General relativity Definition and 999 Threads

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...

CMB photons can be affected by the expansion of the universe through the linear integrated Sachs-Wolfe effect (ISW) [1] and the non-linear ISW effect or also called Rees-Sciama effect [1]. In particular, according to the ISW effect, the photons crossing superclusters would leave them having a...

I cannot get the following out of my head. Suppose this situation. Three frames, with varying velocities, simultaneously intersect their origins at the same time and place, making this point and time x0=0 and t0=0. These frames... let's call them observers. These observers have an agreed upon...

After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative. On the internet I found equation (3), and I have tried to understand where this comes from...

Hi, In Einstein's famous elevator experiment, someone in the elevator cannot tell if the acceleration they experience is from the gravity of a nearby large mass, or from their own change of velocity under the influence of some external force. But if there is an external force accelerating...

I’m trying to understand the Hellings and Downs curve that is being used to argue for the existence of a gravitational wave background ([NANOGrav article][1]). How can it be that the angle between two pulsars is the only variable that determines if the gravitational waves will interfere...

Is energy conserved in general relativity? I have read most of the posts here that address this. But it isn't clear to me, what most people say is that energy is conserved locally but it can't be defined globally, some people say this means that energy is not conserved in GE while others argue...

Other ways of wording this finding about the extended SC (Schwarzschild) spacetime: - in the local frame of a free faller, radial distance is given r coordinate difference - in either Fermi-normal coordinates, or Riemann-Normal coordinates built from a free faller at an event, coordinate...

Hi all, I've been going over some special relativity as it's a topic I never really studied during my younger years and wanted to get to grips with it, especially since it's such a fundamental part of our understanding of the cosmos. I was reading about Einsteins train lightning thought...

Does it exist an invariant way to define acceleration in Newton physics like the proper acceleration in GR ? In Newton physics if an accelerometer attached to an object reads 0 it does not mean it is actually not accelerating (since gravity is a force). To define inertial motion the concept of...

In the Lamda-CDM model of cosmology, dark-energy is explained by a Lamda like curvature of space-time. In this description, space-time is curved in such a way as to cause a gentle outward repulsive force on the large scale, expanding all of the universe over time. This is one cause of the...

Thought experiment: What will happen if you place a mirror behind the event horizon and turn on the light in its direction?

Hi there, have a wonderful next year! I'm here because I have a doubt. I was trying to generalize the Einstein Field Equation for Von Neumann W* Algebra, which is related with non-integer, non always positive degrees of freedom. In particular, with the sum of positive and negative fractal...

A few yesrs ago now I read a First Course in String Theory. In that book strings were part of normal space-time plus for consitency some extra dimensions. Spin 2 partices natually emerged and so did GR. I didnt think anything of it at the time (pun intended), but I recently saw a...

For (a) and (b), since the geodesic is not affinely parametrised, we have that ##t^a\nabla_a t^b = f(\lambda) t^b##, for some function f. As a results, for (a) I get that ##t^a \nabla_a \epsilon = 2 f(\lambda) \epsilon##. And for (b) I get that ##t^a \nabla_a p = f(\lambda) p##. (I can write...

I'm a mediocre physics student (at best) at an Ivy League institution, and I'm passionate about general relativity. My dream is simply to do research in the field, even though I will never be a superstar or pioneer. Finally, I'm planning to complete a Master's in astrophysics or physics...

Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury. My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.

Hi, on Wald's book on GR there is a claim at pag. 43 about the construction of synchronous reference frame (i.e. Gaussian coordinate chart) in a finite region of any spacetime. In particular he says: $$n^b\nabla_b (n_aX^a)=n_aX^b\nabla_b \, n^a$$Then he claims from Leibnitz rule the above equals...

Let me preface this by saying that I do not claim to dismantle relativity. That said, in reading the Feynman I came across an ideal experiment that is not entirely clear to me. The context is the usual two spacecraft moving at relatively constant speed. With the reasoning I give below Feynman...

I have always had certain difficulties in correctly understanding the theory of special relativity; I often apply it to certain situations and they fall apart... Imagine there is a rectangular object placed with its long sides parallel to the ground and its short sides perpendicular to it...

So I've often heard that when GR is applied to the entire observable universe to calculate its curvature, we get a value of zero, meaning that the entire universe is flat. I've got 2 problems with this. The first is that I thought GR was a local theory i.e. it only applies locally. Trying to...

I have a question about the concept of length contraction. The black line from (0, 0) to (1, 0) represents a meter stick in my stationary frame, call it frame A. The blue axes represent my coordinate system with coordinates x and t. The green axes represent the coordinate system of a moving...