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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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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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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157 views

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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How can the interior pressure of compact objects affect cosmology?

This paper suggests that dark energy concentrated in black hole interiors (they use an unconventional BH model) could act like a cosmological constant. Their claim is that to calculate the equation of ...
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Dirac bracket and Poisson bracket, asymptotic symmetry

I am reading the paper arXiv:9906126. https://arxiv.org/abs/gr-qc/9906126 on the symmetry algebra at horizon (see also well known work done by Brown and Henneaux about the asymptotic algebra of AdS$...
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Question about the travel time of a ship “using” a warp drive metric

I - The Warp Drive metric: The Warp Drive is a geometry in a spacetime $(\mathcal{M},g)$ given (in geometrized coordinates $c=G=1$) by the following metric tensor: $$ ds^{2} = -dt^{2}+ (dx-v_{s}f(...
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General Relativity - Confusion between choosing basis (orthonormal & coordinate) and coordinate transformations

I am reading the book 'Gravity' by Hartle and presently I am at the section discussing orthonormal and coordinate bases. I am confused about a few points I had read previously and can't exactly ...
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Confused about the gauge transformation of the amplitude tensor for gravitational waves

Far away from the field sources, where the energy-momentum tensor $$T_{mn}=0 \tag{m,n=0,1,2,3}$$ The linearized EFE becomes $$\Box \bar h_{mn}=0 \tag{1}$$ where $\bar h_{mn}$ is the trace-reverse ...
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100 views

Graviton propagator, and Gauss-Bonnet gravity

Let's say we consider Einstein's Lagrangian from GR. In linearized gravity, we would expand the Ricci scalar to quadratic order in the perturbation parameter to find the propagator. My question is as ...
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Doubt regarding cross sections of a Killing horizon

I was reading Rácz and Wald's paper 'Extensions of spacetimes with Killing horizons' and they are frequently referring to a cross section of a Killing horizon (a three surface where the norm of the ...
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158 views

Action & Energy-Momentum Tensor for Matter Fields

Pg 163 of "Tensors, Relativity and Cosmology" The action integral of a given matter distribution can be written in the form $$I_K=-c\int_\Omega\frac{\rho}{\sqrt{-g}}\frac{ds}{dt}\sqrt{-g} d\Omega ...
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What does it mean for gravity in $(2+1)$ dimensions to be topological?

I have been studying gravity in $(2+1)$-dimensions and I have come across the idea that gravity in this lower-dimensional spacetime is topological but I haven’t been able to find a simple explanation ...
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Evolution of area during the ring-down of a black-hole merger

My training is in nuclear physics, so my intuition is trained by the liquid drop model, in which the nuclear fluid is incompressible. This leads to the wrong intuition when I try to apply it to the ...
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Doubt about energy conditions: the Time-like Convergence Condition

First of all, consider a congruence of smooth time-like geodesics parametrized by proper time $\tau$. So, a tangent vector to a time-like geodesic is indeed a four-velocity up to a factor constant; ...
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108 views

How can one define a covariant rate of change of rest mass for an extended body?

For a point mass I can define a covariant rate of change of its rest mass as $\frac{d}{d\tau}m_0$ where $\tau$ is the proper time. How can I also define a covariant rate of change of rest mass for an ...
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Question about Ricci Rotation Coeficients

Standard General Relativity calculations lies under, indeed, the calculations of three quantities: Christoffel Symbols of second kind, the components of Riemann tensor $R^{\mu}\hspace{1mm}_{\nu \gamma ...
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What is knotted in EM and GR?

I found this paper with beautiful illustrations: Helicity, Topology and Kelvin Waves in reconnecting quantum knots, and this one which seems to describe something closely analogous: New knotted ...
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Is there any use of a quadratic equation of state in FLRW cosmology?

Consider standard FLRW cosmology. Usually, the relation between energy density $\rho$ and pressure $p$ of a cosmological fluid component is linear: \begin{equation}\tag{1} p = w \, \rho, \end{...
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An apparent limitation of the simplified diagrams used to describe space-time around a black hole

If you look at a 2D representation of curved space-time around a black hole, it stretches out into an infinite tube, asymptotically approaching the Schwarzschild radius. image from sciencenews.org ...
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Rindler Coordinates Derivation

In my GR lectures we've derived Rindler coordinates by first showing that the four velocity, which we defined as $$u^{\mu} = (\gamma c, 0, 0, \gamma u),$$ as a function of proper time can be written ...
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100 views

Free-falling stationary observer in curved spacetime?

Let us consider the pseudo-riemannian manifold $(\mathcal{M},g)$ with $\mathcal{M}=\mathbb{R}\times\mathcal{N}$ with $\mathcal{N}$ being a maximally symmetric, 3-dimensional riemannian manifold and $...
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How does one find the Vaidya black hole event horizon?

As for a definition, there are quite precise ones for what an event horizon is. One can define it as the boundary of the causal past of future null infinity, i.e., $\mathcal{H}=\partial J^-(\mathscr{I}...
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Difficult coordinate transformation

I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric $$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\...
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Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
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When is it possible to diagonalize a metric tensor with a coordinate transformation?

The question is a mathematical one, but I believe more likely to find here interested people, as it is relevant for GR. The problem arose for Kerr space-time and Boyer-Lindquist coordinates, where the ...
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Harmonic Oscillator and Shifts in Derivative Operators

What symmetries/symmetry breaking arises from shifts in the derivative operators? To explain what I mean let's study an example. The classical one particle one dimensional harmonic oscillator has the ...
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Do white holes encode the future evaporation of the black hole?

In General Relativity, white holes arise when one formulates Maximal extensions (the process by which coordinate singularities are mapped to new non-degenerate coordinates) to a geometric solution (...
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Elevator moving near the speed of light

Imagine that you are in an elevator that is moving downward at a constant speed that is near the speed of light (let's say close to the surface of the earth: toy problem, needless to say). Now say ...
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Constraint equation for Einstein-Hilbert action in Light-cone gauge

In Light cone coordinate system $(+,-,i)$, where $i=1,2$, the light cone coordinates are defined as $x^{\pm}=\frac{x^0 \pm x^3}{\sqrt{2}}$, if we consider the $+$ coordinate to be our "timelike" ...
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In a globally-hyperbolic spacetime, does every pair of elements have overlapping light cones?

Suppose we have a spacetime $(M,g)$, and denote by $J^+(p)$ the set of points that lie in the causal future of $p$, i.e. $x \in J^+(p)$ iff there is a future-directed timelike curve $\gamma: [0,1]\...
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Why spatial infinity is a point and not an $S^2$?

First a disclaimer, this question already has been asked here, but as pointed out in comments, more detail was required. So this is a more detailed version. Let $(\mathbb{R}^4,\eta)$ be Minkowski ...
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81 views

Why should LTB dust be comoving?

In many research papers about inhomogeneous cosmology, one often considers spherically symmetric (LTB) spacetimes where in the co-ordinate frame $(t,r,\theta,\varphi)$ wherein the metric assumes the ...
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Magnet pulling/repelling another magnet beyond the event horizon

I have read this question: If a Kerr-Newman black hole is like a charged, spinning, heavy magnet, what kind of magnet is it like? If you are carrying a magnet, can you tell when you cross the event ...
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When, and by whom, was the Schwarzschild metric first taken to be valid for all radii greater than zero?

The metric was originally defined to be valid only from the surface of a black hole outward but somewhere along the line it was extended inward to include the region under the event horizon. This ...
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Why doesn't a charge in gravitational field radiate (or free fall)?

I have read these questions: Why do accelerating electrons emit radiation? Why does an accelerated charged particle emit electromagnetic waves? How and why do accelerating charges radiate ...
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Derivation of Equation of Trajectory around a Kerr Black Hole

I was trying to derive equation of motion for test particle around a Kerr black hole. My work is as follows: The Kerr metric is as follows $$ \mathrm ds^2 = -\left(1-\dfrac{2Mr}{\rho^2}\right)\...
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Is there a satisfactory resolution to the Rotating Detector Puzzle?

The Rotating Detector Puzzle refers to the dilemma when trying to study how a rotating detector interacts with the background vacuum. The trail of literature that I found are these papers: p1, p2, p3, ...
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Difference between inertial and gravitational mass in general relativity due to self-contribution of gravitational potential

In the work The relativistic problem of several bodies, Am. J. Math. 59(1), 9 (1937), Levi-Civita addresses the problem of the motion of $n$ bodies in general relativity. He solves simultaneously ...
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Metric vs coframe energy-momentum tensor in metric-affine gravity

Conventions Latin indices represent components in the anholonomic frame and greek ones are for coordinate components. I will call $R_{\mu \nu} := R_{\mu \rho \nu}{}^{\rho}$ (Ricci tensor) and $\bar{R}...
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Nice expressions for geodesics in Penrose diagram for Schwarzschild spacetime?

If $u$ and $v$ are lightlike coordinates for 1+1-dimensional Minkowski space ($ds^2=dudv$), then we can compactify with a coordinate transformation of the form $$U=f(u)$$ $$V=f(v),$$ where $f$ is a ...
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Critical angle in General Relativity

Analogies between optical propagation in different refractive media and the effect of gravity in light geodesics are well established. But in optics one can have total internal reflection if certain ...
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Why does the Unruh effect not cause a physical contradiction?

If the Unruh effect says that the temperature of space depends on one's acceleration, such that an accelerating observer will observe that hir vacuum consists of a thermal bath at some elevated ...
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What would the physical consequences be if space-time was embedded instead of metric?

Suppose instead of using Riemann geometry, Einstein had postulated that 4D space-time was a 4D manifold embedded in a 6D space, and that gravity was a result of the shape of space-time in 6 dimensions....
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Geodesics and actions in affine gravity

It is a well known result in Riemannian geometry that geodesics are also the curves that minimize length (or extremize it anyway, I know what you're gonna say @0celo7), with a very similar result in ...
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223 views

Curl operator in Schwarzschild metric

I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
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Gauge transformation of trace-reversed metric perturbation

This question is in reference to Exercise 30.4.2 in Thomas Moore's A General Relativity Workbook, which asks you to show that a gauge transformation of the trace-reversed metric perturbation $H_{\mu\...
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Hawking Radiation of normal gravitational field

If the event horizon of a black hole can radiate, then is it possible that normal gravitational field also radiates? The radiation will probably be very low, but still there could be some. Also by ...
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Abstract definition of conjugate points

Let $S$ be a Cauchy hypersurface of a globally hyperbolic spacetime $(\mathcal{M},\mathcal{O},\mathcal{A},g,T)$ with unit normal vector field $n$. Define the exponential map on a neighborhood $U\...
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Variation of Einstein-Hilbert action in $D = 2$

I know that Einstein tensor is null in 2 dimensions (1 time and 1 spatial coordinate). But, from the action $$ \int d^2x \sqrt{-g} R $$ how can I prove that this action is pure divergence, i.e. it ...
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The usage of covariant coordinates in relativistic field theories

In the opening chapters of typical QFT books, the covariant coordinates $x_\mu = g _{\mu\nu}x^\nu$ $x^\mu = (t,x,y,z)$ and the differential operator $\partial^\mu = \frac{\partial}{\partial x_{\mu}}=(\...

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