The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.

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Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ...
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Does decomposition of motion rely on Pythagorean theorem?

As an example, when analyzing a simple projectile motion with initial horizontal velocity in Newtonian mechanics, I'm enabled to decompose the projectile motion into the vertical and horizontal ...
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78 views

How to calculate the event horizon and the cosmological radius in a metric?

From reading about general relativity, the event horizon and the cosmological radius are the radius when $f(r)=0$, in the metric $$ ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2} $$ However, ...
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211 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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216 views

Why does the Lorentz transformation have to be a linear transformation? [duplicate]

In my textbook, they say the following statements before doing a proof for the Lorentz transformation: We know that the Galilean transformation $x' = x - vt$ is incorrect, but what is the ...
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Are orbits possible in de sitter space?

Since the de sitter space has constant positive curvature does that mean that objects can't orbit around other objects?
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53 views

Does Schwarzschild metric have cosmological horizon?

Since the space is not expanding in the Schwarzschild metric does that mean that there is no cosmological horizon? Also, what if the Schwarzschild metric was not asymptotically flat and we replace ...
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113 views

What is the degrees of freedom of metric tensor?

As $g_{\mu\nu}$ can be taken to be symmetric, it contains 10 functions of spacetime in 4 dimensions. But, why we call these 10 functions as the degrees of freedom of the metric while they are the ...
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How to invert this metric?

Reading this article i find a result that i am not sure how to obtain (page 3 eq 3). It is about the inversion of a metric of the type $$ g_{\mu\nu}=Al_{\mu\nu}+BH_{\mu\nu}. $$ In order to invert ...
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52 views

Do contractions with Dirac matrices involve a metric?

When figuring out where the spacetime metric enters an equation it is often useful to write all vector indices as covariant indices and write out the inverse metrics that are needed to contract them, ...
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1answer
67 views

Meaning of “physical” and “gravitational” metrics

I've recently been reading some notes (following a paper by J.D. Bekenstein, titled "The Relation between Physical and Gravitational Geometry": http://arxiv.org/abs/gr-qc/9211017) on alternative ...
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439 views

Variation of square root of determinant of metric, $\delta g$ [closed]

I am trying to calculate $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$ where $g = \text{det} g_{\mu \nu}$. We have $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 ...
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How to calculate spacetime intervals on a spacetime diagram?

In SR, the spacetime interval is given by the metric: $ds^2=-dt^2+dx^2$ (where I set $c=1$). To calculate $ds^2$ of a worldline on a spacetime diagram, I measure $dt$ and $dx$ of the line of ...
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161 views

Geometric definition of the Lorentz inner product

In Euclidean space one can define the dot product as projecting one vector to the other and multiply the length of the projected vector with the length of the other vector. This definition doesn't ...
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63 views

What is meant by “the Klein-Gordon equation is unsymmetrical between the temporal and spatial components”, and why is this a problem? [closed]

The Klein-Gordon equation explicitly reads $\left( \frac{\partial ^2}{c^2\partial t^2} - \nabla ^2+\left( \frac{m_0 c}{\hbar}\right)^2\right) \psi =0$ Now I read here on page 8 that: What is ...
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Black hole physics beyond the perturbation theory

Motivated by this question: Perturbation of a Schwarzschild Black Hole How would one deal with the situation where black hole experiences not only small perturbations but major changes to the metric? ...
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Event horizon from the metric

Let us suppose we have a metric of this form $$ds^2=-A(r)dt^2+\frac{dr^2}{B(r)}+r^2d\Omega^2$$ In all documents I can read, I've seen that the event horizon is defined by considering $A(r)=0$ But I ...
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119 views

What is the conformal mode of a metric?

I have a problem in terminology. This article talks about the conformal mode of a physical metric. I know what a conformal transformation is. But what is the conformal mode of a metric?
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How to define $\delta{g_{\mu\nu}}$?

In general relativity, when deriving the field equation using the variational principle we use $\hat{g}_{\mu\nu}=g_{\mu\nu}+\delta{g_{\mu\nu}}$. Does $\delta{g_{\mu\nu}}$ mean the measurement of how ...
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If the measurements of a clock above the earth depend on orientation, then what measurements are correct?

Take a clock in space above the earth (assuming a Schwarzchild spacetime) that works by relaying a light signal a small distance radially; ticking each time the light signal returns. Compare this to ...
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53 views

Is the scale factor Lorentz invariant?

Given that the Minkowski metric does not change under a Lorentz transformation, the scale factor does not change in the special case when it is equal to 1. Is this result true in general? i.e. is the ...
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1answer
80 views

What is the value of the variation stress energy tensor?

If we are living in a portion of space-time where the metric is very close to flat space and we know that the stress energy tensor is negligible at this portion of space-time is it ok to assume that ...
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1answer
79 views

Understanding the nature of metric tensor [closed]

The metric tensor for a flat spatial manifold gives us length on object, or separation between two space points. Similarly, $g_{\mu \lambda} dx{^\mu} dx{^\lambda}$ gives separation between two space ...
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3answers
186 views

General Relativity 2-Body Closed Form

Is there a closed form solution in general relativity to the 2-body orbit problem?
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Normal of a null surface and null junction conditions in general relativity

I am trying to use the null junction formalism in general relativity (as explained in eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.3763&rep=rep1&type=pdf, "Junctions and thin ...
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Black Hole surface area at Schwarzschild radius is half?

I have been interested in black holes for some time, and am still trying to wrap my head around some of their more obscure properties. Now I know that the Schwarzschild radius is $r= ...
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1answer
128 views

Vector product in a 4-dimensional Minkowski spacetime

I'm studying relativity and I lost track of interpretation along the mathematical formalism. What does vector product mean as an event? I mean, how must one interpret the result of the vector product ...
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1answer
82 views

Wave equation on Schwarzschild background

I am trying to follow the solution of the wave equation for a scalar field on Schwarzschild background from http://batteringram.org/science/gr/scalar_wave.pdf. I have a problem on page 2 where they ...
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1answer
72 views

Distance of closest approach

When deriving the gravitational bending angle of light, In this paper, the author introduced $R$ (the distance of closest approach), in equation ($7$), to solve the problem. My question: How is $R$ ...
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1answer
188 views

Does isotropy imply homogeneity?

This question comes from exercise 27.1 in Gravitation by Misner, Thorne and Wheeler. They required the following: Use elementary thought experiments to show that isotropy of the universe implies ...
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1answer
461 views

Variation of Christoffel symbol and Lie derivative

I've also asked this question on Math Overflow; I hope that asking in two separate fora is not a solecism. Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative $$ ...
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0answers
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The null geodesic for given geodesic [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
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What is the null geodesic equation? [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
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What is the radius of the event horizon?

I know that the Schwarzschild radius is given by $$r~=~\frac{2GM}{c^{2}}.\tag{1}$$ However, If we had the metric $$ds^2~=~−A(r,t)dt^2+\frac{dr^2}{B(r,t)}+r^2(dθ^2+\sin^2{θ}dϕ^2),\tag{2}$$ where ...
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Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
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“Center of a black hole is a time”

$\newcommand{\d}[1]{\mathrm{d} #1}$In one lecture (around 1:33:15) of the series of lectures "Theoretical Minimum" of Prof. Susskind he talks about black holes and the Schwarzschild metric: $$\d ...
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1answer
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How is the Lagrangian defined in GR?

Reading about the Schwarzschild metric in general relativity I see that sometimes $$L=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$$ and sometimes $$L=\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ Which is ...
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1answer
104 views

What is the metric of a constant electromagnetic (pure electric or pure magnetic) field?

For example, imagine a magnetic field $B_x$ directing in $\hat{x}$ direction filling all the space. What is its associated metric field? I can construct the electromagnetic stress-energy tensor for ...
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Do anyone know a good software that where I can easily find the metric from the stress-energy tensor? [duplicate]

I'm using SageMath but the obtainment of the metric from the stress-energy tensor is not trivial, i.e., it is not implemented in a predefined function. Do anyone know a good software that where I can ...
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2answers
623 views

Why is the metric tensor symmetric? [duplicate]

I was reading Schutz, A First Course in General Relativity. On page 9, he argued that the metric tensor is symmetric: $$ ds^2~=~\sum_{\alpha,\beta}\eta_{\alpha\beta} ~dx^{\alpha}~dx^{\beta} $$ ...
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How can the universe be flat?

Okay, so I just want to clarify a few things. According to what I have read, we have measured the universe to be flat, and the shape of the universe is directly related to the mass-energy density. ...
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3answers
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Is there a way to see that $ \nabla_\mu g_{\nu \rho} = 0 $ without explicit computation, where $\nabla_\mu$ refers to the covariant derivative?

In books, it is usually said that this is a consequence of the fact that parallel transport preserves dot product. How ?
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1answer
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What happens to the total volume of a chunk of space that is being sucked into a black hole?

Does it increased, decrease, or stay the same? Maybe it explodes to infinity... Here is a similar question: Do black holes have infinite areas and volumes? But it's different because it asks how to ...
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3answers
241 views

Gravitational potential in GR

Why in GR does the $g_{\mu\nu}$ describe potential?
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145 views

Why acceleration cannot vanish everywhere?

In attempt to introduce gr concepts When there are gravitational accelerations present, as for example in the gravitational field of the earth, the space cannot be the flat Minkowski space. Indeed, ...
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139 views

Find the covariant metric tensor from a given contravariant metric tensor

If given $$g^{\mu\nu}=\pmatrix{\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ ...
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Different signatures

I was working out the christoffel symbols, once where the metric that I am using has (+---) signature and another time where it has (-+++) signature because two books had different signatures and I ...
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Stationary v/s Static

Blau, in his GR book, says that a stationary and spherically symmetric metric is automatically static. He says this easily follows from the fact that for a stationary metric, and in spherical ...
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Spherical Symmetric Metrics

In the case where all books try to illustrate a spherical metric, the procedure goes this way: First they impose isotropy in terms of polar coordinates so that one can write: $$ds^2=-A(r)dt^2 + ...
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2answers
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How to show flat FRW metric has a time-like conformal Killing vector?

I would like to derive the fact that the flat FRW metric has a time-like conformal Killing vector. Is there an easy way to do this? @ValterMoretti showed how one can do this for metrics with a ...