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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Why doesn't this proof change indices?

In this pdf, in the second line of the proof, $\sigma$ was plugged in where it appears as $$\frac{\partial x^\sigma}{\partial y^{\rho'}}$$ Meanwhile in converting the coordinates of $g^{\mu'\rho'}$, ...
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29 views

Relation between the curvature of a manifold and the number of covariantly constant vector fields that it admits

Suppose that on a four dimensional manifold we are able to explicitly construct four linearly independent covariantly constant vector fields $K^a_{\mu}$: $$D_{\mu}K^a_{\nu}=0,$$ $a=1,2,3,4$ then it ...
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69 views

How strong is the force of space expansion?

There are many questions about space expansion, its cause, or its effects. But I have the feeling we never get straight and simple answers. I do not expect answers to be simple in general, but I ...
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47 views

Weak Equivalence Principle and universality of free fall

I know how we can derive geodesic equation from varying the action of a test particle with respect to coordinates and i know the fact that particles follow geodesics means that free fall is universal. ...
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34 views

problem with spin connection term

While working out Kaluza Klein compactification, I am getting the unwanted spin connection term $\omega_{c}^{ac}$ .I have tried to show that this is zero.But I am not quite sure of it. What I tried ...
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43 views

Relation involving the Lorentz transformation and the inverse of its transpose

The relation I was referring to in the title is $${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$ where ${\Lambda_a}^b$ is the inverse transpose of $L$, the Lorentz transformation. I was wondering ...
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37 views

What causes the unexpected change in acceleration for flybys of spacecraft?

If the vehicle is not operating on RTG, then thermal recoils of photons shouldn't be considered as in the case of PIONEER. Then what actually accelerate the spacecraft from expected value (expected by ...
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81 views

Is there a general stress-energy tensor for vector fields?

I've been reading about scalar fields in the context of general relativity, and I found this page: https://en.wikipedia.org/wiki/Stress-energy_tensor#Scalar_field. It says that the stress-energy ...
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78 views

Minkowski to Euclidean

When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now ...
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43 views

Caustic and Singularities in General Relativity

What is the relation between the formation of caustics of a family of null geodesics and the existence of an incomplete null geodesic?
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Is there some other name used for “ping rigidity”?

In MTW, p. 398, "Box 16.4 (continued)", there's an interesting sketch (which can also be seen on p. 15 of this excerpt (www.pma.caltech.edu/~ph236/yr2008/readings/MTW_Chapter16.pdf). (It's not the ...
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164 views

Frequency of a photon as related to the change of its frequency

I think that bob maybe already, at least partially, has answered my following question: "Is it true or false that the frequency of a photon is not related to the change of its frequency caused by ...
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66 views

Null geodesics and rotation in stationary axisymmetric space-times

It is well known that in Schwarzschild space-time, a torque-free gyroscope in circular orbit at any permissible angular velocity at the photon radius will, if initially tangent to the circle, remain ...
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freedom of choice of 1-form in canonical representation of generic local field corresponds to gauge choice?

So it is a question in Gravitation Wheeler, Thorne and Misner 4.2 Exercise. Given F=$dp_{i}\wedge dq^{i}$. Using canonical transformation from p to $\bar{p}$ and q to $\bar{q}$, one gets ...
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23 views

Angular diameter distance in an inhomogeneous universe?

Computing the angular diameter distance $D_{A}$ is a well known academic exercise in an homogeneous Universe. But now suppose that we are in an inhomogeneous Universe and that I am interested in ...
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15 views

Why the position of apparent horizon depend on choice of Sigma t of foliation?

Whether a 2d surface is a trapped surface is independent of choice of Sigma t of foliation. Why its outer boundary (apparent horizon) depends on choice of Sigma t of foliation?
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51 views

Conditions that the coordinate must satisfy in order to become local inertial

Consider the coordinate transformation $$ \tilde x^a=x^a+\frac{1}{2}\Gamma^a_{bc}x^bx^c $$ I have shown that at the origin $O=(0,0,0,0)$, $$ \frac{\partial\tilde g_{ab}}{\partial\tilde x^c}=0 $$ ...
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33 views

(Special Relativity) Points that can be seen by an observer

Let the metric be $$ ds^2=(1+gz)^2dt^2-dx^2-dy^2-dz^2 $$ where $g$ is a positive constant. Let an observer be stationary at $x=y=0$ on the surface $z=0$ and look upwards at an angle $\theta$, how ...
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56 views

Exercise Lorentz Transformations

I have 2 inertial frames, both are in relative motion along the x axis moving with constant velocity $v_x$. It is called with Standard figuration where the Lorentz transformations relate the ...
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119 views

Wormholes with event horizons?

Under General Relativity, Lorentzian wormholes (the kind that are traversable) require exotic matter (a kind of unobtainium which is not known to exist). On the other hand, we know black holes exists ...
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39 views

Can we design the orbit of a satellite with Einstein’s curved space?

I think that the most successful theory of gravity is that the orbit of planet can be explained with this theory and the orbit of satellite was designed with this theory. To design an orbit of a ...
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29 views

gravitational waves and orders of magnitude

I have a question about numerical magnitude in eq. 6.110 of this reference. So, using this reference sheet for this PSR1913+16 binary, I get the following numbers: $$M = 2.8 \times 10^{30} ...
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21 views

Angular and luminosity distance in general?

Consider a non-Friedmannian Universe in which we know the trajectories of photons, ie in which we know null geodesics $\left(\eta, x^{1}, x^{2}, x^{3}, a, z\right)$ where : $\eta$ is the conformal ...
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52 views

The border of the System

In General Relativity, if the system accelerates, the inside of the system and the outside of the system will have different speed of time. Where is the boundary of the system? If a human ...
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139 views

How come that an Infinite universe will collapse under gravity?

In this PSE post the issue is about the stability of an infinite universe under Newtonian gravity. Here I'will drop the Newtonian constraint because we know of the finite speed of interactions and I ...
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45 views

Questions about four-momemtum

I am reading a note about Kerr metric, following is some screen shot where I have problems. First of all, I think the author made a mistake. It should be $$ E=-u_\mu k^\mu=u^t\;\;\;L=u_\mu ...
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76 views

Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?

Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which ...
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55 views

Degrees of freedom in physical equations

Say we have the field equation: \begin{equation} f^{\prime}(R)R+3\square f^{\prime}(R)-2f(R)={\kappa}^{2}T, \end{equation} why is the non-vanishing of the second term means that there is an extra ...
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78 views

Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
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51 views

Lorentz transformed Scharwzschild solution

A question that has always intrigued me is: "Imagine a star moving as it evolves into a black hole, Ignore the effect of debris from the supernova. Assume also that before the collapse, the star was ...
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30 views

Do early-time black-hole infallers ruin effective field theory at the horizon for late-time observers?

I have an elementary confusion about black hole physics. The standard consensus is that if I fall into a black hole, at the horizon I don't see any violations of effective field theory for a large ...
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137 views

Has the Reissner-Nordstrom metric ever been experimentally verified?

In contrast to the solution of the conventional Reissner-Nordstrom problem, where the Schwarzschild metric takes on an additional $1/r^2$ term due to the added electric charge, P. Mannheim has in ...
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79 views

Timelike Shell Collapsing into a Black Hole

Does anyone know where I can find the solution for a spherically symmetric thin shell of timelike matter falling into a Schwarzschild black hole? The matter should be pressureless, so that each ...
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94 views

Curvature based derivation of Schwarzchild Metric

I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?
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69 views

How to move from Special to General Relativity

I have understood special relativity nicely, and right now I am trying to learn general relativity from D'Inverno's book. I an finding it rather difficult to understand the pre-requisite math (i.e. ...
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62 views

About divergence of a vector field and geodesic sphere

I have a question. I want to know the difference between the sphere and the geodesic sphere. Another question: given a vector field, $Y$, on a manifold $M$ defined by: $Y(p)=p$ for every point $p \in ...
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85 views

Light cones and reference frames

I'd like to know what does it mean exactly to find a reference frame in which two events occur at the same time or in the same space coordinates. As I picture it if we have two events A and B in a (x, ...
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30 views

concept of density in gravitational lensing

I may just be being very dense (no pun intended) but i'm reading up on gravitational lensing and it seems to require a notion of density (e.g. see here) I'm working on a question involving light ...
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66 views

Problem with relativity of acceleration

In this answer http://physics.stackexchange.com/a/92833/36977 John said that acceleration is not relative in the general theory of relativity. But this is a problem: as we all know, accelerating ...
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49 views

What is the “momentum” referred to in the energy-momentum tensor

What is the "momentum" referred to in the energy momentum tensor from GR? Is it $m\dot{x}$ or is it the canonical momentum $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)$ Also, I ...
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44 views

Hubble's law for the Kasner solution

I'm puzzled with the following question: find an analog of the Hubble's law for the Kasner solution. Kasner metric is a solution to the vacuum Einstein equations ...
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21 views

Method of determining peculiar speed of the galaxy which moves on celestial sphere and emits the light

This question is the continuation of this one. I came up with solution, but I'm not sure that it is correct. Can someone check it? Let's introduce transverse (to the observer) proper speed ...
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44 views

How to find proper speed (relate to homogeneous cosmic background) of the galaxy by given redshift z and observing angular velocity?

The galaxy moves of the celestial sphere. It is given that proper speed is transverse to the observer and it must to find this speed in the moment of light emission. The motion is in the FLRW ...
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Differential equation for speed relate to the homogeneous cosmological background in FLRW metric

How to derive DE for the speed (which relate to the homogeneous cosmological background) of the observer which moves with constant proper acceleration in spatially flat FLRW universe?
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62 views

One more time about the connection of Weyl tensor and gravitational waves

There is differential identity with Weyl tensor and energy-momentum tensor: $$ D^{\lambda}C_{\lambda \alpha \sigma \beta} = 4 \pi G \left(D_{\sigma}T_{\alpha \beta} - D_{\beta}T_{\alpha \sigma} + ...
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91 views

How to use The Schwarzchild Metric formula to get distribution representing “free-fall”

Given formula: How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should ...
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90 views

On motivation for the definition of ADM mass

The ADM mass is expressed in terms of the initial data as a surface integral over a surface $S$ at spatial infinity: $$M:=-\frac{1}{8\pi}\lim_{r\to \infty}\int_S(k-k_0)\sqrt{\sigma}dS$$ where ...
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Interpretation of contribution of gravitational potential energy to the gravitational field

In terms of General relativity we have as a matter of principle that anything that has inertial mass contributes to gravity. All forms of potential energy have inertial mass, it follows that the ...
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51 views

Geodesic in general relativity aproaching ellipse

In the gravity well-like 2d surfaces that are used in documentaries to illustrate the fact Earth orbits the Sun, I don't seem to find any kind of geodesic that will at least resemble an ellipse... ...
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69 views

Closed linear cosmology implies G M / R = c^2?

I have a question about a linear FRW cosmology with $k=+1$. Assuming zero cosmological constant the first Friedmann equation can be written: $$\left(\frac{\dot R}{R}\right)^2 + ...