General relativity Definition and 999 Threads

Yesterday, 9/5/2025, when I was surfing, I found an article The Schwarzschild solution contains three problems, which can be easily solved - Journal of King Saud University - Science ABUNDANCE ESTIMATION IN AN ARID ENVIRONMENT...

Please see the derivation below that, if the metric is time-independent, then the integral over a sufficiently large 4D spacetime region of ##T^0_{\,0} \sqrt{-g}## is independent of time, where ##T^{\mu\nu}## is the energy-momentum tensor of a relevant collection of matter-energy fields. Thus, I...

In Dirac's "General Theory of Relativity", he develops the "comprehensive action principle" in chapter 30. Simply put, he writes a combined action for the gravitational field and all other matter-energy fields ##I=I_g+I'##. Varying this: $$\delta(I_g+I')=\int ( p^{\mu\nu}\delta g_{\mu\nu} +...

In Dirac's "General Theory of Relativity", chapters 26-30, he builds up various action principles from which Einstein's equation ##G^{\mu\nu}=-8\pi T^{\mu\nu}## can be obtained. In chapter 27, he extends the result of chapter 26 (Einstein's vacuum equation) to the case of a dust, where...

In the classical Newtonian theory of gravity, the shell theorem holds. ( https://en.wikipedia.org/wiki/Shell_theorem ) In the beginning of the derivation of the Schwarzschild solution, the spherically symmetric object is replaced by a point mass. The proof that this can be done in curved...

Hello everyone, I've been thinking about the standard physical picture of a neutron star reaching the TOV limit, and I've run into a conceptual question that I can't quite shake. I'd appreciate your perspective on it. The textbook explanation is a beautiful balance of our two great theories...

Background: In flat spacetime (special relativity), Maxwell's homogeneous equations $$\text{curl} \textbf{ E}=-\frac{d\textbf{B}}{dt} \qquad\qquad \text{div} \textbf{ B} = 0$$ can be written in the single equation $$F_{\mu\nu,\sigma} + F_{\nu\sigma,\mu} + F_{\sigma\mu,\nu} = 0 \qquad\qquad (*)$$...

Notation: The flat spacetime metric ##\eta_\mu\nu = \text{diag}(1,-1,-1,-1)##. Thus, the proper time element for a timelike path is ##ds##, and the proper distance element for a spacelike path is ##\sqrt{-ds^2}##. It is well known that ##\int ds## is stationary if and only if the path of...

See the screen shot below from L-L "Classical Theory of Fields" 4th Ed. p. 281. L-L choose a point ##x##, and work in locally inertial coordinates, so at the point ##x## the metric is constant: hence, ##g_{ik,l}=0##. The EM tensor ##T^{ik}## can be written in terms of the metric (and its 2nd...

Hi, I have a question about the behavior of light inside a black hole (inside the event horizon). Is it true that light coming from outside the black hole can be reflected once reaching a mirror inside the black hole? For instance, if an observer inside the black hole hold a mirror would he...

I have this problem: First of all, I would like to have an equation for the proper time. I was using the Eddington-Finkelstein coordinates in the Schwarzschild metric : $$ds^2 = - (1-\frac{2M}{r}) dt^2 + 2dtdr +r^2 d\Omega^2$$ Knowing that the observer falls radially, thus ##d\phi = d\theta =...

Hello everyone!:) This is my first post on here. I'm having trouble understanding what is real for Alice. I looked at different physics websites and there seem to be only a few questions addressing this some of which I don't understand. I thought maybe I'll get some clarity here if I ask my own...

I have problems understanding gravitons. I would understand them in the context of Newtonian gravitation. The Newtonian gravitational field is a force field and the gravitational interaction takes directly place between specific masses. It would therefore make sense to assume the existence of...

Although the speed of light is a 'cosmic speed limit', and is always constant, this seems to be a paradox. Firstly, the units we use to measure it relate to the Earth. How can we measure a metre or a second without the Earth as a reference point? And if we do, what meaning does this have for the...

The action for a free particle is $$I=-m\int{ds} = \int \left(-m\frac{ds}{dt}\right) dt \quad\quad\quad(*)$$ hence the Lagrangian is $$L=-m\frac{ds}{dt}=-\frac{m}{\gamma}=-m\sqrt{1-v^2} .$$ Dirac ("General Theory of Relativity", p. 52) infers this by checking that it gives the correct spatial...

Hi, I'm wondering if it is possible to calculate the volume of a black hole using the Schwarzschild metric. After computing the volume I get the follow integral: $$V = 4 \pi \int_0^r \frac{1}{\sqrt{ (1- \frac{r_s}{r'})}} r'^2 dr$$ This integral diverges at the upper bound. The only way I...

Is the redshift of a photon between two events independent of the path? I assume for this question that the the cosmological constant is zero. Chatgpt writes: "In a gravitational field, the redshift can depend on the path taken by the photon. For example, in a Schwarzschild metric (which...

I've already got "Personal Speculation Warning" and I had a private conversation with @Nugatory, who told me, that I have to publish it in an appropriate peer-reviewed journal before it can be posted for discussion here. I replied with a question, if he is willing to believe, that...

I would like to calculate the principal and Gaussian curvature of the spatial part of the Friedmann-Robertson-Walker (FRW) metric; specifically, the negative Gaussian curvature ##k=-1##. The FRW metric is, \begin{equation*} ds^2 = -dt^2 + R(t)^2 \left( \frac{dr^2}{1-k r^2} + r^2 d\Omega^2...

How did you find PF?: Google search Hello, I am a semi-retired management consultant specializing in procurement and supply chain. Most of my work involves collection and analysis of a company's procurement data and identification of value creation opportunities from initiatives such as...

The conventional picture We often describe two perspectives of infalling matter: Outside observers see the object asymptotically approach but never cross the horizon, with signals becoming increasingly redshifted The infalling object supposedly experiences a finite proper time to cross the...

TL;DR Summary: How can I variate the quadratic Riemann curvature tensor, I tried raising and lowering the indices. Hi, Can you help me with this variation, I tried raising and lowering the indices. I tried for months every possible method to reach the following answer without success.

Is it possible to make an embedding of the ##\phi##=constant slice of a Weyl metric in ##R^3##? In particular, I'm thinking of a metric where the components are both ##\rho## and ##z## dependent.

Looking at this paper, what sort of spatial topology change does the lorentzian metric (the first one presented) describe? Does it describe the transition from spatial connectedness to disconnectedness with time? All I know is that there is some topology change involved, but I don’t see the...

Hi everyone! A few days ago in General Relativity class, the professor introduced the concept of Lie derivative and at the end he mentioned that the Lie derivative was a tensor itself. I've been looking everywhere, but I only find how it acts on vectors, tensors, etc. Does anyone know of any...

Hello! I am a creator who loves the fascinating world of science. I enjoy exploring science from a unique angle, which often leads to new and creative ideas. While some of these ideas can be speculative and challenging to prove, I make sure they align with scientific principles and logical...

Assume a 2D XY grid, and we'll label the obvious directions N, S, E, and W. Assume an object is just going along a straight line from S to N in flat space with no forces acting on it or imparted to it, including rotational. Let's assume some kind of mark on it so we can keep track of...

I was wondering how the notion of a time-independent field translates into the context of General Relativity. In order to specify my confusion, consider a scalar field ##\phi## in Schwarzschild spacetime with usual coordinates ##(t,r,\theta,\phi)##. Its metric is $$g = - f(r) \, dt^2 + f(r)^{-1}...

It is well understood that an infinite monochromatic, circularly-polarized electromagnetic plane wave has no angular momentum density. However, a finite monochromatic, circularly-polarized electromagnetic plane wave packet does have an angular momentum density, arising from effects at the border...

I am trying to prove that in spherically symmetric spacetimes there are no nontrivial time-independent solutions to the Klein-Gordon equation (with mass ##= 0##) (**is this even true?**). My Ansatz is as follows: A spherically symmetric spacetime has metric $$g = g_{tt} \, dt^2 + g_{tr} \, dt...

Using the null geodesic and the Schwarzschild metric, this differential equation for photon trajectory near a mass can be derived, where u is r_s /2r: Though this nonlinear ode is fairly easy to approximate (which I already have), I'm looking for an analytic solution or an approximate...

Consider a halo made up from massive and stable particles like neutrinos* (let's not consider protons which, although we don't have any experimental evidence showing that they are unstable and decaying, there are some GUTs proposing theoretical mechanisms where they could decay over extremely...

This is page 73 of the book. As you can see, the mixed derivatives with the affine connections vanish in the second term. Why does that happen? This is used to prove that the connections are not a tensor, and i figured you could also reason it out even without making those terms vanish. OBS...

I understand that there are vacuum solutions of the equations of general relativity (GR) (i.e. equations with no mass-energy content contributing to the stress-energy tensor), that are studied by physicists interested in GR. I don't understand why these are studied or what purpose they serve...

Core Equation:I am examining the following equation: ## \frac{\partial \mathcal{A}_{\mu \nu}}{\partial g_{\rho \sigma, \tau \pi \lambda}} \frac{\partial g_{\rho \sigma, \tau \pi \lambda}}{\partial g_{\rho^{\prime} \sigma^{\prime}, \tau^{\prime} \pi^{\prime} \lambda^{\prime}}} = \frac{\partial...

In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation ##\delta(I_g+I_m)=0## of the actions for gravity and matter. Here ##p^\mu=\rho v^\mu \sqrt{-g}## is the momentum of an element of matter. He...

Hi, I'd like to discuss in this thread the propagation of Gravitational Waves (GW) in the context of GR. Just to fix ideas, let's consider a FW spacetime. It is not stationary (even less static), however the timelike congruence of "comoving observers" is hypersurface orthogonal. Suppose at a...

I read this question in the second edition of "Exploring Black Holes" by Wheeler, Taylor & Bertschinger. The book can be freely accessed and downloaded from the author's website. Chapter 17 deals with Kerr's solution on the equatorial slice ## \theta = 0 ##. In Doran coordinates, the 2+1 metric...

Would it be possible to eventually have structures made from neutrinos somewhere in the universe, as it is indicated in this question (https://physics.stackexchange.com/questions/80390/are-neutrino-stars-theoretically-possible), like halos of neutrino gas surrounding the center of galaxies...

Would you buy these books?

Hi! I'm going into college during the fall, (Stony Brook University in the US) and I want to research general relativity in the future. I can choose to do an astronomy or math double major alongside my physics degree, and I want to know which degree will best prepare me for a graduate program in GR

suppose the enterprise departs from planet earth on a mission the the other side of the milky way at 90% of the speed of light; since time is dilated while cpt Kirk drinks coffe on planet earth some time passes (let's say 1h). Approaching mars captain Kirk orders dt. Spock to land on Mars with...

Hi, I was thinking about the claim that for instance Sagittarius A* (Sgr A*) black hole is a at 26996±29 light years from the Earth from a GR point of view. Assuming a FLRW model for the Universe, maybe the above meaning is that at a given cosmological time ##t## (the "present" time) the proper...

Black holes accrete mass around them and it falls gradually up to the even horizon where mass is trapped by the black hole forever. However, the rate of mass falling from the accretion disk to the black hole ranges from being very fast to very long-lived, depending on various conditions...

A Relativist's Toolkit (2004) lists the Israel junction conditions as: ##1. [h_{ab}]## ##2. S_{ab}=[K_{ab}]-[K]h_{ab}## Where ##S_{ab}## is the stress-energy tensor of the shell only, and ##[K_{ab}]## and ##[K]## are ##K_{ab}^--K_{ab}^+## and ##K^--K^+## respectively. My understanding is that...

Roy Kerr has recently written a preprint (https://arxiv.org/abs/2312.00841) in which he strongly argues against the possible existence of singularities inside Black Holes. I've read that his arguments are really powerful and that he is most likely right. But, does it mean that Kerr has...

When extending general relativity to include electromagnetism, several authors (e.g. Novello, Sabbata ecc.) assume that the traceless part of the torsion tensor vanishes or is deliberately set to zero. Then, either the trace or axial part of the torsion is used in association with the...

Hi, I'm aware of the measured recession of the galaxies in our universe and the universe expansion itself cannot be understood as an "ordinary" velocity/speed (for instance in the FRW solutions of Einstein's equations). Can you kindly help me clarify this topic ? Thank you.

According to some papers I've found [1], [2] expanding voids can be found inside clouds of denser materials that can cause them to eventually collapse. I have a question about this: Overdensities generally expand up to a given turnaround radius and then collapse. However, as the elements in the...