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Results tagged with general-relativity
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user 156895
A theory that describes how matter interacts dynamically with the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS.
1
vote
Accepted
Why are these vectors perpendicular?
Assuming that $\mathbf F$ is the Faraday tensor, then it is antisymmetric by definition. Therefore,
$$E_\lambda X^\lambda = F_{\lambda \mu}X^\mu X^\lambda$$
is the contraction of an antisymmetric obj …
- 71.4k
3
votes
Would time continue to flow forever in a curved spacetime?
No. Note that curved is not the same as closed. For instance, an infinite plane which is distorted so it has small hills and valleys would have curvature, but would not be closed like a sphere is.
- 71.4k
1
vote
Accepted
Normal vector of hypersurface in Schwarzshild spacetime
Your surface is defined implicitly by the equation $F(t,r,\theta,\phi) = B$. The natural object which defines the orientation of the surface is not a vector but rather a covector $n_\mu$, with compon …
- 71.4k
4
votes
How small is "vanishingly small" when working with the Schwarzschild metric?
Your question pertains to the 2D Euclidean metric in polar coordinates. Given that
$$x = r \cos \phi \qquad y = r \sin\phi$$
It's easy to show that the distance between any two points $(r_1,\phi_1)$ …
- 71.4k
2
votes
Why do we use null geodesics?
Your question is based on a false premise. The equation
$$ \frac{d^2x^\mu}{ds^2} + \Gamma^\mu_{ \ \ \alpha\beta} \frac{dx^\alpha}{ds} \frac{dx^\beta}{ds} = 0$$
is solved by any geodesic, whether it b …
- 71.4k
1
vote
Accepted
Why is a real manifold (used in GR) defined in the way it is?
The images of open sets can be open or closed tough.
Not under homeomorphisms. The image $\psi(O)$ of an open set $O$ under a homeomorphism $\psi$ is also open. One can see this immediately by not …
- 71.4k
0
votes
Falling body geodesics seem counterintuitive
Here is an explicit calculation which might yield some insight. If we take the flat-earth approximation (which in Newtonian mechanics corresponds to $\vec F_g = - mg \hat z$), then an observer standin …
- 71.4k
2
votes
Einstein Summation Problem
If we wanted all possible combinations, it would be a bit silly to use the same letter for both indices.
According to the Einstein summation convention, repeated indices are summed over. That is,
$$a^ …
- 71.4k
2
votes
Accepted
Determinant of metric tensor in Cartesian Coordinates constant in vacuum
$\mathrm{det}(g)$ is not coordinate-independent - it is a scalar density of weight $+2$ (or $-2$, depending on your convention) which generically changes across spacetime. In Minkowski space equipped …
- 71.4k
4
votes
Accepted
Isometry between Minkowski space and Tangent space
Two (metric) manifolds $(M,g_M)$ and $(N,g_N)$ are isometric if there exists a diffeomorphism $\varphi:M\rightarrow N$ such that $g_M = \varphi^*g_N$.
On the other hand, two pre-Hilbert spaces $(V, \l …
- 71.4k
6
votes
Carroll's interpretation of 1-forms
EDIT: I agree with Kostya that there is likely a typo in Carroll's book, and that the equality of first (not second) derivatives is what's required. I haven't thought about this for long, though, so …
- 71.4k
7
votes
Accepted
Why does this derivation of the Einstein Field Equations only work with the Trace-Reversed f...
I will use signature (+---). In the weak-field limit in which $\mathbf g \simeq \boldsymbol \eta + \mathbf h$, the components of the Ricci tensor are given by
$$R_{\mu\nu} = -\frac{1}{2}\left(\square …
- 71.4k
2
votes
Accepted
Torsion tensor definition doubt
If your confusion is with the apparently missing factor of $1/2$, note that
$$\nabla_{[a}\nabla_{b]} \equiv \frac{1}{2}(\nabla_a\nabla_b-\nabla_b\nabla_a)$$
Symmetrization and antisymmetrization brack …
- 71.4k
17
votes
Accepted
"And God said ... and the universe was ..." What does this equation mean?
$\gamma:\mathbb R\rightarrow M$ is a curve whose image lies in the spacetime $M$, so $\gamma(t)$ is the event at parameter value $t$ along the curve. $\gamma'(t) \in T_{\gamma(t)}M$ is the tangent v …
- 71.4k
2
votes
Could gravitational waves be matter waves?
The energy being radiated away by gravitational waves is provided by the orbital energy (kinetic + gravitational potential) of the merging bodies. Conceptually, it is basically the same as the electr …
- 71.4k