All Questions
Tagged with metric-field or metric-tensor
3,672 questions
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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
$$\tilde{g}_{\mu\nu}=a^{2}g_{\mu\nu}$...
- 2,867
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vote
1
answer
87
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Derivation of metric flatness locally
In Einstein Gravity by Zee chapter I.6, he discussed the local flatness of the metric. There are two steps in what he did. First, he showed that the metric at a point can always be written as a flat ...
- 512
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3
answers
105
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What is the physical significance of the electromagnetic tensor $F^{\mu\nu}$ having its indices on top or bottom? [duplicate]
I understand we need to put indices on a four vector on the top or bottom like $A^{\mu}$ or $A_{\mu}$ because of the invariant Lorentz product having its later three terms negative (for $+ - - -$ ...
3
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3
answers
129
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What happens to "time" inside a "warp field" [closed]
I know this is a silly science fiction question, but it does serve a purpose.
If a "warp drive" (much like Star Trek) could be built. If it was essentially used to travel faster than light, ...
- 47
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2
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64
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Why the EOM from the variation of the metric coincide with the EOM from the variation of an ansatz for the metric?
Suppose we have a pure gravitational action $S[g_{\mu\nu}]$ and we obtain, via the usual variation procedure, its associated Equation Of Motion (EOM) for any $g_{\mu\nu}$. After that, we can give an ...
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38
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Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is
$$
\nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda}
$$
It has the following identity:
$$
\nabla_{\mu}U^{\mu} = \frac{\...
- 49
3
votes
1
answer
44
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The interval of Hyperbolic Space (Euclidean Anti-deSitter Space)
I am learning the Group Theory notes from G. Moore.
https://www.physics.rutgers.edu/~gmoore/618Spring2022/GTLect1-AbstractGroupTheory-2022.pdf
7.4.3 Orbits Of The Lorentz Group In $d > 2$ ...
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56
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Deflection of light in linearised gravity
In Sean Carroll’s book, section 7.3, he uses the linearised metric to find the deflection of light around a mass $M$. But a mass $M$ has no energy density around it, so $T_{uv}$ is zero everywhere ...
- 83
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Are these different constructions of Rindler vacuum equivalent?
In a Minkowski spacetime and for a Klein-Gordon field, I have seen two different constructions of the Rindler vacuum in different references, but they don't seem to be equivalent.
The first ...
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Density changes of 3D projections of spacetime in relativity theory?
In relativity theory, the metric tensor $$g_{\mu\nu} $$ describes the gravitational field.
Can one treat the determinant of the metric tensor, normalized by the square of the Jacobian of the ...
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48
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Sign Error in derivation of Faraday 2-form
The electromagnetic field tensor $F_{\mu\nu} $ in component form is:
$$
F_{\mu\nu} = \begin{pmatrix}
0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\
-\frac{E_x}{c} & 0 & -B_z &...
- 4,241
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0
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40
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Raising and lowering indices under conformal transformations [duplicate]
Consider the special conformal transformation (SCT),
$$
x'^{\mu} = \Omega(x)(x^\mu-b^\mu x^2),
$$
where $\Omega(x)=(1-2b\cdot x+b^2x^2)^{-1}$. The covariant form should be
$$
x'_{\mu} = \Omega(x)(x_\...
- 11
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3
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124
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How to show that $\partial x^\sigma / \partial x_\nu = g^{\nu\sigma}$?
I just want to confirm that $$\frac{\partial x^\sigma}{\partial x_\nu} = g^{\nu\sigma} .$$
You cannot proceed directly:
$$\frac{\partial x^\sigma}{\partial x_\nu} = \frac{\partial (g^{\rho\sigma} x_\...
- 467
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110
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General relativity and velocity of light
Let us consider a generic metric: $$ds^2 = g_{00}c^2dt^2 + g_{0i}cdx^idt + g_{ij}dx^idx^j$$
I wish to find the speed of light in a particular direction $x^i$, say with $i=2$.
For light, the proper ...
- 1,053
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42
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In “black hole topological sector”, why does the regularity at centre require being locally isometric to flat disc?
Sorry for having such a long and confusing title.
The question arose while reading Thermodynamic ensembles and gravitation by J D Brown et al (https://dx.doi.org/10.1088/0264-9381/7/8/020).
In Section ...
- 120
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63
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Components of Killing vectors being zero
The following is a question from a GR worksheet given by my professor:
Does the following line element:
$$ ds^2 = -dt^2 + e^{\alpha t}(dr^2+r^2\vartheta^2+r^2\sin^2\vartheta d\varphi^2), \qquad [\...
- 172
2
votes
2
answers
55
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Cosmic expansion: FLRW metric vs Distance
What is the conceptual difference between the FLRW metric expansion of spacetime and the observed increasing of cosmic distances between two light-emitting sources according to the Hubble Law?
- 986
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320
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Proper time is invariant but undefined at the event horizon in schwarzschild metric. How can we remove this problem with a coordinate transformation?
We know that the line element gives us the proper time taken to traverse “nearby” coordinates.This is an invariant as well, so how can we remove the issue of undefined proper time at the event horizon ...
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vote
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 $\Delta s^2=-(c\Delta t)^2+\Delta x^2$ ...
- 711
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1
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84
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How to manipulate this equation with the metric tensor? (index notation) [closed]
I try to follow a derivation in special relativity using index notation. According to the solution, from this equation:
\begin{equation}
\eta^{\mu\tau}v'_{\tau}=\Lambda^{\mu}_{\;\sigma}\eta^{\sigma\nu}...
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1
answer
57
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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: $$ds^2 = d\...
2
votes
0
answers
101
views
Interpretation of the components of the Einstein tensor? (for visual thinkers)
I am trying to understand what the different components of the Einstein tensor correspond to, and how they can be interpreted in physics.
So we consider the tensor:
$$G_{\mu\nu} = \begin{pmatrix}G_{tt}...
- 1,227
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1
answer
56
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Question about comoving frame
From what I understand, in an expanding universe given by
$$ds^2=-dt^2+e^{2Ht}dx_{com}^2,$$ where I believe that $dx_{com}$ denotes a comoving spatial line element, the distance between a comoving ...
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0
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40
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On the Gauge-fixing for the case of the Polyakov string action
In the book "String theory and M-theory" by Becker-Becker-Schwarz, the author says that
"reparametrization invariance of the string sigma-model action $$S_{\sigma}=\frac{-T}{2}\int d^2 ...
- 614
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40
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Four-vector algebra - Notation confusion [duplicate]
I am trying to understand what are the consequences of the ordering (and placement in general) of the subscripts and superscripts for the matrix representation of different elements in four-vector ...
- 1,624
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1
answer
51
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Can Martire and Bobrick's general warp metrics all produce CTCs?
It's well known that the Alcubierre metric allows for the creation of CTCs under the assumption that warp bubbles with arbitrary relative velocities can be created (Everett, Physical Review D, 1996).
...
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45
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Equation of motion for $X^{\mu}$ (geodesic equation)
The action for a relativistic particle of mass $m$ in a curved $D$-dimensional is $$\tilde{S}_0=\frac{1}{2}\int d\tau (\dot{X}^2-m^2)$$ for particular gauge and $\dot{X}^2=g_{\mu\nu}(X)\dot{X}^{\mu}\...
- 614
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103
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Spherical symmetry of Schwarzschild solution
I was reading Regge-Wheeler's 1957 paper titled "Stability of the Schwarzschild Singularity", where in the section II, under the heading "Spherical Symmetry; Specialization to $M = 0$&...
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37
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Ricci scalar on TOV metric
Hi i computed the ricci scalar for the Tolman-Oppenheimer-Volkoff metric and obtained $R=\frac{8\pi G}{c^{4}}(\varepsilon - p)$ where $\varepsilon$ is the energy density and $p$ the pressure. I can't ...
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1
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94
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Where this definition $T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-h}}\frac{\delta S}{\delta h^{\alpha \beta}}$ come from?
In the book "String theory and M-theory" by Becker, Becker and Schwarz, the author says that the Nambu-Goto action $$S_{NG}=-T\int d\sigma\, \tau \sqrt{(\dot{X}\cdot X')^2 -\dot{X}^2X'^2}$$ ...
- 614
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1
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108
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Lorentz scalar from the second derivative of the metric tensor
This might be a very simple question. Is the following second derivative of the metric tensor a Lorentz scalar: $\partial_{\mu}\partial_{\nu}g^{\mu\nu}$ ? I know that for a vector field, $\partial_{\...
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2
answers
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How to raise and lower indices as a physicist would handle it?
Show that Einstein's equation
$$G^{\mu\nu}=R^{\mu\nu}-\frac12\mathcal{R}g^{\mu\nu}=\frac{8\pi G}{c^4}T^{\mu\nu}\tag{1}$$
can be written in the form
$$R^{\mu\nu}=\frac{8\pi G}{c^4}\left(T^{\mu\nu}-\...
- 476
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3
answers
239
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Why Lorentz transformations instead of general linear transformations?
I am trying to understand why, in physics, we look specifically at Lorentz transformations, instead of the larger group of general linear transformations.
To fix the terminology: let spacetime be ...
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42
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The different choises of a metric with the next form: $ds^{2}=-Adt^2+Bdr^2+Cd\Omega^2$
In some papers the starting point is a metric with the next form:
$$ds^{2}=-A(t,r)^{2}dt^{2}+B(t,r)^{2}dr^{2}+r^{2}d\Omega^{2}.$$
Like in the Schwarzschild metric where they choice A=exp and B=exp ...
0
votes
1
answer
63
views
How to combine Vierbein fields?
As far as I understand each Vierbein field $e^{a}_{\mu}$ and $e^{b}_{\nu}$ can be represented by a $4\times4$ matrix that can act on the Minkowski metric $\eta_{ab}$ to give the curved spacetime ...
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4
answers
2k
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How can coordinates be meaningless in General Relativity?
I am fairly new to the subject of General Relativity. While looking for answers to some questions I had about it, I came across this post:
Whose coordinates are the Schwarzschild coordinates?
One of ...
- 437
3
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1
answer
77
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Practical question related to reference frames in general relativity
Sorry if my question requires clarification. I am having trouble conveying exactly what my problem is.
I'm trying to code a ray tracer that works in curved spacetime. In principle, this just entails ...
- 437
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2
answers
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 ...
- 184
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2
answers
73
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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 $\theta$ ...
- 191
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1
answer
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
$$g = (- dt^2 + dx^2 + \sin^2 x \ d\Omega^...
- 685
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1
answer
52
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How to calculate the functional derivative of a product?
Let $\omega_{J}^{I}=\omega^{I}_{\nu~J}dx^{\nu}$ be a one-form connection with values in the Lorentz group $SO(3,1)$ and $$B_{J}^{I}=B^{I}_{\mu\nu~J}dx^{\mu}\wedge dx^{\nu}$$ a two-form with values in ...
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76
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How to demonstrate the form of the FLRW metric from homogeneity and isotropy?
In most introductions to the standard cosmological model, the metric of FLRW is stated to be corresponding to:
$$ds^2 = -c^{2}dt^2+a^{2}\left(t\right)\gamma_{ij}dx^{i}dx^{j}$$
which means (if I'm not ...
- 1,227
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vote
1
answer
66
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Question about electric flux in curved spacetime
I just started physics II (electricity and magnetism) and in learning about Gauss's Law, I came across this definition on Wikipedia:
$$\Phi_E=c\oint_{S}F^{\kappa0}\sqrt{-g}\ dS_{\kappa}$$
Where $c$ is ...
- 121
2
votes
0
answers
119
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How to get the field equations to match Oppenheimer's equations in his 1939 paper proving stars could collapse? [closed]
The famous 1939 paper by Oppenheimer (and Snyder) that proved that stars could collapse into Black Holes has me puzzled. I can’t get the signs to match in his first 2 field equations for the same ...
- 33
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votes
0
answers
78
views
A confusing tensor index calculation [duplicate]
We define Lorentz transformation as $\Lambda$ satisfying:$$\eta_{\alpha\beta}=\eta_{\mu\nu}\Lambda^{\mu}_{\space\space\alpha}\Lambda^{\nu}_{\space\space\beta}$$
And its inverse$$(\Lambda^{-1})^{\mu}_{\...
- 184
0
votes
1
answer
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 ...
- 239
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0
answers
77
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Non-metric gravity calculations
According to "Gravity and Strings" by T. Ortin (2015), the non-metricity tensor is calculated as
$$
Q_{\rho\mu\nu}\equiv\nabla_\rho g_{\mu\nu}=\partial_\rho g_{\mu\nu}-\Gamma^\beta_{\rho\mu}...
- 7
2
votes
1
answer
140
views
On the Background Independence condition
In General Relativity, one has that the equations of motion for any matter distribution are given by extremizing the following action:
\begin{equation}
S[g] = \int\left[\frac{1}{8\pi}(R - 2\Lambda) + ...
- 563
5
votes
1
answer
208
views
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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-3
votes
1
answer
103
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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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