A theory that describes how matter produces and responds to 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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Parity invariance of Einstein-Hilbert Lagrangian

How can we show that the Einstein-Hilbert action is Parity invariant? $$ S_{EH}=\int \sqrt{-g}R d^4x $$
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Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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Should a radiation-filled Universe be scale invariant?

Imagine a spatially flat Universe, without cosmological constant, filled only with EM radiation. As Maxwell's equations without charges or currents are scale invariant then should this Universe be ...
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Confinement of charged tachyons in AdS spacetime

It is well known that the negative cosmological constant of AdS spacetime can act like a confining potential. That is, in contrast to asymptotically flat spacetime, in an asymptotically AdS spacetime ...
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23 views

Stability condition for AdS background (when gravity coupled to matter fields)

In finding the stability condition for AdS background (when gravity coupled to matter fields), why the conserved energy should be positive?
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72 views

Is general covariance a symmetry?

Is general covariance a symmetry? If it is ,what is its symmetry group and corresponding generator?
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186 views

Wald problem 4 of chapter 4

I'm trying to derive equation 4.4.51 in Wald's GR book (the second order correction in $\gamma$ term for the Ricci tensor): where $g=\eta+\gamma$. So ...
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56 views

Dark Matter and Modified Gravity

Could someone please explain briefly or refer me to an article or manuscript that shows how f(R) modified gravity theories can be used to explain the problem of Dark Matter, particularly Galaxy ...
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49 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
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123 views

Geodesic distance in de Sitter space

Consider $N$ dimensional de Sitter space embedded in $N+1$ dimensional Minkowski space: $$\eta_{\mu\nu}X^\mu X^\nu=1, \hspace{1cm}\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$$ where I set $H=1$ for ...
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86 views

Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ ...
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138 views

The difference between an apparent horizon and event horizon?

I'm currently writing a project on minimal surfaces and general relativity - however I don't understand the difference between the apparent and event horizon? They ultimately both seemed to be defined ...
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Gravitational redshift of temperature and electrostatic potential

Consider a charged black hole in four-dimensional Minkowski spacetime, with charge $Q$, mass $M>Q$: $ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Omega_2^2$, with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$. ...
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116 views

No hair theorem and black hole entropy

The no hair theorem says that black holes rapidly converge to a state that is completely described just by their mass, spin and charge. Black hole thermodynamics says that the black hole entropy is ...
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Further explanation of the Penrose Conjecture

I'm currently a third year maths undergrad, writing a dissertation on the application of minimal surfaces in space. I have recently come across the Penrose Conjecture that the mass of a spacetime is: ...
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Homeomorphism between the space of all Ashtekar connections and spacetime?

Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, where $\mathcal{M}$ is spacetime. Then, let ...
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Trapped Surfaces. Any good articles?

I'm currently writing a dissertation on trapped surfaces as minimal surfaces. I have exhausted all of the resources I have, and the internet is pretty limited (in that it is fairly repetitive on just ...
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On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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332 views

Are there any good video lectures for learning general relativity at the level of Hobson?

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
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153 views

Gauss-Bonnet term in Physics

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
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Is the equivalence principle in General Relativity an approximation?

I read in web that Einstein used the principle of equivalence to explain General Relativity but we know the gravitation is approximately equal in all of rested frame in gravitional field. In ...
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149 views

Schwarzschild metric in a different coordinate system

In PADMANABHAN, Gravitation (Foundations and Frontiers), Cambridge, p $304$, exercice $7.6$, an example of the Schwarzschild metric in a different coordinate system is given : $$\mbox{d}s^2= ...
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132 views

Dirac equation in curved space-time with Torsion

I am looking for pedagogical references in which Dirac equation in space-time with curvature and torsion were discussed.
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122 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
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Is it mathematically possible or topologically allowable for cutouts, or cavities, to exist in a 3-manifold?

A few weeks back, I posted a related question, Could metric expansion create holes, or cavities in the fabric of spacetime?, asking if metric stretching could create cutouts in the spacetime manifold. ...
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Gravitational effects and metric spaces

Could somebody please explain something regarding the Nordstrom metric? In particular, I am referring to the last part of question 3 on this sheet -- about the freely falling massive bodies. My ...
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are pinch-off bubbles valid solutions to general relativity?

are bubbles of spacetime pinching-off allowed solutions to general relativity? With "pinch-off bubble" i really mean a finite 3D volume of space whose 2D boundary decreases until it reaches zero and ...
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704 views

Derivation of the Gauss-Codazzi equation

I'm interested in the derivation of the Gauss equation (Gauss-Codazzi). Usually we consider the definition of the Riemann tensor on the hypersurface. $$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c$$ ...
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216 views

Modification of de Donder gauge

The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge ...
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47 views

What is the metric of Vaidya black-hole horizon?

The metric of a Vaidya black hole in outgoing/retarted null coordinates are $$ds^2=-\left(1-\frac{2m(u)}{r^2}\right)du^2-2dudr+r^2\Big(d\theta^2+\sin^2\theta d\phi^2 \Big)$$ The eveolving horizon ...
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If $S$ is a closed achronal set in a spacetime, any timelike curve starting at a point in $I^+[S]$ and ending at a point in $I^-[S]$ interset $S$?

Suppose $S$ is an achronal set in a spacetime $M$. And $S$ is closed. At the same time, any null geodesic of $M$ intersects $S$. Then, why does any timelike curve from $I^+[S]$ to $I^-[S]$ intersect ...
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Calculation of Einstein Equation

I have a 3d system with Lagrangian $$e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q)$$ From this I want to calculate the Einstein equation by ...
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Deformation of light-cone

In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable ...
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In KK theory, is proper time defined using the 5 dimensional or the 4 dimensional line element?

Let's consider five dimensional KK theory. This is Klein's metric $\hat{g}_{AB}= \begin{pmatrix} g_{00}+A_{0}A_{0}&g_{01}+A_{0}A_{1}&g_{02}+A_{0}A_{2}&g_{03}+A_{0}A_{3}&A_ 0\\ ...
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Irreducible representation for the massless particle with helicity 2 and the Weyl tensor

As it can be shown, the equations for the irrep with zero mass and helicity 2, -2 respectively can be given in a form $$ \tag 1 \partial^{\dot {b}a}C_{abcd} = 0, \quad ...
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How to calculate the 2-point function of gravitons?

I'm curious about how to calculate the 2-point function of graviton, but there are no textbooks of general relativity covering this problem. So how to calculate it? In which book can I find the ...
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Avoiding Pseudo-tensors when addressing global conservation of energy in GR

Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way ...
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71 views

How to properly construct the electromagnetic tensor in curved space-time? (Part II)

In this question, I am testing what was previously discussed. I can't seem to get my results to match D'Inverno's electromagnetic tensor for a charged point (page 239 of his book - Introducing ...
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How to prove that a time-oriented spacetime possesses a nowhere vanishing timelike vector field?

Penrose gave a very brief proof to this question. Since the spacetime is paracompact, there exists a positive definite metric called $h_{ab}$. Then, the nowhere vanishing time-like vector field $V^a$ ...
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Local symmetry and General Relativity

First I want to consider an example of 1D motion. Lagrange equation: $$ \frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0 $$ If we transform $ L \rightarrow L+a $ ...
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63 views

A question on spin algebra

In scattering theory, one can form a lorentz invariant quantity by $\epsilon_{\mu 1 2\nu}P^{\mu}_{1}P^{\nu}_{2}$ which is really $1\otimes 1$ 's spin 0 state. Is there such a kind of argument to show ...
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Observers in (Schwarzschild-) de Sitter spacetime

In (pure) de Sitter spacetime, the cosmological horizon is said to be ‘observer dependent’. I imagine that as the observer always being in the center of that horizon. Another (spacelike separated) ...
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77 views

A Subtle Connection Between Time Dilation in SR and GR - Why is this so?

I've been reading a book on General Relativity lately (Gravitation and Cosmology, Weinberg), and I was reading about the weak field approximation. It derived the time dilation in a weak gravitational ...
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45 views

Geodesic Deviation between Test Particles from Gravitational Wave

I'm having trouble understanding how Carroll (Spacetime and Geometry p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, ...
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Physical Interpretation of four velocity in GR

I'm confused about the physical interpretation of the four-velocity $U^\mu=\frac{dx^\mu}{d\tau}$ in General Relativity. I know that it is a tangent vector to a particle's "worldline", but what does ...
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Is quantum gravity, ignoring geometry, the theory of a fictitious force?

This question is motivated by this question and this one, but I will try to write it in such a way that it is not duplicate. In short, I don't understand the motivation for a "quantum theory of ...
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What is the effective physical difference between a massive region of a polarized vacuum and a region of curved space-time?

What is the effective physical difference between a large region of curved space-time and an equally large region of a polarized vacuum? Consider the fact that vacuum polarization amounts to an ...
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An error in Gravitation by Misner Thorne and Wheeler?

I was studying on Gravitation the PPN formalism. Since in equation (39.41) pag. 1087, the term $1 + \dfrac{v^2}{2}+(2+\gamma)U = 1 + \dfrac{v^2}{2}+3U$ (the second in GR) looked odd, I tried ...
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$M^{+}_4$ Randall-Sundrum Brane Calculation

The basic Randall-Sundrum model is given by the metric, $$\mathrm{d}s^2 = e^{-2|\sigma|}\left[ \mathrm{d}t^2 -\mathrm{d}x^2-\mathrm{d}y^2 - \mathrm{d}z^2 \right]-\mathrm{d}\sigma^2$$ where $\sigma$ ...
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General covariance and global Poincaré algebras

Reading an article (page 7) I read this: Just as ordinary general covariance may be regarded as the local gauge symmetry corresponding to the global Poincare algebra and local gauge invariance ...