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.

learn more… | top users | synonyms (1)

0
votes
0answers
28 views

How is gravitational time dilation different from time dilation due to differences in speed? [duplicate]

This is what I understand from what I've been reading online: In the derivation for the gravitational time dilation equation, $$t = t_0 + \sqrt{1-\frac{2GM}{rc^2}}$$ we use the special relativity ...
2
votes
1answer
53 views

Are the Schwarzschild metric and the Geodesic Equation relevant in the context of the Earth?

The geodesic equation used in general relativity is the following: $$ {\mathrm d^2 x^\mu \over \mathrm ds^2} =- \Gamma^\mu {}_{\alpha \beta}{\mathrm d x^\alpha \over\mathrm ds}{\mathrm d x^\beta ...
5
votes
1answer
53 views

Signature of $f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \mathbb{R}$, $f(\omega, \omega') = \omega \wedge \omega'$

Define$$f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \Lambda^4(\mathbb{R}^4) \cong \mathbb{R}, \quad f(\omega, \omega') = \omega \wedge \omega'.$$ What is the signature of $f$? ...
4
votes
0answers
145 views
+50

Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
7
votes
1answer
213 views

Homotopy proof of the lack of foliation of the Gödel metric

A common proof of the lack of foliation of the Gödel universe, apparently mostly copy pasted from Hawking and Ellis, goes thusly : A closed timelike curve must cross a spacelike hypersurface ...
4
votes
1answer
88 views

Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
4
votes
0answers
33 views

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught? [on hold]

Inspired by this question here. The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - ...
2
votes
0answers
27 views

Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial ...
0
votes
0answers
16 views

torsion tensor proof [on hold]

I looked up torsion tensor derivation on 2 different books, and encountered 2 different situations, so my mind has been confused. For the first image, I could totally understand how torsion tensor was ...
0
votes
0answers
25 views

How conclusive is “Gravitational red-shift Gedanken”?

The gedanken goes as you take a particle of mass $m$ at a height $H$. Then let it fall to gain the velocity (approximately)$\sqrt{2gH}$ when it reaches the ground. Convert the particle into a photon ...
2
votes
1answer
58 views

How do you actually use the geodesic equation?

The geodesic equation used in general relativity is the following: $$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}. $$ It states that the ...
7
votes
2answers
130 views

Do gravitational waves have entropy?

We know, according the current understanding of black holes and General Relativity, as well as quantum fields in General Relativity, that black holes have an entropy proportional to the area of the ...
2
votes
2answers
33 views

Is it possible to express “free”-ness of a time-like world line without referring to “tangent space” (but only directly to causal relations )?

I don't know much about tangent spaces, or tangent vectors, "as such"; nor about affine parametrization (which seems to be closely related to the notion of tangent vectors, as far as I understand for ...
-3
votes
0answers
56 views
4
votes
5answers
592 views

What is the significance of Planck force?

I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4/G = 1.21031359\times 10^{44} \, \mathrm{N}$, where $c$ is the speed of light and $G$ is ...
2
votes
1answer
35 views

Are the quasinormal modes scalar quantities?

I am studying the so-called quasinormal modes (QNMs) in the context of the AdS/CFT correspondence and I got stuck. For instance, if I choose a weird patch of coordinates for the, say, ...
7
votes
3answers
623 views

How to show that every Killing vector field is a matter collineation?

Various texts make this claim, but no proof is given. Explicitly, let $L$ denote the Lie derivative. Suppose $L_X g_{ab} = 0$ for some vector field $X$, called a Killing vector field. Suppose that ...
6
votes
2answers
388 views

Wave packet in curved spacetime

It is known that the classical equation of motion for a scalar field wave packet on a curved spacetime background gives the geodesic trajectory (the e.o.m. is $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). ...
3
votes
1answer
135 views

Distance between two galaxies of different redshift

Let $Q_1$ and $Q_2$ two different objects in the Universe (we can think to two galaxies or quasars), that we observe from the Earth at different angular position $(\alpha_1,\delta_1)$, ...
1
vote
1answer
44 views

Why is the Einstein Static Universe represented as an infinite cylinder when it seems like only half a cylinder?

The Einstein static universe metric is $$ds^2=-dt^2 + d\chi^2 + \sin(\chi)^2d\Omega^2$$ where $-\infty<t<\infty$ , $0<\chi<\pi$ and $d\Omega^2$ is the metric on a $S^2$. It describes the ...
6
votes
3answers
117 views

Two Robertson-Walker observers, at what time will a light signal be received?

Here is a question I have that is inspired by this question here. The spacetime metric of a radiation-filled, spatially flat ($k = 0$) Robertson-Walker universe is given by$$ds^2 = - dT^2 + T[dx^2 + ...
6
votes
1answer
239 views

Null geodesics in uniform gravitational field metric

I'm trying to understand the null geodesics in the metric: $$\mathrm{d}s^2 = -(1+gz)^2 \mathrm{d}t^2 + \mathrm{d}z^2 + \mathrm{d}x^2$$ In particular I'm wondering if the following intuition is ...
1
vote
2answers
84 views

Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
6
votes
2answers
94 views

When is stress-energy tensor defined as variation of action with respect to metric conserved?

In General Relativity Einstein's equation implies that stress-energy tensor on its RHS is conserved (has vanishing divergence), due to the Bianchi identity. Considering variational principles leading ...
16
votes
2answers
896 views

How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?

In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure ...
4
votes
2answers
127 views

$C^\infty$, nonvanishing parallel vector field along geodesic, orthogonal to tangent

The following question(s) showed up in my admittedly basic undergraduate research in general relativity/cosmology, and I was wondering if anybody could me with it. Let $(X, g)$ be a $n$-dimensional ...
1
vote
2answers
59 views

Two Black Holes held stationary by EM forces

If two black holes with large enough mass (so that the tidal forces are minimal and the intersection is large) that are held apart by like charges (saddle point stability). Imagine the black holes in ...
3
votes
1answer
54 views

Torsion in kerr black holes

In General Relativity, we generally assume that the derivative operator is torsion-free, i.e., second covariant derivatives commute on functions. However, in Kerr black holes, spacetime is dragged ...
9
votes
3answers
920 views

Could a ship equipped with Alcubierre drive theoretically escape from a black hole?

Could a ship equipped with Alcubierre drive theoretically escape from a black hole? Also, could it reach parts of the universe that are receding faster than the speed of light from us?
12
votes
2answers
817 views

How can we recover the Newtonian gravitational potential from the metric of general relativity?

The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration: ...
106
votes
5answers
14k views

Did the Big Bang happen at a point?

TV documentaries invariably show the Big Bang as an exploding ball of fire expanding outwards. Did the Big Bang really explode outwards from a point like this? If not, what did happen?
3
votes
1answer
68 views

Metric that is Minkowski plus sum of null vectors

In GR exercises I've often seen metrics of the form $g_{ab} = \eta_{ab} + k_ak_b$ where $k_a$ is null with respect to $g$ (or equivalently $\eta$). I'm happy doing calculations with such metrics, but ...
3
votes
1answer
57 views

Schwarzschild metric black hole

Schwarzschild metric solution presents two singularities. An apparent one at $r=2GM$ and a real one at $r=0$. It is known that everything freezes at the event horizon from an outside observer point of ...
0
votes
1answer
36 views

Do the energies of cosmic rays approach infinite at the event horizon of a black hole?

Let's assume an observer orbits close to a black hole, he is not alone, massive cosmic rays, like electrons and protons and other kind of space dust comes from the outer space and may hit him. Since ...
1
vote
0answers
28 views

Going to the Einstein frame in f(R) theories

First of all thank you for your time! I have a question that I can't solve. In every review that I read, I find that when you want to go to the Einstein frame in a $f(R)$ theory what you have to do ...
5
votes
2answers
290 views

Non-trivial scalar quantity

Is there any scalar quantity made of only the Christoffel symbols, determinant of a metric and tensors, not derivatives? In other words, can we construct a scalar quantity which cannot be written in ...
1
vote
1answer
55 views

Is it contradictory with any theory or experimental result to have a negative gravitational force mass?

I am aware that there are many similar questions here about this in this site, but most answers concentrate on negative inertial and gravitational energy. My question is more specific. QM together ...
5
votes
1answer
95 views

Trying to understand Newtonian limit of GR

First ever post - please be kind. I'm trying to understand how General Relativity becomes equivalent to Newton's laws of motion, plus Newton's law of gravitational attraction in the limiting case of ...
235
votes
15answers
21k views

How does gravity escape a black hole?

My understanding is that light can not escape from within a black hole (within the event horizon). I've also heard that information cannot propagate faster than the speed of light. It would seem to ...
4
votes
3answers
127 views

If an astronaut had stationed in International Space Station for the duration of mission, 17 years, would he be older?

Today the NASA International Space Station started the 100000 orbit after 17 years in the space. I just wonder if there were a team of astronauts which were in the Lab for all the duration of last 17 ...
4
votes
0answers
26 views

Orthogonal geodesics to hypersurfaces [migrated]

Say we have a Riemannian manifold $(M, g)$ with vector field $Y$, obeying: $g(Y, Y) = 1$; and the $1$-form $\varphi(X) = g(X, Y)$ is $d$-closed, $d\varphi = 0$. I know that the integral curves of ...
1
vote
0answers
11 views

Lapse Function and Shift Vector in Minkowski and de Sitter

I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way: The $y$-axis represents time while the x-axis ...
4
votes
1answer
50 views

Schwarzschild metric, acceleration of ball before it's dropped [duplicate]

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + ...
2
votes
3answers
99 views

How can geometrized units have more than one constant equal to 1?

I can understand how you could manipulate units to make a certain constant equal to $1$, like $c$ or $G$, et cetera. But how can you make it so two constants (in this case $c$ and $G$) are equal to ...
0
votes
0answers
61 views

Interpretation of black-hole infalling and apparent geometry of the event horizons [closed]

Consider a citizen of the cosmos crosses the event horizons of a black hole. For sake of discussion let's suppose is a Kerr black hole, so that his fate is not sealed the moment he crosses. Does at ...
1
vote
0answers
39 views

Showing classical spin tensor is time independent for free particle

Reading through Weinberg's gravitation book, the following definition is given for the spin tensor (Pauli-Lubanski psuedovector): $$ S_\alpha = \frac{1}{2}\epsilon_{\alpha\beta\gamma\delta} ...
1
vote
1answer
41 views

What relative effects be for object with near light speed velocity in compactified dimensions?

What relative effects be for an object with near light speed velocity in compactified dimensions? Does gravity increase the same as for an object with near light speed velocity in usual spacial ...
-4
votes
0answers
52 views

Can wormholes transfer fire to a different place or time, when it is made stable? [closed]

Can a wormhole transfer fire? I have tried searching the answer but I couldn't find it. Mostly all wormholes collapse so quickly that not even light can make through it; but a wormhole can be made ...
0
votes
1answer
44 views

Wald's Equation 3.3.6

I have an issue with Eq. 3.3.6 of Wald's General Relativity. There he would like to prove that for Gaussian normal coordinates, the geodesic tangent field remains orthogonal to all coordinate basis ...
7
votes
1answer
284 views

Does frame dragging apply to linear motion?

Firstly I will admit I do not understand the real cause of rotational frame dragging and some of the math heavy explanations are too complicated for me. To me frame dragging looks like unsubstantiated ...