11
votes
What is the evidence that gravitational fields don't sum up as a superposition?
Black hole solutions would not exist in a linear theory of gravity. This is because black holes are vacuum solutions, not sourced by any matter, and there are no static vacuum solutions that die off ...
- 33.3k
9
votes
Accepted
Where is the Lorentz signature enforced in general relativity?
The Lorentz signature is just part of the theory; for example in a weak-field limit, we should reduce to special relativity, which is described using Lorentz signature (in order to talk about light, ...
- 106
6
votes
Physical significance of metric compatibility
There is a formulation of GR where metric compatibility follows from the equations of motion. If you think of the Einstein-Hilbert action as being a functional of both the connection and the metric (...
- 33.3k
6
votes
What is the evidence that gravitational fields don't sum up as a superposition?
Einstein's equations are
$$
G_{\mu\nu}[g] = R_{\mu\nu}[g] - \frac{1}{2} g_{\mu\nu}R[g] = 8\pi G_N T_{\mu\nu} \tag{1}.
$$
where $g_{\mu\nu}$ is the metric of the spacetime. The Ricci scalar is given by
...
- 20.6k
6
votes
Where is the Lorentz signature enforced in general relativity?
As a simplified, purely mathematical answer, the signature of the metric is baked into the initial/boundary conditions. First of all note that it cannot change along a smooth enough connected space-...
- 305
5
votes
Accepted
Contravariance and covariance of vectors
That's because you're using the 'fake' version of gradient. The true version for a scalar field $F$ is given as:
$$ (\nabla F)= \frac{\partial F}{\partial Z^i } e^i$$
Here $e^i$ is the dual basis and ...
- 5,558
4
votes
What is the evidence that gravitational fields don't sum up as a superposition?
Gravitational wave (GW) observations of binary black holes (BH) may provide experimental tests of superposition of spacetimes, as defined by you.
Each BH is described by a spacetime metric, but the ...
- 5,370
4
votes
Accepted
Riemann curvature tensor in an inertial frame
The fact that a function's first derivative vanishes at a point does not mean that its second derivative vanishes at that point. Note that for $f(x)=x^2$, $f'(0)=0$ but $f''(0)=2$.
- 50.9k
4
votes
Accepted
Could 2D spacetime be seen as an embedded manifold?
If you mean "embedded in $\mathbb{R}^3$ with a Euclidean metric", then the answer is no.
Suppose that such an embedding exists. Consider the neighborhood of a point $P$ on the submanifold. ...
- 37.2k
4
votes
How do we figure out what is the right geometry of space?
Don't we we already have a geometry of space time as soon as we write down the Minkowski metric?
Yes, but that's putting the cart before the horse. Einstein's equations are a system of differential ...
- 50.9k
4
votes
Are there types of spacetime that have no symmetries?
I assume you're actually asking about isometries of the metric $\phi^* g = g$ in the context of GR (physical spacetime symmetries). The 'gauge symmetry' of GR, diffeomorphism invariance, is an ...
- 2,848
4
votes
Where is the Lorentz signature enforced in general relativity?
Quoted from Pseudo-Riemannian manifold - Lorentzian manifold:
A principal premise of general relativity is that spacetime
can be modeled as a 4-dimensional Lorentzian manifold
of signature $(-,+,+,+)$...
- 25.9k
3
votes
Where is the Lorentz signature enforced in general relativity?
The Lorentzian signature of the metric is "baked into" the local causal structure (the set of "light cones", one at each event) of spacetime, which plays a role in the initial ...
- 8,048
3
votes
Could time be a secondary effect due to curvature of space?
You can't have curvature in 1 dimension: it is a line and always locally flat, with $R^a{}_{bcd} = R^0{}_{000} =0$. So it doesn't make sense to ask about 'time being curved'.
If you want to consider ...
- 2,848
3
votes
Invariance of spacetime interval by Schutz
I would suggest you read Landau & Lifshitz argument for invariance of $ds^2$, particularly the Wikipedia link because it states clearly the theorem which is being proved, and it fills in a few ...
- 2,502
3
votes
Accepted
Invariance of spacetime interval by Schutz
The claim is that if the relationship between coordinates in the two inertial frames is linear and if $\Delta S^2 = 0$ necessarily implies that $\Delta S'^2 = 0$, then in general we must have that $\...
- 50.9k
3
votes
Understanding EFE: RHS linear, LHS not?
Consider the equation
\begin{equation}
x^2 + 1 = x
\end{equation}
The left hand side is nonlinear in $x$, while the right hand side is linear, but this is still a valid equation.
Actually the ...
- 33.3k
3
votes
Accepted
I'm confused about the number of Killing vectors in Schwarzschild metric
A brute force (and ugly) derivation of the Killing fields of Schwarzschild metric
The Schwarzschild metric is
\begin{equation}
ds^2 = -\left(1-\frac{R_{\text{S}}}{r}\right) \text{d} t^2 + \left(1-\...
- 501
2
votes
Physical significance of metric compatibility
Metric compatibility follows from the Einstein equivalence principle, which asserts that the laws of special relativity hold for local non-gravitational phenomena in a freely falling laboratory. Here, ...
- 3,195
2
votes
Accepted
Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$
Just use the standard properties of the tensor in solving for C,
$$
\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho} = C \varepsilon^{\lambda\rho} ~~~~~\leadsto \\
\epsilon _{\lambda \rho} \...
- 47.3k
2
votes
Understanding EFE: RHS linear, LHS not?
This is just a small comment to buttress Andrew's answer (since it was a long string of comments on his answer!). Let us review the definitions of basic geometric gadgets used in Einstein's field ...
- 2,645
2
votes
Writing the metric of 4D de Sitter space as a 5D metric
What you want is the induced metric. I am not going to solve this explicitly for you since this is a very standard homework/exam exercise. But let me sketch the identical computation for a sphere.
A ...
2
votes
Accepted
Is this version of Einstein field equations linear?
Einstein's equations are not linear in the metric tensor $g_{\mu\nu}$, which is the field we want to solve for$^\dagger$. This means that if $g^{(1)}_{\mu\nu}$ and $g^{(2)}_{\mu\nu}$ are two solutions ...
- 33.3k
2
votes
Does the stress energy tensor scale with the metric tensor?
What you're asking about is a specific case of Weyl transformations of the form
$$ g_{ab} \rightarrow e^{-2\phi(x)}g_{ab}
$$
where $\phi$ is a constant. Under these, the Ricci scalar is not invariant (...
- 2,848
2
votes
Change of Metric Under Coordinate Transformation
I think that you are trying to compute the Lie derivative of the metric. If so, there should be no factor of 1/2. Under an infinitesimal shift $x^\mu\to x^\mu +\eta^\mu$ we have $g \to g+ {\mathcal ...
- 41.3k
2
votes
Where is the Lorentz signature enforced in general relativity?
4-dimensional Lorenzian manifold was the most natural way to express the invariance of light velocity (Maxwell's wave equation), in terms of metric $$ds^2=c^2 dt^2-dx^2-dy^2-dz^2$$ For Herman ...
- 989
2
votes
Are there types of spacetime that have no symmetries?
Let me rephrase your question.
You are asking if there exists a Riemannian (or a Lorentzian) manifold $(M,g)$, such that any smooth vector field $X$ on $M$ satisfying the Lie derivative equation $$\...
1
vote
Could time be a secondary effect due to curvature of space?
Are there cases where time is curved independent from space?
To my knowledge not. Einstein field equations (EFE) are just about this dependency. In case of Schwarzschild exterior solution the metric ...
- 989
1
vote
How to calculate Proper Distance as an arc length in Schwarzschild metric?
For a fixed $r$ and $t$,
$$ds^2 = r^2(d\theta^2 +\sin^2\theta\ d\phi^2)\ ,$$
which is the same arc length as in Euclidean space.
Then if $\phi = f(\theta)$ then
$$s =r \int \left(1 + \sin^2\theta\ \...
- 111k
1
vote
Accepted
Linear Momentum in General Relativity
Is there a metric that describes this case?
Yes. It is the Schwarzschild metric (valid outside of the gravitating body if we are talking about something like a star). When written in the form where ...
- 12.5k
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