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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2
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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
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1answer
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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 $f$? ...
2
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0answers
28 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 ...
-3
votes
1answer
94 views

Calculating speed in four dimensions [on hold]

If you are moving at $c$ in 3D space and $c$ in time axis too, What would be your total speed? Edit: Since question has been voted to be closed, I shall make an Edit. In 4D world all objects move ...
3
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1answer
60 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 ...
4
votes
1answer
27 views

Lie Derivative of Kahler 2-form

Suppose there is a Killing vector $k$ on a Kahler manifold $M$. By definition, $k$ generates isometries of the metric. That is, $L_kg=0$, where $L$ is the Lie derivative. At the same time, there is a ...
0
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1answer
42 views

Physical reasons for metric definition in special relativity [duplicate]

I am working through "General Relativity" by Wald, and am currently going through the brief section on Special Relativity. The spacetime metric is defined as $\eta_{ab} = \sum\limits_{\mu, \nu=0}^3 ...
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2answers
65 views

Trouble understanding spacetime and invariant interval

First, how is the invariant interval useful? How can it help us understand things around us in the universe? Second, I know that they changed time into space or better say SPACETIME in order to ...
3
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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 ...
5
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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 ...
4
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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 + ...
4
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1answer
57 views

How can you tell if spherical-like coordinates are locally flat across the origin?

In general relativity, with spherical-like coordinates in a radial gauge, I have a metric that looks like: $$-g_{tt}\mathrm{d}t^2 + g_{rr}\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ ...
0
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2answers
70 views

Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$

We have a covariant derivative of a covariant tensor: $$ A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha} $$ The covariant derivative of a contravariant tensor is: $$ ...
0
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1answer
45 views

Norm of the momentum 4-vector

The norm of the momentum 4-vector is $\mathbf{P}.\mathbf{P}$ $= (\gamma mc, \gamma mv).(\gamma mc, \gamma mv) = \gamma mc^2 - \gamma mv^2$ But why is $\gamma mc^2 - \gamma mv^2 = mc^2$?
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0answers
26 views

AdS boundary global vs Poincare'

Is the global boundary of AdS the same of the boundary written in Poincare' coordinates?
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0answers
24 views

Volume form of the AdS_{4} Space

Regarding the unit radius $AdS_{4}$ space, the metric in global coordinates, is given by: $$ds^{2}_{AdS_{4}}=\frac{1}{\cos^{2}{\rho}}[dt^{2}-d\rho^{2}-\sin^{2}\rho d\Omega_{2}^{2}]$$ where ...
76
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1answer
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Is there such thing as imaginary time dilation?

When I was doing research on General Relativity, I found Einstein's equation for Gravitational Time Dilation. I discovered that when you plugged in a large enough value for $M$ (around $10^{19}$ ...
1
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2answers
70 views

Straight line null geodesics in Minkowski, De Sitter and Schwarzschild

I'm trying to understand which part of the following metric determines whether photons travel on a "straight" line (thinking of $(t,r,\theta,\phi)$ as a flat background), the metric I'm considering ...
4
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0answers
22 views

Gravitational lensing and cosmic strings

Say we have a straight cosmic string lying along the $z$-axis, with energy-momentum tensor $$T_{\mu\nu}=\mu\delta(x)\delta(y)\operatorname{diag}(1,0,0,-1)\tag{1}\label{1}$$ for some small positive ...
1
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2answers
48 views

Motivation for usage of 4-vectors in special relativity

I understand that if one considers a 4-dimensional space-time from the outset then 4-vectors are the natural quantities to consider (as opposed to 3-vectors as in Newtonian mechanics), since the ...
3
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1answer
72 views

Homogeneity and isotropy and derivation of the Lorentz transformations

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad ...
0
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0answers
27 views

Inertial coordinate systems [duplicate]

In Newtonian mechanics, by the following two assumptions: (i) The time is absolute. (ii) The length is absolute. it is easy find the relations betweem two coordinate systems with uniform motion ...
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2answers
75 views

Definition of the Lorentz transformations [closed]

Until very recently I believed that the Lorentz transformations were defined as "the transformations that carry one inertial reference frame into another". In Wikipedia's page we find something along ...
0
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1answer
55 views

Questions about null geodesic [closed]

Show for the null geodesic in 3D flat spacetime using polar coordinates so the line element is $ds^2=-dt^2+dr^2+r^2d\phi^2$. Do light rays move on straight lines? My question is that I only learned ...
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0answers
35 views

Interpreting meaning of coordinates given a metric

I was working problem 3.6 in Carroll's GR textbook and was given the following metric, which is a good approximation to the metric outside the surface of the Earth. $ds^2=-(1+2 \Phi(r))dt^2 + ...
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4answers
165 views

How do we know the Schwarzschild solution contains an object of mass $M$?

The Schwarzschild metric is $$ds^2 = - \left( 1 - \frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$ In Carroll's GR book, it is claimed that $M$ is the mass of the ...
6
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2answers
187 views

What manifold is spacetime?

In General Relativity, spacetime is a $4$-dimensional manifold with one Lorentzian metric tensor defined on it. In the Special Relativity case what manifold is spacetime is quite clear: it is ...
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0answers
31 views

Geodesic tangent vector in a Riemannian 4-space

I am doing a question in Lewis Ryder's introduction to General relativity. I am very close to the answer but not quite there. The question is: A Riemannian 4-space has metric $$ds^2 = ...
3
votes
1answer
50 views

Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
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1answer
32 views

Conformal Transformation: Minkowski sheet to cylinder

What conformal transformation can I make to 2d Minkowski with metric $ds^2=-dt^2+dx^2$ to show that it is conformal to a cylinder?
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0answers
32 views

What is the Metric Tensor? [duplicate]

I was studying Einstein's Field Equation, and this was the most common symbol. Can you explain what it is and how it could be used?
6
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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
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1answer
49 views

Orthogonal of tangent vector in Rindler coordinates [closed]

For 2D space time from $(t,x)$ to $(u,v)$ the transformation are $$t = u \sinh(v)$$$$x=u\cosh(v)$$ Asking to show that two families of curves $u = \textrm{constant}$ and $v = \textrm{constant}$ ...
4
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2answers
125 views

Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
0
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1answer
63 views

Straight lines in general relativity

This question stems from a possibly misguided attempt to understand General Relativity. I am about to leave High school for college, I do however have a rudimentary understanding of tensors, and I ...
3
votes
1answer
126 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
5
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0answers
129 views

Can special relativity be derived from the invariance of the interval?

As far as I know, the classical approach to special relativity is to take Einstein's postulates as the starting point of the logical sequence, then to derive the Lorentz transformations from them, and ...
8
votes
1answer
144 views

Acceleration of particle “held in place” at $x = 1$ [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = -\cosh x\,dt^2 + dx^2.$$Consider a particle that is "held in position" at $x = 1$. What ...
4
votes
4answers
190 views

Distance in General relativity

I read a few lines about general relativity and one of the first equations is the one defining the eigentime of a time - like curve. But observers should also be able to measure length, right? So is ...
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0answers
47 views

Gauge invariance in gravitational field

I have read that the linearized equation for the metric fluctuations $h_{\mu\nu}$, namely: $$ \partial^2h^{\mu\nu}-\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha}) ...
1
vote
1answer
27 views

Deriving Electromagnetism energy-stress tensor in GR [closed]

Please find the mistake in the following calculations. We have $L=-F^{\mu\nu}F_{\mu\nu}$, and try to derive the energy-stress tensor using $\delta(-g)^{1/2}=\frac{1}{2}(-g)^{1/2}g^{\mu\nu}\delta ...
5
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2answers
60 views

Wald's General Relativity, section 6.3 Page 144

I cannot understand how he reaches the conclusion in equation 6.3.36 and 6.3.37; even the terminology is somewhat confusing. This is a problem of bending of light under gravitational field. This is ...
0
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1answer
56 views

Classical Limit of Schwarzschild Metric

The orbit of a test particle orbiting a black hole can be described by the Lagrangian $$\mathcal{L} = -\frac{1}{2}\left(-\left(1-\frac{2 G m}{c^2 r}\right) \dot{t}^2 + \left(1-\frac{2 G m}{c^2 ...
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0answers
51 views

Finding an explicit expression for inverse of Lorentz transformation ${\Lambda^\mu}_{\nu}$

I'm studying in a module titled "Symmetries and Action Principles in Physics." I'm having a small trouble with the notation, I think, of special relativity. We have in the text In fact we may ...
0
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1answer
64 views

Scalar fields in AdS$_3$

I'm looking at lecture notes on AdS/CFT by Jared Kaplan, and in section 4.2 he claims that the action for a free scalar field in AdS$_3$ is $$S=\int dt d\rho d\theta ...
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0answers
18 views

Gibbons-Hawking-York boundary term expanded at second order in the fluctuation

Does anyone know a general form for the Gibbons-Hawking-York boundary term expanded at quadratic order in the fluctuation of the metric? Assume to define the fluctuation of the metric $g_{\mu \nu}$ ...
4
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2answers
82 views

Maxwell's equations from differential forms

I found the following in some lecture notes I took some time ago: $$ \mathbf{E}=-\text{grad}\Phi-\partial_t\mathbf{A}\\ \mathbf{B}=\mathrm{rot}\mathbf{A} $$ These are the electromagnetic fields ...
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2answers
37 views

Curvy space in and around massive objects [closed]

If space curves around massive objects, what happens to the space within the massive objects?
4
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2answers
84 views

What's the metric of the Standard Non-Time-Orientable Spacetime

If you've read any spacetime topology, you know that spacetime. It is the amazing rotating lightcone identified after half a rotation. And outside of De Sitter space with some identifications, it is ...
4
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1answer
78 views

Negative mass thin shell collapse

Suppose we have a collapsing light-like (ingoing) shell with negative mass and decreasing further. The shell is radiating and so the exterior region is that of the outgoing Vaidya solution. $$ds^2 = ...