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Questions tagged [general-relativity]

A theory that describes how matter interacts dynamically with the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS.

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Usages of transition maps on general relativity

On the initial courses of topology and differential geometry, we learn again and again about charts, and atlas, and transition maps. I feel that transition maps are a very powerful idea, because they ...
UnkemptPanda's user avatar
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Covariant derivatives of geodesic distance

The geodesic distance is defined as follows: $$\sigma(y_1,y_2)=\int_0^1 d\lambda \sqrt{\frac{d x_\mu dx^\mu}{d\lambda d\lambda}}$$ with the boundary condition $x(0)=y_1$ and $x(1)=y_2$ and $\lambda$ ...
grodta's user avatar
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Two contradictory derivations of Killing equation

In David Tongs lecture notes he derives the Killing equation by showing that the charge $Q=\xi_\mu \frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}$ is conserved $$ 0=\frac{\mathrm{d}Q}{\mathrm{d}\tau}=\frac{\...
Silas's user avatar
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Spatial Curvature of Universe at recombination vs now

From my understanding, we use the CMB data to measure the spatial curvature of the universe today. Why is it the value for today if the CMB data reflects the universe at recombination (380K years ...
KaraboMadisa's user avatar
-3 votes
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On definition of time in non-inertial frames [closed]

Strictly speaking we can only compare times and lengths in inertial frames of reference because the measuring tools are in the same conditions and aren't subject to G-force. So they are identical ...
Марат Рамазанов'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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Energy of the gravitational field within a sphere of radius $R$ in the Schwarzschild metric

The Landau-Lifshitz energy-momentum pseudotensor $t^{μν}$ is defined by $$16πt^{μν} = -2G^{μν} - g^{-1} \left[ -g \left( g^{μν}g^{αβ} - g^{μα}g^{νβ} \right) \right]_{,αβ}$$ where $g=\text{det}[g^{μν}]...
Khun Chang's user avatar
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Are Christoffel symbols in Schwarzschild metric symmetric?

Schwarzschild metric is following $$ ds^2 = (1-\frac{r_s}{r})c^2dt^2 -\frac{dr^2}{1-\frac{r_s}{r}} - r^2d\theta^2 - r^2\sin^2(\theta)d\phi^2 $$ Metric tensor has only diagonal terms non zero from this ...
Lexorde's user avatar
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What is the gravitational field of a hole in an infinite perfect crystal?

Or equivalently and more interestingly: In the early universe when there was uniform H/He gas everywhere, gravitational field was close to 0 everywhere. Every test particle was pulled from all sides ...
Alien from future's user avatar
3 votes
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Analog between Electromagnetism and Gravity

Feynman makes an analogy between EM field and gravity field in his Feynman's Lectures on Gravitation. The vector field representing EM potential would couple to the current source(vector) in the ...
Ting-Kai Hsu's user avatar
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Memory effects from black hole migration in the early universe

Let's say that there is a black hole at the center of a galaxy. When this galaxy is merging with another galaxy, it moves through space. If I am looking at a black hole now, there is a certain region ...
Kaushal Timilsina's user avatar
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1 answer
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David Tong, notes on General Relativity, pg. 25

I was studying GR using David Tong's notes, which I find very compelling and easy to read. The material is explained clearly to a fault, and I would recommend anyone picking up GR to at least skim ...
Filipp's user avatar
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Varying $\Box h_{\lambda\kappa} \Box h^{\lambda\kappa}$ with respect to $h_{\mu \nu}$ [closed]

I'm trying to gain a working understanding of the basic calculus of variations used in field theories, and I'm a little stuck trying to understand a step I've seen in a derivation. I'm sure my ...
cosmologia's user avatar
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Solving TOV equations that describes neutron stars in modified f(R, T) gravity

Sorry for the long post, tldr at bottom. I'm trying to use standard RK4 code in C/C++ to solve a coupled system of 2 modified TOV equations in f(R,T) gravity and reproduce some of the results of this ...
hidenori's user avatar
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Problem with linearized gravity on flat background in spherical coordinates

I am solving the linearized Einstein equations with a flat background in spherical coordinates, i.e $ds^2 = -dt^2 + dr^2 + r^2 d\Omega^2$ and writing $h_{\mu\nu}$ in terms of spherical harmonics. ...
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1 answer
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Question on spatiotemporal dimensionality about the contradictions of time being a dimension

We can axiomatically see that all spatial dimensions have a fundamental rule where they can either move back or forwards infinitely. However, the temporal dimension started when the universe began and ...
Mason Kang's user avatar
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GR and Riemann Surfaces -- does the complex plane have anything to do with it?

I have only the vaguest understanding of Riemann Surfaces -- my sense is that Einstein used them in General Relativity because of their shape. But Riemann Surfaces I think are not just deformations of ...
releseabe's user avatar
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Is tetrad postualte independent of gauge field?

Here is what I know, $g_{\mu \nu} = e^{a}_{\mu} e^{b}_{\nu} \eta_{ab}$ and the tetrad postulate is \begin{equation} D_{\rho} e^{a}_{\nu} = \partial_{\rho} e^{a}_{\mu} - \Gamma^{\lambda}_{\rho \nu}e^{a}...
trying's user avatar
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As per Friedmann Equations, is big-bang singularity necessary?

The Friedmann Equations do not directly require that the scale factor $a(t)$ was zero in the beginning. Since Einstein's static universe is still a valid solution, is it possible that before the Plank ...
Nayeem1's user avatar
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Viable values for the $K$ parameter in the FLRW metric

The FLWR metric is sometimes given as $$c^2 d\tau^2 = c^2 dt^2 - \frac{a(t)^2}{(1-KX^2)} dX^2. $$ I am not interested in the tangential motion so I set $d \Omega = 0$ although it is of interest in ...
KDP's user avatar
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2 votes
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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
1 vote
2 answers
39 views

Cone vs. small circle parallel transport

I'm having trouble reconcile the following two seemingly contradicting conclusions (in 2d space): A cone is flat, because you can unfold it and it's a flat 2d surface. A cone as shown in the picture ...
Cosmo's user avatar
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Confused about Weinberg's result of gravitational time dilation

I am reading Weinberg's Gravitation and Cosmology. In section 3.5, the author got a result$$\frac{dt}{\Delta t}=(-g_{00})^{-1/2}\tag{3.5.2}.$$Here $dt$ is the time interval of a stationary observer in ...
rioiong's user avatar
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9 votes
1 answer
913 views

Does the cosmological constant entail a mass for the graviton?

If I consider the Einstein equations into the form $$ R_{\mu\nu}=\kappa \left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)+\Lambda g_{\mu\nu} $$ and then linearize them, we should get by moving to ...
Jon's user avatar
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1 vote
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Does gravity accelerate you towards the geodesic of light between you and the mass?

If there's a planet far away, you will accelerate straight towards it due to gravity. If you place a Schwarzschild black hole right in the middle between you and the planet (the distance between the ...
Zach's user avatar
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1 vote
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53 views

Orthogonal complement of null vector [closed]

I am trying to solve excercise 8.5 d) from Straumanns book on general relativity (here V is an n-dimensional Minkowski vector space): Prove that the orthogonal complement of a null vector is an $(n-1)$...
Lenti's user avatar
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3 votes
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Negative (absolute - not potential) energy of the gravitational field; how to generalize to GR?

Alan Guth gives a thought experiment to show that a gravitational field has negative energy. (See the picture below.) Consider a thin spherical shell of elastic, compressible matter, of radius $R_o$. ...
Khun Chang's user avatar
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ADM Formalism for the Effective String Theory

We will consider the following effective action of string theory in leading order of $\alpha'$: $$S=\frac{1}{2\kappa^2_0}\int d^{D}X\sqrt{-G}e^{-2\Phi}\left[R-2\Lambda-\frac{1}{12}H_{\mu\nu\lambda}H^{\...
Daniel Vainshtein's user avatar
-14 votes
1 answer
118 views

Do satellites in orbit create Relativity paradoxes? [closed]

Can someone point out the flaw in this very realistic scenario below? I will start by stating established first principles of the applicable orbital and relativistic conditions. Then I will describe ...
Anakin Skywalker's user avatar
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0 answers
31 views

Why are laser beams not affected by gravitational waves in Michelson interferometer? [duplicate]

Michelson interferometers are used to detect gravitational waves. This means two laser beams are sent in long arms and reflected at the end. Usually, they annihilate each other and no signal is ...
OpenSourceOrDie'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
4 votes
0 answers
92 views

Lorentz force error in the present 2024 version of the gravitoelectromagnetism Wikipedia page? [closed]

I noticed that the Gravitoelectromagnetism (GEM) Wikipedia page has been edited recently. The factor of 4 in the GEM Lorentz force equation is now missing. But the GEM field equations are identical to ...
rdryne's user avatar
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2 votes
3 answers
126 views

Can a light signal from Earth reach a galaxy outside the Hubble Horizon?

Is this video on the FLRW metric (timestamp 29:00 minutes) mistaken in its claim that a light signal from Earth cannot catch up with a galaxy outside the Hubble horizon, due to the horizon receding at ...
KDP's user avatar
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1 vote
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+100

Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion

I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding. The starting form of action (3-Dimensional) is: $$ S_{...
StarPlatinumZaWardo's user avatar
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72 views

Where is the mass in a Black Hole without a "central" curvature singularity?

Not all black holes have a curvature singularity at their center (an example). But in principle, I thought that the curvature singularity was a direct result of the fact that the mass is concentrated ...
Aleph12345's user avatar
2 votes
1 answer
73 views

How to properly combine kinetic and gravitational time dilation effect?

I developed a time dilation calculator that includes both kinetic (Lorentz Factor) and gravitational (Schwarzschild Metric Formula) factors to assess the time difference between Earth and satellites. ...
Eliot Mallamo's user avatar
8 votes
5 answers
1k views

Do you always experience the gravitational influence of other mass as you see them in your frame?

You see a galaxy far away. That galaxy is attracting you with a certain amount of gravity. I'm wondering if the gravity influence of the galaxy on you, as measured by you, always ends up being what ...
Zach's user avatar
  • 171
0 votes
1 answer
148 views

Energy-momentum tensor and equation of motion in Einstein-Dilaton theory [closed]

I am following this paper (see eq. 19-22) and trying to derive the equation of corresponding to Einstein-Dilaton gravity (ignoring the Maxwell part for now) \begin{align} S_{\text{E-D}} = \int d^4 ...
Faber Bosch's user avatar
3 votes
2 answers
86 views

Tug of war between observers in frames with different rate of time

You have a very dense hollow sphere of matter. Observer A is inside the sphere inside a rocket. Observer B is in an identical rocket outside the sphere where the ring's gravity is negligible. They are ...
Zach'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
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Independence of the equations resulting from the action principle $\delta (I_{\text{gravity}} + I_{\text{other fields}}) = 0$

In Dirac's "GTR", Chap. $30$, he discusses the "comprehensive action principle" and shows that variation of the combined action of the Hilbert-Einstein action plus all other matter-...
Khun Chang's user avatar
2 votes
1 answer
109 views

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 ...
Nayeem1's user avatar
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1 vote
3 answers
100 views

Do clocks tick faster when gravitational forces are weaker?

A professor last year taught us that "gravity slows clocks," when teaching about the relationship between gravity and time. This led me to think about places, such as intergalactic space, ...
William Solomon's user avatar
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0 answers
52 views

Action principle dependent on spacetime-topology?

Consider the Lagrangian density $$L(\phi, \nabla \phi, g) = g^{\mu \nu} \nabla_{\mu} \phi \nabla_{\nu} \phi$$ If one varies the action as usual, then one finds the equation $$\delta S = \int_{\mathcal{...
Octavius's user avatar
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1 answer
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What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$

What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$ In John Dirk Walecka's book 'Introduction to General Relativity',...
Jianbingshao's user avatar
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0 answers
43 views

Radiation energy and momentum relation [duplicate]

Why is $\rho = 3p$ for radiation? What is the intuition behind this? If we had only 2 spatial dimentions, would it be $\rho = 2p$? (I came across this relation while studying the state of the universe,...
Nayeem1's user avatar
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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
3 votes
3 answers
180 views

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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0 answers
25 views

Do the Komar/ADM mass equations also hold in 2+1D?

All definitions I have come across for the ADM mass require asymptotic flatness, which always is defined for 4 dimensional spacetimes. I was wondering if these formulae in 3+1D hold in 2+1D aswell?
DifferentialgeometryCrusher123's user avatar
2 votes
1 answer
74 views

Doppler Effect and the concept of relative velocity in GR

While reading Sean Carroll's book on General Relativity, I understood that the concept of velocity is ill-defined over large distances in arbitrarily curved manifolds, like the one used to describe ...
V Govind's user avatar
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