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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Induced metric on the boundary of a manifold

The Gibbons-Hawking-York term which supplements the Einstein-Hilbert action is, $$S_{GH} = \frac{1}{8\pi G} \int_{\partial M} d^3 x\sqrt{-h} \, K$$ where $\partial M$ is the boundary of the manifold ...
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1answer
79 views

Stress energy tensor and the covariant derivative of the 4-momentum

Another basic question. I have usually seen the stress energy tensor $T^{ij}$ described as the flow of the 4-momentum field $p^i$ along direction $x^j$ in spacetime with $p^0$ as energy and $x^0$ as ...
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106 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
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36 views

How to calculate the minimum number of extrinsic dimensions of a metric tensor?

The Question How does one calculate the minimum number of dimensions of an extrinsic space that can be used to define the metric tensor \begin{align} g_{mn} = \dfrac{\partial y^k}{\partial x^m} ...
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39 views

Positive Mass Theorem [duplicate]

I'm a third year maths undergrad doing a project on minimal surfaces. However I'm really struggling to understand what the PMT is trying to explain? Could anyone help explain this (as simply as ...
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5answers
779 views

Naive visualization of space-time curvature

With only a limited knowledge of general relativity, I usually explain space-time curvature (to myself and others) thus: "If you throw a ball, it will move along a parabola. Initially its vertical ...
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28 views

gravitational lensing [duplicate]

I had read somewhere that a star, whose light passes very close to the sun and reaches the earth produces 4 images of the same star (left, right, top and bottom) in a telescope due to gravitational ...
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1answer
106 views

General relativity, gravity and spacetime curvature [duplicate]

There is a very fundamental flaw in the common explanation given of the space-time curvature due to massive objects. It is said that a massive object curves space time just like a bowling ball on a ...
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55 views

Reissner-Nordström Black Holes

The Reissner-Nordström black holes are described by the metric, \begin{align} ds^2 = -\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)dt^2 + \frac{1}{1-\frac{2M}{r}+\frac{Q^2}{r^2}}+r^2d\Omega^2 ...
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151 views

Mistake in Briefer History of Time by Stephen Hawking [closed]

I was reading A Briefer History of Time by Stephen Hawking and Mlodinow. I found something silly. On page 36 at the bottom, it says the following : If, say, the sun suddenly disappeared, Maxwell's ...
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2answers
167 views

How long does it take for a black hole to form?

The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the ...
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90 views

The difference between an apparent horizon and event horizon?

I'm currently writing a project on minimal surfaces and general relativity - however I don't understand the difference between the apparent and event horizon? They ultimately both seemed to be defined ...
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1answer
86 views

Can Information Travel Faster Than The Speed Of Light? [duplicate]

Many believe that nothing can travel faster than speed of light, not even information. Personally, i think theoretically information can. Consider this following imaginary experiment: Imagine we are ...
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1answer
38 views

Smaller mass in gravity well?

When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from ...
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0answers
25 views

Physical difference between the derivative of the Hubble parameter and the second derivative of the scale factor?

In Carroll's GR book, he discusses the difference between $\dot H$ and $\ddot a$, stating that they are the answers to two different questions, pg. 349. He seems to imply that if one set up two ...
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2answers
85 views

Speed of gravity in cosmological codes and ephemeris generation

There are few questions in Phys.SE concerning the speed of gravity, and the answers are traditionally that the speed of gravity equals to the speed of light. But in that case I have three more ...
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2answers
276 views

Can special/general relativity be derived from the standard model?

Can special/general relativity be derived from the standard model? For example the time dilatation in strong gravitation? My feeling is yes, but I am not quite sure.
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164 views

Why does gravity attract non-metallic objects?

Why does gravity attract non-metallic objects as magnetism does? I understand why gravity, because of mass of an object, works. But earth has a magnetic field, and the moon does not. Indeed, many ...
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24 views

How is the scale factor from the FLRW equation used with Volume?

I'm trying to put a spreadsheet together that shows the co-moving volume of the universe from the time soon after the Big Bang through the present and then as predicted into the future. I am pretty ...
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1answer
57 views

Regarding the possibility of Closed Timelike Curves

I've been looking a lot at Closed Timelike Curves, and how if a theory allows for these curves it doesn't respect causality. I understand that about the curves themselves (Grandfather Paradox), but ...
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36 views

Gravitational redshift of temperature and electrostatic potential

Consider a charged black hole in four-dimensional Minkowski spacetime, with charge $Q$, mass $M>Q$: $ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Omega_2^2$, with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$. ...
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3answers
175 views

D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for ...
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1answer
119 views

How can a black hole have spin?

How is it possible, or even meaningful, to say that a black hole has spin? (Tangentially, if the singularity is assumed to be a point, it must have either zero or infinite angular momentum, in both ...
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379 views

How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?

The Question How does one prove that Rindler's definition of the covariant derivative of a covariant vector field $\lambda_a$ as \begin{align} \lambda_{a;c} = \lambda_{a,c} - \Gamma^{b}_{\ \ ca} ...
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93 views

Positive Mass Theorem

I'm currently a third year undergrad writing about Minimal Surfaces. In particular, trapped surfaces and black holes. What does the Positive Mass Theorem have to do with this? And does the theorem ...
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1answer
58 views

Is conformal time observable?

The standard FRW metric with cosmic time is $$ ds^2 = -dt^2 + a^2(t)(\gamma_{ij}dx^i dx^j),$$ and we can measure $t$ as the proper time for comoving observers. Thus it makes sense to talk about the ...
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42 views

Information paradox and spacelike slices

I'm reading S. Mathur's paper on the information paradox and I can't seem to understand the reason why we choose spacelike slices. Is it because we want to have a global timelike vector so that we ...
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1answer
35 views

Why will choice of coordinates impose functional relations on the metric?

I am reading Steven Weinberg's Gravitation and Cosmology. On page 10 he says: In $D$ dimensions there will be $D(D+1)/2$ independent metric functions $g_{ij}$, and our freedom to choose the $D$ ...
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1answer
83 views

Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature ...
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1answer
93 views

Intuition for actions written as integrals over spacetime

Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action ...
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33 views

Meaning of $k$ in Sachs-Wolfe formula for angular power spectrum

I've seen the formula for the angular power spectrum of the CMB written as $$C_\ell = \frac2\pi \int\left|\Theta_\ell(k) \right|^2 k^2dk, $$ where $\Theta_\ell(k)$ is the temperature contrast at a ...
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1answer
52 views

Sobolev norm for Schwarzschild metric

Considering a static spacetime of the metric form \begin{equation} \mathrm{d}s^{2}=-V^{2}\mathrm{d}t^{2}+h_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j} \end{equation} with a timelike killing field ...
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99 views

Hypersurface Normal

Could anyone explain why $$n^{a}n_{a}=\pm1$$ where $n^{a}$ is the normal to the hypersurface
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2answers
126 views

Unitary representations of the diffeomorphism group in curved spacetime

In (special) relativistic quantum mechanics there is a standard argument that says that the (rigged) Hilbert space of states $H$ should be equipped with a projective unitary representation $U$ of the ...
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1answer
115 views

Why do galaxies “dissappear?”

So, this is a dumb question but a bit of information confused me lately. Before, I figured galaxies were no longer visible by us because their luminosity decreased in an inverse square manner. ...
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104 views

How explain this perturbing equation about the 43 arcseconds?

The planetary orbits have been studied as ellipses but the solar system is in motion in relation to the distant stars. Their path is along the tip of an helix and the ecliptic plane is a convenient ...
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3answers
259 views

Comparing predictions and reality for the gravitational attraction due to light beams

While doing some on-the-side reading, I stumbled across this question: Do two beams of light attract each other in general theory of relativity?. Great question and a great, easily understandable ...
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49 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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1answer
86 views

Setting $\delta R =0$ on boundary of hypersurface

Does requiring $\delta R=0$ on the boundary of hyper-surface create any restrictions or problems in deriving the field equations from Einstein-Hilbert Action?
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121 views

Space and the size of infinity [duplicate]

If you could build a spaceship and keep travelling in one direction, what would finally happen? One answer is that you would never ever reach the end. But this sounds purely platonic space and comes ...
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1answer
64 views

Time Dilation Effects from simply being on a spinning planet orbiting a star in a rotating galaxy in an expanding universe.

I am a layman, so take this with a grain of salt. I saw a TV show the other day which showed a Russian Cosmonaut who had spent more time in space than any other human. The relativistic effects of ...
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1answer
57 views

Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's ...
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2answers
195 views

How does time dilate in a gravitational field having a relative velocity of v with the field?

Consider a Mass on earth. The time dilation on the surface of Earth is $$T' = T \sqrt{1 - \frac{2GM}{rc^2}}$$ Now if the mass is moving around the earth at velocity of v w.r.t Earth, what will be ...
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2answers
709 views

Can Flow of time become still?

According to theory of time dilation, flow of time slows down significantly at the speed of light.Is there any conditions practically or theoretically when flow of time is reduced to zero means it ...
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1answer
91 views

How Would an Alcubierre Drive “ride” a Wave of Spacetime?

The colloquial explanation is that the spacetime in front of a ship contracts and the spacetime behind expands. I see how one could think that this would bring you closer, but I don't see that it ...
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160 views

How to prove that a spacetime is maximally symmetric?

In Carroll's book on general relativity, I found the following remark: In two dimensions, finding that $R$ is a constant suffices to prove that the space is maximally symmetric [...] In higher ...
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2answers
156 views

If NASA could send a camera into a black hole, could we then see what's inside the black hole?

Inspired by Stephen Hawking I recently tripped upon an idea of what is really inside a black hole. I thought if NASA (or any other space agency) could send a super protected camera into a black hole, ...
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Klein Gordon eq. expressed with Killing fields

I have a question on the reformulation of the Klein Gordon equation in terms of Killing fields. Suppose we have a static spacetime with timelike Killingfield $\xi^{\mu}$ (e.g. Schwarzschild). Then ...
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1answer
83 views

What's wrong with Schwarzschild equations?

I don't know much about black holes physics and so I find the Schwarzschild equations with a few contradictions. In particular I am trying to understand this little puzzle. The Schwarzschild ...
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95 views

What do we learn from gravity in three spacetime dimensions?

The last decades there has been a lot of research going on in the the area of three dimensional gravity. The motivation, I understand, is threefold: Whereas gravity is not perturbatively ...