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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31 views

We all accept/believe in C. But what happens when light travels away from the center of gravity of a heavy star? Does it slow down?

I'm basically trying to understand a mysterious characteristic of the universe. Why light has to travel at C. I understand and accept that from experiment. Not arguing that it does not. Just saying ...
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Is the relativistic time dilation compatible with the gravitational field of a moving massive planet?

Assume that a satellite orbits a massive planet/star at a distance $r^\prime$ away from its center in a circular path in plane $x^\prime y^\prime$. Suppose that the gravitational field is $g^\prime$ ...
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How the variation of Gauss Bonnet term is done?

I am trying to do the variation of Gauss Bonnet Invariant which is $G=R^2+R_{abcd}R^{abcd}-4R_{ab}R^{ab}$ and having problem in doing the variation of $δ(R_{abcd}R^{abcd})$. Can anyone please give ...
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Motivation for tensor theory of gravity

In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a ...
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Gravitational wave energy fluxes and bracket averaging notation

Typically the rate of energy loss due to gravitational waves is given in the form (e.g. MTW or Eq 33 of these notes), $$ \dot{E} = - \frac{1}{5} \langle \dddot{I}_{jk} \dddot{I}_{jk} \rangle$$ where ...
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67 views

Generating new solutions of the Einstein equation by active transformation, and the physical interpretation of the new ones

Given a manifold $\mathcal{M}$ with coordinates $\psi : \mathcal{M} \rightarrow \mathbb{R^4}$ , $\Psi(p)= (r,\theta ,\phi,t)$ for $ p \in \mathcal{M}$ Suppose we have the active transformation $F : \...
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Does light comeback again?

I know light follow the space time curvature.I listen about gravity lensing.Is there any proof of total reflection of light due to gravity like total internal reflection of light? If this possible ...
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What is the best comparison to understand how space is formed without resorting to the design of the temporary space blanket? [on hold]

For Einstein, space is like a blanket that curves according to the mass of the bodies, so objects should move up, sideways, and down very little, according to the mass of the bodies. This ...
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28 views

Variation of the Ricci tensor “squared” and antisymmetrization of the derivatives

I'm dealing with some extension of GR, with action: $S=\int d^4x\Big[\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})$ Varying this action gives: $\delta S=\int d^4x\Big[\delta\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})...
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Comparing Theory of Relativity with Newtonian gravity [duplicate]

How does Einstein's General Theory of Relativity compare with Newtonian gravity? How does the curvature of spacetime affect both gravity and the path of light in a planetary system?
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Is there somesort of “recipe” to check if an spacetime exibit the Unruh effect?

Firstly I don't know for sure if this question makes a solid sense, concerning the physical phenomena. I) Unruh Effect Now, we have then the Unruh effect which appears just by considering the ...
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Undertsanding gravitational time dilation in terms of four-vectors

I am struggling to understand how to calculate gravitational time dilation using four-vectors, based on my understanding of them. Specifically, I would appreciate if someone could point out the ...
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Why covariant derivative is partial derivative in SR?

I'm new in these topics and i've been confuse at some relations between the limit of SR for GR. In cartesian coordinates, basis do not change, so \begin{equation}\Gamma^{\mu}_{\alpha\beta}=0 \quad \...
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In a gravitational wave, why is the effect on path length due to the change in path negligible?

This is about Example 16.1 of Hartle's Gravity book, which considers a wave traveling along the $z$-direction given by $$ds^2=-dt^2+(1+f(t-z))dx^2+(1-f(t-z))dy^2+dz^2,$$ where $f(t-z)\ll1$. Of this ...
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If I surrounded a black hole with gravitometers and observed the gravitational field as mass passed the event horizon, what would I observe?

By surrounding a black hole with gravitometers I would be able to get a 3-D map of the gravitational field. If I observed this field as an object approaches the event horizon, what would I see? Would ...
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What is the Komar mass of the de-Sitter spacetime?

The Komar mass of some spacetime is defined as an integral (volume or surface, depending on its formulation): https://en.wikipedia.org/wiki/Komar_mass The de-Sitter metric in static coordinates is ($\...
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A form $F$ is simple if and only if $F\wedge F=0$?

Gravitation by Charles W. Misner, Kip Throne and John Wheeler page 93 Box 4.1 point 5 b. Applications: a. "In four dimensions, all 0 forms, 1- forms, 3-forms, and 4-forms are simple. A 2-form $F$ is ...
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Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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What does it mean when we say the kinematic space of the time slice of Ads3 is ds2?

I have been going through this paper Integral Geometry and Holography the authors in page 19 demonstrate the idea of kinematic space using $Ads_3$, they start off with a hyperboloid model and show ...
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How should be 'time' understood? [duplicate]

Is time an invention or discovery? How to relate time with the gravity via general relativity theory?
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Radial null geodesics in Schwarzschild de Sitter space

I am currently studying the geodesics of different type of spacetimes and I'm not sure if I'm doing it in the correct way for Schwarzschild de Sitter space (SdS). The metric in SdS is given by: $$ ds^...
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Metric (in)dependence of the electromagnetic field strength

In GR, the vector potential is defined as $A^\mu$ which is a contravariant vector. Then lowering the indices requires the metric $A_\mu =g_{\mu\nu} A^\nu$, using this vector one defines the field ...
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Direct derivation of point-like particle metric in GR

The usual way to derive metric of a point mass in general relativity is (to my knowledge) based on assuming specific form of the metric that reflects spherical symmetry and independence on "time" (...
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What is the correct form of Dirac equation?

Usually the Dirac equation in curved space is written as $$i\Gamma^{\mu}D _{\mu}\Psi-m\Psi=0,$$ where $\Gamma_{\mu}$ are curved space gamma matrices and $D_{\mu}$ is covariante derivative. This ...
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How to derive the Einstein-Skyrme equations?

I would like to derive the Einstein-Skyrme equations. The action can be read as \begin{equation} S[g,U] = \int d^{4}x\sqrt{-g}\biggl[R + \frac{K}{4}Tr\bigg(A^{\mu}A_{\mu} + \frac{\lambda}{8}F_{\mu\...
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An uncertainty principle within GR? [closed]

The core principle in quantum mechanics is the Heisenberg's uncertainty principle, that is $$\Delta(\text{position})\times \Delta(\text{momentum})\geq \text{positive constant}.$$ That principle ...
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Modern form of Brown-Henneaux formula

Almost every paper mentioning Brown and Henneaux's matching of asymptotic symmetries of AdS$_3$ with the Virasoro algebra of a $1{+}1$-dimensional CFT summarizes their results in the formula $$c=\frac{...
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Bohr Radius of Uniformly Accelerating Hydrogen Atom

Suppose we have a rocket-ship that is uniformly accelerating along its bow-stern axis (connecting line). Also, assume we have a hydrogen atom in the rocket. Mathematically (theoretically), does ...
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Is really there exists any kind constant that can explain expanding universe and increasing time?

I know far galaxies moving out from us I.e universe expanding I.e increasing space.There also time always growing up.From four dimensional analysis space time has covariance. I want to know is there ...
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Quantum waves as waves of space itself?

Are there any interpretations of quantum mechanics were quantum waves are waves of space itself? I'm not talking about gravitational waves, more like pilot-wave theory. There seems to me like it would ...
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Geodesic Equation from Coordinate Transformation

Let $\xi^a$ be the usual coordinates and $x^\mu$ the new coordinates, both flat. Now we know that since the metric is flat, $$ \frac{d^2\xi^a}{d\tau^2} = 0 $$ $$ \Rightarrow \ \frac{\partial}{\...
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Diffeomorphic but physically inequivalent spacetimes

In the last few years there has been a considerable endeavor in understanding the asymptotic symmetries of quantum gravity on Minkowski Spacetime. This has been tied to a study of the BMS group that ...
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Problem understanding boundary AdS problem with its time-like boundary

In Anti De Sitter space, light rays coming from the boundaries are important since they reach particles geodesics. This is explicit in the picture of the cylinder. The information in a slice of it ...
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Action in Electromagnetism expressed in differential geometry and tensor notation

$$ S = -\frac{1}{4} \int F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int F \wedge *F$$ Trying to figure out why this identity holds true and getting stuck.
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Explicit form of the Christofffel Symbol used in Geodesic Equation

One way to motivate the Christoffel symbol is to consider the partial derivative of a tensor, $T_\alpha$ $\frac{\partial T_\alpha}{\partial x^\gamma}=\frac{\partial^2 x^\beta}{\partial x^\alpha \...
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Is The Third Time Derivative of Position Relative, and if it is, Can it be represented in 6 Dimensions?

Einstein’s Theory of Special Relativity only applied to objects at constant velocities. This could be represented in a four dimensional Minkowski Space. From what I understand, Einstein compared ...
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How to find a normal to an hypersurface

I have to apply the Israel junction conditions in a region in which a hypersurface with O(3) symmetry separates two spacetime with Schwarzschild metric (with masses $M_+$, the exterior one, and $M_-$, ...
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Fluid dynamical analogs to General Relativity - perfect fluids and bulk viscosity

i have problems understanding the following two articles : https://arxiv.org/abs/0705.3882 - Viscous Spacetime Fluid and Higher Curvature Gravity https://arxiv.org/abs/0907.3180 - Viscous ...
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Counting independent components of the Riemann curvature tensor

In 4D spacetime, we may choose a locally inertial frame at point P, that is we always have a transformation such that $g_{{\mu'}{\nu'}}(P) = \eta_{{\mu'}{\nu'}}$ and its first derivatives vanish. ...
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How to get this metric tensor perturbation equation? (Gravitational wave)

(I need some time to come back to re-edit this post) LHS of equation 7.37 is gauge invariant The Twisted H symbol is comoving hubble parameter, the h_ij is metric perturbation, I do not know how to ...
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How to justifiy that $\rho_{\text{rad}} \approx \rho_{\text{mat}}$ at recombination time?

In standard cosmology, the recombination time is estimated to be $t_{\text{rec}} \approx 380~000~\mathrm{years}$ after the Big Bang, when matter and electromagnetic radiation becomes decoupled. It's ...
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Bounds on the size of the normal neighbourhood

One of the most important feature of general relativity is the existence of the convex normal neighbourhood, a neighbourhood $U$ on which the exponential map is a diffeomorphism between $\exp^{-1}(U)$ ...
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How can I show that the inverse of the induced metric $h_{\alpha \beta}$ is $h^{\alpha \beta}$?

So I was reading through Becker, Becker, Schwarz and there is a line in the second chapter that states that $h^{\alpha \beta} = (h_{\alpha \beta})^{-1}$ where $h_{\alpha \beta}$ is defined as: $$h_{\...
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What would the eternal black hole look like?

The white hole and black hole regions in a Kruskal diagram are said to be actually two different locations. Given the problems with white holes it might be a silly question but, hypothetically, what ...
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Different versions of the Robertson-Walker Metric

One form of the Robertson-Walker metric is $$ds^2 = c^2dt^2 - a(t)^2[d\chi^2+ S_k(\chi)^2(d\theta^2 + \sin^2\theta ~d\phi^2)]\tag{1}$$ $$\\$$ Considering curvature, where k = 0 , +1, -1 for flat, ...
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Leibniz Rule for Covariant derivatives

I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be, $\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
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Basic question about units of velocity and speed of a curve on a smooth manifold

Frederic Schuller says that velocity has units in Hertz in The WE-Heraeus International Winter School on Gravity and Light. He says: \begin{align} [v^a]&=\frac 1 T \\ [g_{ab}]&=L^2 \\ \Big[\...
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Changes to a Length of Physical Ruler Caused by Gravity vs Caused by Cosmological Expansion of Space

I read here (Feynmann Lectures, Lecture 42) that "Just as time scales change from place to place in a gravitational field, so do also the length scales. Rulers change lengths as you move around." (...
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Notion of Present

Can't I sync all watches in spacetime and call this time slice the present? In Carlo Rovelli's book he tried to explain that the notion of the present is local only, which I could not follow.
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About space/timelike intervals and event horizon of observed universe

How does interval between us (e.g earth) and the spacetime outside of observable universe (above the event horizon) can be described in terms of timelike and spacelike intervals? Is it just spacelike ...