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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.
21
votes
Do we know anything about the age of the universe?
The rough idea is that under the assumptions contained in the cosmological principle, the application of Einstein's equations leads us to the equation
$$d(t) = a(t) \chi$$
where $d(t)$ is called the p …
- 71.4k
19
votes
Accepted
Vectors as functions?
However, I have read that the modern way to learn these concepts is to think of vectors as multilinear functions of covectors
This is actually not quite true, though the distinction is subtle.
In th …
- 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
17
votes
Accepted
Parallel transport: Lie derivative vs covariant derivative
Consider the expression $\nabla_{\mathbf X_p} \mathbf T$, for some arbitrary tangent vector $\mathbf X_p$ and tensor field $\mathbf T$. Since a tangent vector is always defined at a point $p$, we are …
- 71.4k
16
votes
Schwarzschild metric with negative mass
Yes, the Schwarzschild solution with $M<0$ is still a vacuum solution to the Einstein equations. Bondi explored some ideas about negative mass in GR in this paper, in which he says
As long as relativ …
- 71.4k
16
votes
Is there a version of the Einstein field equations that uses the Riemann curvature tensor in...
Well, as you know the Ricci tensor/scalar are built from the Riemann tensor, so it's not entirely clear to me what you're asking. You can of course write
$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R g_{\m …
- 71.4k
13
votes
Accepted
Is 4-momentum a vector or a 1-form?
Note: A $k$-form is a smooth, totally-antisymmetric $(0,k)$-tensor field. A one-form is therefore a covector field, but the momentum of a particle is not a field, so this answer has been revised to us …
- 71.4k
12
votes
How to know If the given space is torsionfree?
Both torsion and curvature are properties of a connection, not an intrinsic property of a metric manifold itself. For a given manifold $\mathcal M$ with pseudo-Riemannian metric $g$, you could equall …
- 71.4k
12
votes
Accepted
Basis Vectors as Partial Derivatives Issues
I begin with a brief(ish) review, and then answer your question directly afterward.
Tangent Vectors
In your standard differential geometry treatment, given a smooth curve $\gamma:[-1,1]\rightarrow \ma …
- 71.4k
12
votes
Are the Christoffel symbols all zero in gravity-free space?
No, not necessarily.
Flat space (which is what I assume you mean when you say gravity-free space) is special because it's possible to choose a global coordinate system in which all of the Christoff …
- 71.4k
11
votes
Einstein field equations in empty space, question about non-zero curvature
Surely regardless of the mathematics, in empty space it should still be forced zero by the fact that nothing is there?
The moon orbits the Earth despite the fact that it is, for all intents and p …
- 71.4k
10
votes
Which comes first, basis vectors or coordinates?
You can do either one.
Any choice of coordinates $x^i$ on a neighborhood $U\subseteq M$ induces a basis $\frac{\partial}{\partial x^i}$ for the tangent space $T_pM$ at each point $p\in U$.
Given a …
- 71.4k
10
votes
How can two black holes merge without violating No Hair?
The no-hair theorem refers to stationary black hole solutions to the Einstein equations. If the black holes in question are undergoing dynamics, then the no-hair theorem doesn't apply. Specifically, m …
- 71.4k
9
votes
Accepted
Why the basis of vectors and one-forms can not be related through the metric as a vector and...
First I'll state three quick preliminaries so we're both on the same page, and then I'll answer the question. In the following, I'm going to use tildes to distinguish one-forms and their components f …
- 71.4k
9
votes
Accepted
Is the raised Levi-Civita symbol a tensor density of weight 1?
To me, a more conceptually direct derivation goes as follows. The Levi-Civita symbol is a symbol, not a geometrical object. We can denote its value in a coordinate system $x$ as $\tilde \epsilon_{(x) …
- 71.4k