5
votes
Accepted
Interpreting a constraint on a simplified static spherically symmetric metric
OP's metric is an example of ultrastatic spacetime, i.e. it is a direct product of a Riemannian manifold (“space”) and $\mathbb R$ (“time”) and thus $-g_{tt}$ can be set to unity everywhere.
This ...
- 16.5k
4
votes
Accepted
The variational derivative of the metric with respect to inverse metric
The correct expression is (1). The key point is that the derivative $\partial_\mu$ acts on the entire expression within the functional derivative, not just the delta function.
To see this, we start ...
- 136
3
votes
Do all the geodesic equation of Schwarzschild metric should have the same unit?
The Schwarzschild metric in spherical coordinates $(t,r,\theta,\phi)$ is
$$ds^2=\left(1-\frac{r_s}{r}\right)c^2dt^2
-\left(1-\frac{r_s}{r}\right)^{-1}dr^2
-r^2(d\theta^2+\sin^2 \theta\ d\phi^2).$$
...
- 41k
2
votes
Accepted
Strange coordinate transformation in this proof of Birkhoff's theorem?
As already pointed out by the user @Pipe in a comment, you have a typo in your equation labelled ($\ast$). In the paper you reference it reads,
$ds^2 = \Big( F - 2 \partial_u Y \Big) \, du^2 + 2 \xi \,...
- 1,741
2
votes
Lorentz scalar Lagrangian in curved spacetime
First, the main answer to your question is that in general relativity, we want to be able to express the action in a manner that is invariant under changes of coordinates. There are fancy ways to ...
- 55.3k
2
votes
Deriving differential equation for the path of a particle in potential $U(r)$ using Maupertuis’ principle
The $\mathbf r$ is the position variable of the particle, such that $\mathbf r(t)$ forms its trajectory. Mathematically, this function is a curve in $\mathbb R^3$, parametrized by the free parameter $...
- 3,142
1
vote
Accepted
How do you obtain the coordinates of 3D space from the FLRW metric?
If your issue is just interpreting spherical coordinates, note that you don’t actually need to define an additional spatial dimension to properly set stuff up. It is perfectly-fine to call the spatial ...
- 3,347
1
vote
Deriving differential equation for the path of a particle in potential $U(r)$ using Maupertuis’ principle
The line element
$$ (\mathrm{d}\ell)^2~=~\sum_{i,k=1}^na_{ik} \mathrm{d}q^i\mathrm{d}q^k $$
in eq. (44.10) has metric tensor components $a_{ik}$ coming from the kinetic term
$$ T~=~\frac{1}{2}\sum_{i,...
- 213k
1
vote
Full model of gravitational waves?
First of all there is no known closed-form solution for the two body problem in general relativity (except in some very specific edge cases). If you want to solve the full non-linear problem, you have ...
- 12.9k
1
vote
Accepted
Wormhole metric by identification
If you integrate $e^{\phi_{-}} dt_{-} = e^{\phi_{+}} dt_{+}$
you obtain (18.40), i.e. $e^{\phi_{-}}t_{-} = e^{\phi_{+}}t_{+}$. This means $t_{+} = t_{-}e^{\phi_{-} - \phi_{+}}$ and so identifying $(t_{...
- 1,084
1
vote
How to model a Black Hole inside a Black Hole?
the Schwarzschild metric is the only spherically symmetric vacuum in GR. So this leads me to believe that the spacetime around a Black Hole inside a Black Hole is still a Schwarzschild spacetime, ...
- 109k
1
vote
Questions For the Expansion of Gravitational Waves
Normalizing $h$ as in $h/M_{\rm Pl}$ has the effect that when you expand the Einstein-Hilbert action
$$
S = \frac{M_{\rm Pl}^2}{2}\int d^4 x \sqrt{-g} R
$$
the kinetic term for $h$ is canonically ...
- 55.3k
1
vote
Does the generic wave equation imply the universe has a Minkowski spacetime?
It does not imply a Minkowski spacetime if the hypothesis is just as in the question.
However, it would imply it if making stronger the hypothesis as follows.
If we assume that
$\Psi$ is a scalar
in ...
- 78k
1
vote
Are standard and isotropic forms of Schwarzschild metric truly equivalent?
I want to offer a possible resolution to the issue raised in this post:
The two spacetime geometries represent the same system of a static mass with spherically symmetric spaces but in different ...
- 51
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