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Results tagged with general-relativity
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user 130091
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.
0
votes
Equivalence principle question
I'm not sure if the O.P. was satisfied with the previous answers or not but I feel that there is something that wasn't answered, at least in the way that the question asked for it. The reason why the …
- 4,737
0
votes
0
answers
41
views
Is the normal to an embedded surface generally undefined up to a connection?
Let there be an embedding $\phi:S\rightarrow M$ where $M$ is a d-dimensional manifold and $S$ is a codimension-k submanifold. the space of all tangent vectors to the embedded surface $\phi(S)$ is sim …
- 4,737
5
votes
4
answers
2k
views
Motivation for the use of 1-forms in General Relativity
During a course I took on General Relativity, the professor started with an introduction on differential geometry. Vectors were properly motivated: he said that since the differential manifold doesn't …
- 4,737
2
votes
2
answers
638
views
Laws of physics and diffeomorphism covariance
Up to my limited understanding, diffeomorphisms on a space-time manifold can be viewed as changes of coordinates. While studying general relativty, I read that the theory has diffemorphism covariance …
- 4,737
1
vote
Accepted
Why do we need 3 variables to parametrize $\scr{I}^\pm$ in a Penrose diagram?
I'm just answering myself to close this thread, but the answer is basically what people said in the comments:
the regions $\scr{I}^\pm$ are reached by travelling on a light ray (i.e. a null geodisic) …
- 4,737
1
vote
2
answers
125
views
What's the correct relativistic lagrangian when calculating the Hamiltonian?
Introduction:
The action of a test particle in curved spacetime is
\begin{equation}
S=-m\int d\tau
\end{equation}
because we can't do a variation on proper time without changing the boundary terms, on …
- 4,737
0
votes
1
answer
262
views
Proof that 4-velocity is normalized in curved spacetime
Whenever I try to find an explanation for the normalization of the four-velocity
\begin{equation}
g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-1
\end{equation}
I'm always shown a proof in Minko …
- 4,737
0
votes
0
answers
62
views
Expression for chain rule of a geodesic equation solution
I'm doing some work on General Relativity and I found that there's an identity that -if true- would really simplify my calculations. I feel it has to be true but I haven't been able to prove it. It go …
- 4,737
2
votes
0
answers
80
views
Degrees of Freedom in the Newman-Penrose Formalism
In the Newman-Penrose formalism one encodes the ten degrees of freedom of the Weyl tensor $C_{\alpha\beta\mu\nu}$ in the five complex scalar potentials $\Psi_0$, $\Psi_1$, $\Psi_2$, $\Psi_3$ and $\Psi …
- 4,737
1
vote
0
answers
84
views
Different notions of embedded metric in a hypersurface
Let there be a metric tensor $g_{\mu\nu}$ on a manifold $\bf{M}$ of dimension $d$. I want to find the metric on an hypersurface $\Sigma$ of dimension $d-1$ parametrized by some coordinates $y^i=y^i(x^ …
- 4,737
2
votes
2
answers
601
views
How to integrate a tensor in curved spacetime?
I've read "We can only define the integral of a scalar function. The integral of a vector or tensor field is meaningless in curved spacetime" on many books and lectures on General Relativity (For exa …
- 4,737
0
votes
Diffeomorphisms and pullbacks
I think I have an answer for my own question. I also want to clarify what was confusing me.
Diffeomorphisms
A diffeomorphism is an isomorphism $\phi:M\rightarrow N$ that is invertible, continuously d …
- 4,737
1
vote
1
answer
66
views
Does the divergence theorem require the covariant derivative to be metric compatible?
I know this is more of a mathematical question, but it arises in the context of general relativity and uses its language so I thought it would be best to ask it here. I understand that the divergence …
- 4,737
2
votes
0
answers
74
views
How does one draw the Penrose diagram for an FLRW universe with three different epochs?
Let's model the universe with the FLRW metric
$$ ds^2 =-dt^2 +a(t)^2\big(d\chi +R_k(\chi)^2 d\Omega^2\big)$$
where $a(t)$ is the scale factor and $R_k(\chi)$ is $\chi$ for a spatially flat ($k=0$) spa …
- 4,737
3
votes
2
answers
188
views
Why do we need 3 variables to parametrize $\scr{I}^\pm$ in a Penrose diagram?
In the figure we can see the Penrose diagram for Minkowski space
If I understand correctly, $i^-$ and $\scr{I}^-$ have coordinates $r=\infty$ and $t=-\infty$ while $i^+$ and $\scr{I}^+$ have coordi …
- 4,737