All Questions
Tagged with spacetime metric-tensor
571
questions
-2
votes
0
answers
46
views
Does Mass Actually Displace Space-Time, or does Mass only Distort it?
1. Question
Given the plethora of space-time illustrations, there is a sense that space-time is actually being displaced by mass, (planets). But on its face, this doesn't really make sense because ...
- 93
2
votes
3
answers
462
views
Question on special relativity
I am trying to learn special relativity. If we consider two inertial reference frames with spacetime co-ordinates $(t,x,y,z)$ and $(t',x',y',z')$ and let there be 2 observers who measure the speed of ...
- 576
2
votes
0
answers
60
views
Under what circumstances can a 4D singularity occur in General Relativity?
I've tried to find on the literature about 4D (single point) singularities, but most of the theorems about singularities pertain to either space-like or time-like singularities, which always have some ...
- 159
-1
votes
1
answer
71
views
What happens if we differentiate spacetime with respect to time? [closed]
Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
1
vote
2
answers
133
views
Is the FRW metric, based on spatial homogeneity and isotropy, rotationally and translationally invariant? If so, how?
The spatial part of the Minkowski metric, written in the Cartesian coordinates, $$d\vec{ x}^2=dx^2+dy^2+dz^2,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
- 11.8k
2
votes
1
answer
79
views
A few doubts regarding the geometry and representations of spacetime diagrams [closed]
I had a couple questions regarding the geometry of space-time diagrams, and I believe that this specific example in Hartle's book will help me understand.
However, I am unable to wrap my head around ...
- 23
2
votes
1
answer
72
views
Confusion about local Minkowski frames
This is sort of a follow-up to the question I asked here:
Confusion about timelike spatial coordinates
The important context is that we imagine a metric that, as $t\rightarrow\infty$, approaches the ...
- 160
2
votes
1
answer
89
views
Confusion about timelike spatial coordinates
I'm pretty new to general relativity, and I'm self-studying it using Sean M. Carroll's text on the subject. In Section 2.7, he introduces the notion of closed timelike curves. He gives the example of ...
- 160
1
vote
0
answers
25
views
How to derive Feffermann-Graham expansion for AdS Vaidya geometries?
Introduction
The Feffermann-Graham expansion for an asymptotically AdS spacetime [0] looks like Poincare AdS but with the flat space replaced by a more general metric i.e.
$$ds^2=\frac{1}{z^2}(g_{\mu \...
- 785
3
votes
1
answer
55
views
Time component of four-velocity
While reading through Spacetime and Geometry by Sean Carroll, I came across the following passage:
"Don't get tricked into thinking that the timelike component of the four velocity of a particle ...
- 442
4
votes
3
answers
199
views
Change of variables from FRW metric to Newtonian gauge
My question arises from a physics paper, where they state that if we take the FRW metric as follows, where $t_c$ and $\vec{x}$ are the FRW comoving coordinates:
$$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$
...
- 293
0
votes
1
answer
83
views
What objects are solutions to the Einstein Field Equations?
The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
0
votes
2
answers
65
views
What is $r$ in a metric signature in general relativity? If $v$ and $p$ are the time and spatial coordinates?
The Wikipedia article on metric signatures says that the signature of a metric can be written $(v,p,r)$, where $v$ is the number of positive eigenvalues, $p$ is the number of negative eigenvalues, and ...
- 4,509
0
votes
1
answer
76
views
How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?
In the presence of a gravitational field, the spacetime metric, $$ds^2=\eta_{ab}dx^a dx^b,$$ should be changed to, $$ds^2=g_{ab}(x)dx^adx^b.$$ What are the convincing physical arguments that motivate ...
- 11.8k
3
votes
1
answer
143
views
Clarification on Representing Distances and Trajectories in Minkowski Spacetime
In the context of Minkowski spacetime, where the metric has a signature of (-, +, +, +), the $x-t$ plane (spacetime diagram) is commonly used to visualize events and their evolution in both space and ...
- 489
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
5
votes
1
answer
398
views
Linearity of Lorentz Transformation proof
I was reading this article and got to the part where the homogeneity of space and time leads to the linearity of the transformations between inertial frames.
In particular, the function $x^\prime=X(x,...
- 93
0
votes
1
answer
54
views
From infinitesimal interval invariance to finite interval invariance in SR
In Landau and Lifshitz's The Classical Theory of Fields, on page 5 about interval invariance between different frames, it reads
Thus, $$ds^2=ds'^2,\tag{2.6}$$ and from the equality of the ...
- 613
0
votes
1
answer
66
views
Is the invariance of the 4-dim scalar product the fundamental law behind time dilatation and length contraction?
The Lorentz Group is defined as the group of all transformations that leaves the 4-dim. scalar product invariant. An implication of this definition is that the absolute value of the first matrix ...
- 189
0
votes
0
answers
56
views
Why is spacetime pseudo-Riemannian manifold?
Forgive me for asking, This is a relatively naïve question, though, one i've had for a while now. I know that a pseudo-Riemannian manifold is a differentiable manifold with a metric tensor that is ...
1
vote
1
answer
85
views
Metric of Einstein static universe (ESU) black hole
The Einstein static universe (ESU) has metric
$$ g = - dt^2 + d\chi^2 + \sin^2 \chi d\Omega^2 $$
With
$$ t \in \mathbb{R}, \chi \in (0,\pi) .$$
Is there a metric that describes an eternal black hole ...
- 743
1
vote
2
answers
88
views
Meaning of principal definitions of SR in relativistic quantum mechanics
I have just started with relativistic quantum mechanics in my advanced quantum theory class and we only had a very short intermezzo on special relativity. I feel like I don’t have enough knowledge on ...
- 189
2
votes
1
answer
319
views
Why cant a repulsive event horizon of negative mass be theoretically constructed?
An event horizon appears in the Schwarzschild metric when considering a positive point mass in General Relativity.
But for a negative point mass in the negative mass Schwarzschild metric, which ...
- 476
2
votes
2
answers
189
views
Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
- 107
-3
votes
1
answer
84
views
Time-ordering and Minkowski metric's negative sign [closed]
I'm coming at the following question from a mostly lay perspective (i.e. barely-undergrad physics), so please bear with the weirdness of it if possible.
I've generally been uncomfortable with the ...
- 163
0
votes
0
answers
60
views
How would you reparametrize a worldline in terms of proper time in 2-dimensional Minkowski spacetime?
In a 2-dimensional Minkowski spacetime i.e. $x^\mu=(t,x)$, you can define the metric simply by the Minkowski metric, $ds^2=-dt^2+dx^2$, and the Christoffel symbols vanish. If you have a worldline ...
- 51
7
votes
2
answers
1k
views
Why does the Minkowski matrix appear in the free particle action?
It is usual to write the "kinetic" part of the SR action as the Minkowski space-time interval, here $(-,+,+,+)$, times $mc$
$$
S_{kin} = -\int_{\tau_1}^{\tau_2}mc\sqrt{-\eta_{\mu\nu}\dot{x}^{...
- 391
3
votes
2
answers
297
views
Time in the negative mass Schwarzschild solution
I have read that for the Schwarzschild metric solution with $M<0$, something odd happens with the time coordinate. For the constants of motion, $dt/d\tau=e(1 - 2GM/r)^{-1}$ with $M<0$ and $e$ a ...
- 476
0
votes
1
answer
60
views
Approximating curved spacetime with a grid of cartesian metric tensors?
Let's assume a universe with only some ($n$) single point masses $m_i$ in it. The point masses have initial positions in space-time, $x_{i0}$.
The spacetime between them is curved due to general ...
- 63
1
vote
0
answers
74
views
Why can't the metric have more than one timelike coordinate? [duplicate]
In one of his lectures, L Susskind stated that he cannot make sense of a metric with more than one timelike dimension. I also have trouble imagining it, but is there a good mathematical or physical ...
- 795
3
votes
1
answer
90
views
Kruskal Diagram: 2D projection?
Is a Kruskal diagram a 2D flat space projection of Schwarzschild space-time diagram? If not, isn't it true that one could not draw one accurately on a paper?
BTW, I am not referring to Penrose ...
- 1,161
3
votes
1
answer
64
views
Proportional null vectors [closed]
the past few days I've been studying special relativity and was just now making some exercices on it. One exercice was the following:
Let $U$ and $V$ be two null vector is a $d$-dimensional Minkowski ...
- 29
7
votes
2
answers
2k
views
Do the Einstein Field Equations force the metric to be Lorentzian?
In GR, we are working with Lorentzian metrics, which are examples of a pseudo-Riemannian metrics. That is, we are trying to find pseudo-Riemannian $g_{\mu\nu}$ that are solutions to the field equation ...
- 453
2
votes
2
answers
156
views
Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ can (up to constant) be given by a distance between the two time-like separated events $z$ and $w$ where $...
- 125
1
vote
0
answers
67
views
Chose coordinates where $g_{01}=g_{02}=g_{03}=0$ to disentangle space and time?
$g_{\mu\nu}$ is the metric tensor. It describes the curvature of spacetime in general relativity. The choice of coordinates is completely arbitrary. It should be possible to find and chose coordinates ...
- 93
2
votes
6
answers
2k
views
Is it possible to describe every possible spacetime in Cartesian coordinates? [duplicate]
Curvature of space-time (in General Relativity) is described using the metric tensor. The metric tensor, however, relies on the choice of coordinates, which is totally arbitrary.
See for example ...
- 93