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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Some sort of conservation equation

As far as I know, in General Relativity, an expression of the kind $\nabla_{\mu} X = 0$ states that, associated to $X$, there exist a charge which is conserved. The first example that comes to mind is ...
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0answers
173 views

Gravitational redshift of Hawking radiation

How can Hawking radiation with a finite (greather than zero) temperature come from the event horizon of a black hole? A redshifted thermal radiation still has Planck spectrum but with the lower ...
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1answer
134 views

Killing vector argument gone awry?

What has gone wrong with this argument?! The original question A space-time such that $$ds^2=-dt^2+t^2dx^2$$ has Killing vectors $(0,1),(-\exp(x),\frac{\exp(x)}{t}), ...
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1answer
139 views

Assuming space is infinite can our observable universe be an island amongst an archipelego?

According to recent measurements our observable universe is roughly 93 billion light years in diameter; also it appears (according to WMAP measurements) that spacetime is flat. Supposing space is ...
3
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2answers
588 views

Geodesic equations

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2 dimensional space such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ the timelike geodesics ...
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1answer
241 views

Does the actual curvature of spacetime hold energy?

My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy ...
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2answers
426 views

Excluding big bang itself, does spacetime have a boundary?

My understanding of big bang cosmology and General Relativity is that both matter and spacetime emerged together (I'm not considering time zero where there was a singularity). Does this mean that ...
9
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1answer
305 views

Our Universe Can't be Looped? [duplicate]

With reference to the Twin-Paradox (I am new with this), now information of who has actually aged comes from the fact that one of the twins felt some acceleration. So if universe was like a loop, and ...
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0answers
55 views

How to keep the clock of a spaceship synchronised to the clock of an observer? [duplicate]

I read that the clocks of GPS satellites seem to run slower than the clock of stationary observer, because of their speed (special relativity) and seem to run faster than the clock of stationary ...
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0answers
143 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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1answer
197 views

Evaluating the Ricci tensor effectively

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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3answers
727 views

Why Can We Observe Space Curvature / Warping At All?

I don't understand why we are able to see and measure curvature / warping of space at all. Space as I understand it determines distances between objects, so if space were "compressed" or warped, ...
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220 views

Is period of rotation relative?

My question is inspired by the following answer by voix to another problem: "There is a real object with relativistic speed of surface - millisecond pulsar. The swiftest spinning pulsar currently ...
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2answers
125 views

Are there problems solvable with Newtonian physics, GR and QM?

First I must let you know that I don't have much understanding of neither GR nor quantum mechanics, and therefore this question. I've mentally pictured Newtonian physics, GR and quantum mechanics all ...
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0answers
58 views

metric extension outside the light cone

Could anyone explain what "extending the solution" beyond the past light cone means? Say, for example, if I have a metric (no coordinate singularities), how can I extend it to the outside of the past ...
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2answers
337 views

Negative potential energy of gravity

Does the negative potential energy in the gravitational field have to be considered in calculating the total mass of the system in question (because of $E=mc^2$)? If so it seems to me that the ...
3
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1answer
200 views

Why does the Kruskal diagram extend to all 4 quadrants?

Why is it that the Kruskal diagram is always seen extended to all 4 quadrants when the definitions of the $U,V$ coordinates don't seem to suggest that the coordinates are not defined in, say, the 3rd ...
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1answer
1k views

Christoffel symbol for Schwarzschild metric

I know that the christoffel (second kind) can be defined like this: $$\Gamma^m_{ij} = \frac{1}{2} g^{mk}(\frac{\partial g_{ki}}{\partial U^j}+\frac{\partial g_{jk}}{\partial U^i}-\frac{\partial ...
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1answer
399 views

When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
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0answers
129 views

Ising Hamiltonian for relativistic particles

An Ising system is described by the simple Hamiltonian: $$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units ...
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0answers
86 views

Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric ...
7
votes
3answers
274 views

Do velocity and acceleration time dilation factors add?

For a spinning space station such as in 2001, A Space Odyssey, what would be the time slowing in the perimeter of the spinning space station with respect to the center axis of the station? The ...
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2answers
793 views

Is time going backwards beyond the event horizon of a black hole?

For an outside observer the time seems to stop at the event horizon. My intuition suggests, that if it stops there, then it must go backwards inside. Is this the case? This question is a followup ...
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1answer
2k views

Stress energy tensor of a perfect fluid and four-velocity

In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : ...
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2answers
236 views

What is a sudden singularity?

I've seen references to some sort of black hole (or something) referred to as a sudden singularity, but I haven't seen a short clear definition of what this is for the layman.
3
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1answer
352 views

The most general form of the metric for a homogeneous, isotropic and static space-time

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
3
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1answer
242 views

Sign crazyness on the stress energy tensor?

I would like to know on what depends the sign of the stress energy tensor in the following formula : $T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$ In my case the metric is equal to ...
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3answers
408 views

Are gravitational time dilation and the time dilation in special relativity independent?

There are two kinds of time dilation: One because the other clock moves fast relative to me (special relativity). Another one because the other clock is in a stronger gravitational field (general ...
4
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1answer
131 views

“WLOG” re Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem ...
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1answer
118 views

Gravitational time delay and contraction of matter [duplicate]

How can any matter contract to its Schwarzschild radius if gravitational time dilation clearly states that all clocks stop at that point. So any contraction any movement would stop. If that is so why ...
2
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0answers
187 views

Do we expect that the universe is simply-connected? [duplicate]

I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe cannot be simply-connected, since the ...
2
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1answer
852 views

General relativity and the conservation of momentum

I'm trying to understand the conservation of momentum in general relativity. Due to the curvature of space-time by matters and energy, the path of a linear motion appears to be distorted. Therefore ...
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63 views

Singularities in Schwarzchild space-time

Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element $$ ...
6
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2answers
512 views

First and second fundamental forms

I'm writing notes about the 3+1 formalism in general relativity, for myself. Inevitably I came across the notions of first and second fundamental forms. Mathematically, it is clear how these objects ...
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0answers
135 views

Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ ...
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0answers
641 views

How to calculate Riemann and Ricci tensors for a sphere? [closed]

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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1answer
639 views

Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement In space of constant sectional curvature, $K$ is ...
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0answers
202 views

Stress-energy tensor of point particle when the trajectory is a transcendental equation?

I'm working through Carroll's GR book, and Problem 7.8 is not coming together. I'm missing something idiotically simple, but I'm not sure if I can cleanly write a stress-energy tensor for a point ...
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2answers
529 views

Is Earth's orbit around the sun affected by the ~8 minutes light delay?

Gravitational change occurs at the speed of light. As a consequence, we experience on Earth the gravitational attraction of the sun based on its position relative to us ~8 minutes ago. How does this ...
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1answer
125 views

A physical sense of an Inertial frame

Definition clarification needed, please: I am hoping to get physical sense of an "inertial frame". Do inertial reference frames all have zero curvature for their spacetime? So is an inertial frame ...
2
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2answers
403 views

Local inertial coordinates/Fermi normal coordinates

It is said that we can introduce local inertial coordinates/Fermi normal coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do ...
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5answers
619 views

How universal gravitation falls short

As a non physicist I can understand how Newtonian mechanics falls short in cases of high velocity etc. and is properly generalized by the special theory of relativity. What is not clear to me is how ...
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0answers
176 views

The interior of a cylinder as an Einstein manifold

The interior of a curved cylinder is an Einstein manifold (the Ricci Curvature Tensor is proportional to the Metric $R_{\mu\nu}=kg_{\mu\nu}$) since it has a constant curvature. Using the metric $$ ...
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1answer
50 views

Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial ...
6
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2answers
593 views

Does non-mass-energy generate a gravitational field?

At a very basic level I know that gravity isn't generated by mass but rather the stress-energy tensor and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ ...
4
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1answer
174 views

Wald problem 11.4

Consider a stationary solution with stress-energy $T_{ab}$ in the context of linearized gravity. Choose a global inertial coordinate system for the flat metric $\eta_{ab}$ so that the "time direction" ...
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1answer
299 views

Cancelling special & general relativistic effects

We know that for a GPS we need to make a correction for both general and special relativity: general relativity predicts that clocks go slower in a higher gravitational field (the clock aboard a GPS ...
11
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1answer
338 views

General definition of an event horizon?

Horizons are in general observer-dependent. For example, in Minkowski space, an observer who experiences constant proper acceleration has a horizon. Black hole horizons are usually defined as ...
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1answer
71 views

Change of variables in an interval expression

This question is a continuation of How to calculate a scalar curvature fast? . Let's have Lorentz-Fock spacetime with an interval $$ d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat ...
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3answers
214 views

Combining metric tensors/curvature tensors

I was thinking about the following scenario: Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...