All Questions
Tagged with metric-tensor spacetime
587 questions
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1+1D spacetime quantitative description
Suppose we try to imagine a 1+1D analogy of spacetime (1 dimension for space + 1 dimension for time). Lets say there is a point mass $m$ at $x = 5$. So the world line would be the line $x = 5$ in $x$-$...
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How to find 4-acceleration scalar product in terms of $ds$ spacetime interval?
We know 4-velocity $$U^i =dx^i/ds$$ where $$ds=\sqrt{dx^idx_i}$$ so we have 4-acceleration $$A^i=dU^i/ds$$
Then we have $$A^iA_i=\dfrac{dU^i}{ds}\dfrac{dU_i}{ds}$$
How should I proceed to find this ...
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1
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59
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Why can coefficient $a$ between spacetime intervals depend on absolute relative velocity between the systems?
I read Landau & Lifshitz' Classical Theory of Fields book (page 14-15) (see pic below) and I was confused when I saw in proof that coefficient $a$ between spacetime intervals $(ds)^2$ and $(ds')^2$...
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3
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134
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What exactly is the role of the Lorentzian metric within spacetime? [closed]
I learned that twodimensional spacetime diagrams and fourdimensional spacetime manifolds are provided with Lorentzian (pseudo-Riemannian) metric. However, regarding a spacetime diagram with a light ...
- 3,151
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2
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125
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Why in physics we work with metric without defining topology/smooth structure first? [closed]
I come from more mathematical background, so I always get confused how physicists
define spaces straight by writing metric tensor without specifying all point-set infromation on which this metric ...
- 149
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460
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Can we use local proper time instead of global coordinate time in special relativity?
In special relativity we have:
$$dt^2-dx^2-dy^2-dz^2 = d\tau^2$$
Where $d\tau$ is the change in local proper time of the object (the change in the clock ticks).
I've always wondered why we can't just ...
- 287
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4
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194
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Time-like separated and Space-like separated events
I am trying to get an intuition about Time-like separated and Space-like separated events. I understand the definition of these terms, but I lack intuitive understanding of what these concepts ...
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112
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Does an expanding universe naturally induce a Lorentzian metric in local nonexpanding coordinates?
When we consider the expansion of our spatial universe in time the standard method is through a conformal transformation of the spacetime metric as:
$$g_{\mu\nu}\rightarrow a(\tau)^{2}g_{\mu\nu}$$
...
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0
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36
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Conformal time and conformal coordinates in a NON-COMOVING frame?
Lets consider an uniformly expanding universe with scale parameter $a(\tau)$ where $\tau$ is comoving time. Then we can write the metric as conformally transformed
$$\tilde{g}_{\mu\nu}=a^{2}g_{\mu\nu}$...
- 2,867
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3
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129
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What happens to "time" inside a "warp field" [closed]
I know this is a silly science fiction question, but it does serve a purpose.
If a "warp drive" (much like Star Trek) could be built. If it was essentially used to travel faster than light, ...
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General relativity and velocity of light
Let us consider a generic metric: $$ds^2 = g_{00}c^2dt^2 + g_{0i}cdx^idt + g_{ij}dx^idx^j$$
I wish to find the speed of light in a particular direction $x^i$, say with $i=2$.
For light, the proper ...
- 1,053
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3
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182
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How to deal with square root of negative number when calculating spacetime interval?
I was reading Hartle's Gravity: An Introduction to Einstein's General Relativity and I was doing an exercise from page 57 that asks me to use the metric $\Delta s^2=-(c\Delta t)^2+\Delta x^2$ ...
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Proper Time in Eddington Finklestein Coordinates not calculable?
I've been trying to calculate proper time in EF coordinates:
$$ds^2 = (1-\frac{2M}{r})d^2\upsilon -2d\upsilon dr - r^2(d^2\theta +sin^2\theta d^2\phi)$$
As it is on a timelike worldline: $$ds^2 = d\...
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2
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117
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How to decompose the metric tensor orthonormally?
In fact I don't know how to explain my question properly.So I would like to introduce the following condition.
[width=0.5,scale=0.5]
Now let's consider two questions.
1)When observer A want to do ...
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74
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Max distance in 2d and 3d positively curved space
On a 2d positively curved space (surface of a sphere), using polar coordinates, we have $ds^2 = dr^2 + R^2 sin^2(r/R) d\theta^2$. I understand we can calculate circumference by going around $\theta$ ...
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46
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Why is AdS-boundary considered timelike?
I was wondering why the conformal boundary of (compactified) AdS is said to be timelike. Consider the conformal compactification of AdS spacetime with metric
$$g = (- dt^2 + dx^2 + \sin^2 x \ d\Omega^...
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69
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Meaning of $\mathbb{R}^{1,3}$ and $\mathbb{R}^{3,1}$ notation for Minkowski spacetime
In David Tong's lecture notes on general relativity (page 12) he denotes Minkowski spacetime as $\mathbb{R}^{1,3}$. Also, on Wikipedia, I found that both $\mathbb{R}^{1,3}$ and $\mathbb{R}^{3,1}$ are ...
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In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?
I've been working on studying Special Relativity and General Relativity for the past few years. As I think we all know, GR gets a lot more complicated than SR, and my knowledge is limited. I am very ...
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Importance of orthogonality in Minkowski space [closed]
I am currently studying Minkowski space. Orthogonality in this space is new to me. I have seen in a blog post, in 1 that states that, orthogonality is important in this space.
It will be helpful, if ...
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Why cant I measure proper length on spacetime curvature with the following formula? [closed]
I'm struggling right now from the definition of proper length along spacetime curvature, it is said as I found online the length that object covered on his spacetime rest frame , so why cant I use the ...
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3
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483
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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 ...
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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 ...
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81
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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 ...
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153
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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,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
- 12.3k
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1
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86
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A few doubts regarding the geometry and representations of spacetime diagrams [closed]
I had a couple questions regarding the geometry of space-time diagrams, and I believe that this specific example in Hartle's book will help me understand.
However, I am unable to wrap my head around ...
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86
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Confusion about local Minkowski frames
This is sort of a follow-up to the question I asked here:
Confusion about timelike spatial coordinates
The important context is that we imagine a metric that, as $t\rightarrow\infty$, approaches the ...
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114
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Confusion about timelike spatial coordinates
I'm pretty new to general relativity, and I'm self-studying it using Sean M. Carroll's text on the subject. In Section 2.7, he introduces the notion of closed timelike curves. He gives the example of ...
- 437
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How to derive Feffermann-Graham expansion for AdS Vaidya geometries?
Introduction
The Feffermann-Graham expansion for an asymptotically AdS spacetime [0] looks like Poincare AdS but with the flat space replaced by a more general metric i.e.
$$ds^2=\frac{1}{z^2}(g_{\mu \...
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68
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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 ...
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3
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216
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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:
$$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$
...
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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: ...
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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 ...
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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 ...
- 12.3k
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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 ...
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2
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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 $\...
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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 ...
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73
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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 "...
- 12.3k
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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 ...
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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 ...
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73
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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 ...
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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
$$g_{\alpha(s)}\big(\...
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What is the problem with two time dimensions? [duplicate]
I am reading a book "General relativity: The theoretical minimum" by Leonard Suskind.
In page 168-169, the author explains the reason why we don't consider the case with two time dimensions ...
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2
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589
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Help with the Minkowski space-time metric
I've been trying to learn how to multiply two tensors in order to go from
$$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, ...
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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 ...
- 4,157
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2
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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 ...
- 475
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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 ...
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0
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81
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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
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1
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130
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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....
- 2,029
1
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0
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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{...
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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,...
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