EFE
First, note that for scalar functions, the covariant derivative reduces to the partial derivative. So for scalar functions, it is true that if the covariant derivative is zero at a point, then the ...
- 42.9k
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