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Questions tagged [metric-tensor]

The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.

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Second-order perturbations of gauge field in GR

When expanding a Lagrangian $\mathcal{L}[g_{\mu\nu},A_\mu,\chi]$ to second order in perturbations, the metric is expanded like $$g_{\mu\nu}\to g_{\mu\nu}+\delta g_{\mu\nu}+\frac{1}{2}\delta g_{\mu}^{\,...
furious.neutrino's user avatar
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Finding coordinate transfromations by line element

I’m confused on how to generally approach these coordinate transformations: I initially thought we can set $dT^2=(dt-b/2dx)^2$ and $a^2dX^2= (a^2+\frac{b^2}{4})dx^2$. This way, when carrying out the ...
David's user avatar
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In which direction is the relation between the time-component of celerity and the Lorentz factor defined?

Celerity (a.k.a. proper velocity) is defined as $w^\alpha=\frac{\mathrm{d}X^\alpha}{\mathrm{d}\tau}$, where $\mathrm{d}X^\alpha=(\mathrm{d}t,\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$ and $\mathrm{d}\tau$ ...
controlgroup's user avatar
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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 ...
elika kohen's user avatar
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Is the tensor product involved in the metric a symmetric product?

The expression of the FRW metric in Cosmology in usually written as: $$ds^2=-dt^2+a^2(t)d\vec{x}^2$$ where $c=1$. However, $dt^2$ is a shortening of $dt\otimes dt$, that is, of the tensor product of $...
Wild Feather's user avatar
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Question coming from Cosmological Perturbation

We consider the following scalar perturbation on the FRW metric: $$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$ where $\...
Shivam Mishra's user avatar
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Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?

I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$. In doing this, letting the derivative ...
user410662's user avatar
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 ...
morpheus's user avatar
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Is there a metric, a solution to Einstein's field equations, for a single body in a space of uniform non-zero density?

The Swarzschild metric describes a single body in an empty space with zero density, while the FLRW metric is presumably for a space with uniform non-zero density but no single body. But is there a ...
John Hobson's user avatar
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Derive Minkowski metric from Lorentz transformation

I am trying to learn special relativity. My goal is to prove that given the fact that a 4-vector $\mathbf{x}$ is transformed as $\mathbf{Lx}$, between two inertial reference frames where $\mathbf{L}$ ...
morpheus's user avatar
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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 ...
UnkemptPanda's user avatar
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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 ...
Kimaya Deshpande's user avatar
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Extrinsic Curvature Calculation on the Sphere

Given the following 2+1 dimensional metric: $$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
Daniel Vainshtein's user avatar
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Interpretation of degenerate metrics

I was studying the metric tensor and saw all about degenerate metrics. I would like what is the physical or geometrical intuition of a degenerate metric. What is the meaning of $g(v,w) = 0$ for a ...
JL14's user avatar
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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 $\vec{a}$ ...
Solidification's user avatar
1 vote
1 answer
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Weyl transformation of induced metric

Consider the Weyl/conformal transformation in four dimenions $$\tilde{g} \enspace = \enspace \Omega^2 g \quad \Longrightarrow \quad \sqrt{-|\tilde{g}|} \enspace = \enspace \Omega^4 \sqrt{-|g|}$$ The ...
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Physical intuition for the Minkowski space?

As the title suggests, I am looking for physical intuition to better understand the Minkowski metric. My original motivation is trying to understand the necessity for distinguishing between co-variant ...
user10709800's user avatar
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1 answer
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Checking inverse metric and Christoffel symbols for the Kerr metric against references

I am trying to cross-check the Christoffel symbols and other very laborious geometric components in several metrics. In particular the Kerr metric is notoriously complex and results in expressions ...
UnkemptPanda's user avatar
2 votes
1 answer
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Saddle Shaped Universe

The universe, as described by FLRW metric, if $k = -1$ is clearly a 2 sheet 3-hyperboloid described by $x^2+y^2+z^2-w^2=-R^2$. So where does the more common saddle shaped picture of the open universe ...
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Embedding diagram of $\phi=\mathrm{constant}$ surface in cylindrically symmetric spacetime

I'm trying to generate an embedding diagram for the $\phi=\mathrm{constant}$ hypersurface in a cylindrically symmetric spacetime. I think I'm supposed to start with $$A(p,z)dp^2+A(p,z)dz^2=dw^2+dp^2+...
user345249's user avatar
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3 answers
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Photonic black holes

"Can a photon turn into a black hole?" - usually the answer to this question is - it can't, because it has zero rest mass. However, when we derive the Schwarzchild Metric initially the $2M$ ...
Nayeem1's user avatar
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Homogeneous and Isotropic But not Maximally Symmetric Space

Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
Nayeem1's user avatar
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4 votes
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Constant curvature on a sphere?

$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
Nayeem1's user avatar
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2 votes
1 answer
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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 ...
amansas's user avatar
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1 answer
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?

I want to experiment with this relation (from Dirac's "General Theory of Relativity"): $$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$ using the electromagnetic Lagrangian $L = -(...
Khun Chang's user avatar
1 vote
1 answer
108 views

What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?

The Hamiltonian constraint of General relativity has the following form \begin{align} \frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{...
Faber Bosch's user avatar
2 votes
1 answer
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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 ...
Aidan Beecher's user avatar
2 votes
1 answer
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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 ...
Aidan Beecher's user avatar
2 votes
0 answers
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If a slice of a 4 dimensional metric violates an energy condition, does the 4 dim metric violate it aswell?

I 'm currently studying analogue gravity see this paper for a review. Here a 2+1 dimensional metric is derived: $$ ds^2 = -dt^2 + (dr - \frac{A}{r} dt)^2 + (r d\theta - \frac{B}{r} dt)^2 $$ Now it ...
DifferentialgeometryCrusher123's user avatar
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2 answers
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How to prove $ g^{\mu\nu}\Lambda^{\rho}{}_{\mu}\Lambda^{\sigma}{}_{\nu}=g^{\rho\sigma} $ for the inverse metric?

In Srednicki's book, we have \begin{align*} g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma=g_{\rho\sigma} \end{align*} and let $ \Lambda \to \Lambda^{-1} $, use the relationship $ (\Lambda^{-1})^\...
liZ's user avatar
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1 answer
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Conformal equivalent to Schwarzschild metric

Consider Schwarzschild spacetime in Eddington-Finkelstein coordinates $(v,r,\theta,\phi)$ $$g \enspace = \enspace -f(r) \, dv^2 + 2 \, dv \, dr + r^2 \, d\Omega^2 \quad , \qquad f(r) = 1 - \frac{2m}{r}...
Octavius's user avatar
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Are the quantity $\frac{\delta \sqrt{-g}}{\delta g_{\mu \nu}}$, $\frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} $ are computable? [duplicate]

From $\delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g} g_{\alpha \beta}\delta g^{\alpha \beta} = \frac{1}{2}\sqrt{-g} g^{\alpha \beta} \delta g_{\alpha \beta}$, can we compute \begin{align} &\frac{\...
phy_math's user avatar
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1 vote
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Can a vector tangent to a spacelike surface be null?

I'm studying the peeling-off behaviour of zero rest-mass fields, as described in Penrose's paper. In it, he talks about the boundary $\mathscr{I}$ of the conformal completion of an asymptotically ...
Smikkelma's user avatar
2 votes
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What is the topology of a spacelike cross-section of a null hypersurface?

I'm studying the peeling-off behaviour of zero rest-mass fields, as described in Penrose's paper. In it, he talks about the boundary $\mathscr{I}$ of the conformal completion of an asymptotically ...
Smikkelma's user avatar
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77 views

Graviton metric interaction

In most discussions on quantum-gravity, graviton is considered as a perturbation that is being added linearly to a flat metric (the $h_{\mu\nu}$ term in $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$). ...
physics_2015's user avatar
2 votes
0 answers
52 views

Doubt about metric tensor definition in Weinberg's Gravitation and Cosmology

In Weinberg's Gravitation and Cosmology, by using the Equivalence Principle, the author defines the metric tensor as (equation 3.2.7) $$ g_{\mu \nu} \equiv \frac{\partial\xi^\alpha}{\partial x^\mu} \...
lcostal's user avatar
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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 \...
Sanjana's user avatar
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3 votes
1 answer
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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 ...
V Govind's user avatar
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2 votes
1 answer
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Building a Lorentzian metric from an induced metric in Euclidean space

As an exercise, I wanted to apply the tool of taking some "time-evolving" surface embeddable in Euclidean space, defined parametrically as $X_0(u, v, t), X_1(u, v, t), X_2(u, v, t)$, and ...
UnkemptPanda's user avatar
6 votes
1 answer
160 views

Radial reparametrization ansatz in Schwarzschild metric derivation

The standard derivation of Schwarzschild solution (and Birkhoff's theorem) seem to begin with the most general spherically symmetric static metric $$ds^2 = -U(\rho) dt^2 + V(\rho) d\rho^2 + W(\rho) \...
UnkemptPanda's user avatar
1 vote
1 answer
77 views

Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice

In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
Alex's user avatar
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3 votes
0 answers
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Connection between the metric tensor and mass

The general expression of a line element in a space with metric tensor $g_{\mu \nu}$ is $$ds = \sqrt{ g_{\mu \nu} dX^{\mu} dX^{\nu} }$$ If we consider a curve $X^{\mu}(\tau)$ parametrised by $\tau$, ...
pll04's user avatar
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Why does the line element expression contain only second order differential terms? [duplicate]

The general expression of the line element $ds^2$ is $$ds^2 = g_{ij}dX^{i}dX^{j},$$ where $g_{ij}$ is an element of the metric tensor. Is there a rigorous proof of why there are no terms in the ...
pll04's user avatar
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2 votes
3 answers
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Sign conventions for the Lagrangian from the EM Lagrangian density

In Chapter 13.6 of the 3rd edition of Goldstein's Classical Mechanics, Goldstein proposes the Lagrangian density of the electromagnetic field as: $$\mathcal{L} = -\frac{F_{\lambda \rho} F^{\lambda \...
tugboat2's user avatar
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1 answer
104 views

Interior Solution for Black Hole in Particular

This paper seems to suggest that the interior metric for a black hole in particular (a.k.a not a different kind of spherically symmetric non-rotating body) is just the exterior Schwarzschild metric ...
user345249's user avatar
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: $$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$ ...
Wild Feather's user avatar
1 vote
0 answers
68 views

In the frame field construction in GR, how do you get the vector field dual to a co-frame?

I am trying to understand the frame-field construction in General Relativity. We basically have four point-wise orthonormal vector fields, one of them being timelike and the other three being ...
Moustafa M. Kamel's user avatar
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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: ...
emilio grandinetti's user avatar
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Derivation of measure for summation over surfaces, including the polyakov action

In his 1981 paper "Quantum geometry of bosonic strings" Polyakov defines a measure for the summation over continuous surfaces. This measure must count all surfaces of a given area with the ...
Jens Wagemaker's user avatar
3 votes
1 answer
81 views

Circumference of ellipse in post-Newtonian metric

The post-Newtonian metric, in harmonic coordinates, is: $$\tag{1} \mathrm{d}s^2=-\left(1+\dfrac{2\phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\dfrac{2\phi}{c^2}\right)\mathrm{d}\mathbf{x}^2$$ where $\...
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