All Questions
Tagged with spacetime metric-tensor
571
questions
3
votes
4
answers
364
views
Regarding the signature of special relativity
in special relativity we add time as a dimension and replace euclidean space $ \mathbb{R}^4 $ with a pseudo-euclidean space $ \mathbb{R}^{1,3} $ of signature $ (1,3) $ by defining a quadratic form $\...
- 309
0
votes
1
answer
54
views
Proof of the invariance of $c$ using the Lorentz group
Apologies if this question was already asked a few times but i could only find proofs of the invariance of $ ds^2 $.
Is there any way of proving the 2nd postulate (that $c$ is invariant in all ...
- 309
0
votes
2
answers
70
views
Do we have notion of a proper time for any two timelike separated arbitrary events?
Consider two infinitesimally close, timelike separated but otherwise arbitrary events $P$ and $Q$ with coordinates $(t,\vec{x})$ and $(t+dt,\vec{x}+d\vec{x})$. For example, imagine event $P$ is "...
- 11.8k
0
votes
1
answer
173
views
Can momentum exist in a null direction?
CONTEXT (skip to "my question is"):
As I understand it, and correct me if I'm wrong, an orbit trades momentum between the X and Y directions. But spacetime can have negative and even null ...
- 278
0
votes
1
answer
100
views
Proof of Invariance of Spacetime Interval?
I was going through Spacetime Physics by Taylor and Wheeler and came to a point where they showed a proof of Invariance of Spacetime Interval. You can find the proof Here and Here is the second part ...
- 121
0
votes
0
answers
72
views
How to mathematically describe the process of spacetime curvature?
I guess as a result of the energy-momentum tensor $T_{\mu\nu}$ coupling to a flat Minkowski metric, $\eta_{\mu\nu}$, the flat metric can become that of a curved spacetime, $g_{\mu\nu}$. How can one ...
- 357
1
vote
0
answers
32
views
Example of lightlike curve that's not a geodesic in Lorentz spacetime [duplicate]
Let $(M,g)$ be a 4 dimensional Lorentz spacetime. A smooth curve $\alpha:\ I\to M$ is called lightlike if $\alpha'(s)\in TM_{\alpha(s)}$ is lightlike for all $s\in I$, which means
$$g_{\alpha(s)}\big(\...
- 143
3
votes
1
answer
124
views
What is the problem with two time dimensions? [duplicate]
I am reading a book "General relativity: The theoretical minimum" by Leonard Suskind.
In page 168-169, the author explains the reason why we don't consider the case with two time dimensions ...
- 311
5
votes
2
answers
577
views
Help with the Minkowski space-time metric
I've been trying to learn how to multiply two tensors in order to go from
$$g_{\mu\nu} dr^\mu dr^\nu$$
to
$$c^{2}\,dt^2-dx^2-dy^3-dz^2$$
But I can't figure it out.
$g_{\mu\nu}$ is a $4\times4$ matrix, ...
- 179
1
vote
0
answers
48
views
JT gravity metric - solution to the dilaton equations of motion
I am reading Closed universes in two dimensional gravity by Usatyuk1, Wang and Zhao. The question is not too technical, it is about the solutions to the equations of motion that result from the ...
- 3,992
1
vote
2
answers
149
views
Does the interval pseudometric say that elapsed time is negative spatial distance?
Quick review (skip it):
In the formula from 8th grade, you figured out the length of the long side of the triangle
using this equation:
And in three dimensions:
This gives the length of the line ...
- 278
1
vote
1
answer
82
views
Why is there a negative sign in the formula for proper time [duplicate]
I recently read in the footnotes of 'The Elegant Universe' by Brian Greene about the formula for proper time, defined as
$d\tau^2=dt^2-c^{-2}(dx_1^2+dx_2^2+dx_3^2)$.
I am new to the subject of Special ...
- 49
1
vote
0
answers
62
views
Confused about spherically symmetric spacetimes
I'm following Schutz's General Relativity book and I am confused about his description and derivations of a spherically symmetric spacetime. I searched online and found that using Killing vectors is a ...
- 151
0
votes
1
answer
125
views
Event horizon in stationary spacetime
In the case of non-stationary spacetimes finding the event horizon is no easy task.
The stationary case should somehow be less involved or so it is in some well known cases, such as the Kerr spacetime....
- 1,989
1
vote
0
answers
20
views
Rescaling the null coordinates
Given a $4$-dimensional spacetime described by four coordinates $(t,r,\theta,\phi)$, we usually define the null coordinates by,
\begin{equation}
u = \frac{t-r}{2}, \quad v = \frac{t+r}{2}
\end{...
- 404