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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Norm of summation of vectors

If we have a vector $\partial_v$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$ g^{vv}\partial_v\partial_v$. My question what if we have ...
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1answer
102 views

Confusion with anti-symmetric tensors!

I want to prove the following fact for two antisymmetric tensors: $$ 2G^{+\rho (\mu} H_{\,\,\, \rho}^{+\nu)} = -\frac{1}{4}\eta^{\mu \nu} G^{+\rho \sigma}H_{\rho \sigma}^{+}. $$ When I try to do it I ...
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1answer
33 views

Raising and lowering indices of the Levi-Civita epsilon symbol in two dimensions

In two dimensions, what is the relation between $\epsilon^a{}_b$ and $\epsilon_{ab}$ where $a, b$ take the values $\{1,2\}$? By that I mean, how does the sign change in that case? In four dimensions ...
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60 views

Is the metric-induced topology relevant at all in a (psuedo) Riemannian manifold? [duplicate]

A (pseudo) Riemannian manifold is a tuple: $$(M,g)$$ where $M$ is a smooth manifold (in particular, a topological space with an atlas) and $g$ is a (pseudo) Riemannian metric tensor. It is apparent ...
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1answer
31 views

A particular coordinate transformation of a metric tensor

So, this was a problem set question for my GR class due yesterday, and I can't for the life of me solve it, it seems I am missing something very trivial. Either the given answer is wrong, or I am. ...
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77 views

Can some components of metric be Finslerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
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84 views

Physical interpretation of a certain Hamiltonian

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
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11 views

Determination of Ricci tensor and Scalar curvature from vielbeins [migrated]

Consider the following metric: $ds^2=h(r)\bigg(dr^2+r^2\big(d\theta^2+\sin^2\theta ~d\phi^2+(d\psi+\cos\theta ~d\phi)^2\big)\bigg)$ We can try to compute the Ricci scalar of this metric by using ...
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50 views

Computing the Ricci Tensor for a Spherically Symmetric Spacetime

For a homework question, we are given the metric $$ds^2=dt^2-\frac{2m}{F}dr^2-F^2d\Omega^2\ ,$$ where F is some nasty function of $r$ and $t$. We're asked to then show that this satisfies the Field ...
2
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1answer
55 views

Variation of the metric with respect to the metric

For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did): \begin{equation} \frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= ...
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1answer
56 views

A true singularity at $t=0$, coordinate independent Big Bang

Consider a flat Robertson-Walker metric. When we say that there is a singularity at $t=0$, clearly it is a coordinate dependent statement. So it is a "candidate" singularity. In principle there is ...
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2answers
65 views

Proper time in general relativity

For general relativity, Wald's GR states that timelike curves, with the norm $g_{ab}T^{a}T^{b} < 0$, can be parameterized by the "proper time" $$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$ This ...
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2answers
64 views

Meaning of general covariance

Quoting from Wald's GR: In the context of special relativity, the principle of general covariance states that the spacetime metric $\eta_{ab}$, is the only quantity pertaining to spacetime ...
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0answers
60 views

(Scalar) Ricci flatness of a metric

What is the physical meaning to vanishing Ricci scalar $R=0$ of a metric in general relativity? Note that this is not the same questions as the geometric meaning of $R_{\mu\nu}=0$ which has been asked ...
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36 views

Uniqueness of the Einstein tensor

This is related with an exercise 17.4-a in MTW Here what i want to show is the Einstein tensor $G_{\alpha\beta} = R_{\alpha\beta} - \frac{1}{2} R g_{\alpha \beta}$ is the only second-rank, symmetric ...
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1answer
54 views

Calculate the mass of a Schwarzchild black hole with Komar integral

In Wald's GR, Komar integral is Eq. (11.2.9): $$M=-\frac{1}{8\pi}\int_S\epsilon_{abcd}\nabla^c\xi^d$$ $S$ can be chosen as a 2-sphere, the boundary of a spacelike hypersurface $\Sigma$ such that the ...
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30 views

“Projection of metric” vs. “projection of curvatures” [migrated]

Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by ...
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0answers
50 views

Finding the metric [closed]

If given the metric: $$ds^2=e^{2U}(dt+w_idx^i)^2-e^{-2U} d\overrightarrow{x}^2$$ where $w = w_idx^i$ is one form How to find the metric in order to find the inverse metric? The new thing about this ...
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1answer
45 views

Most general second-rank symmetric tensor in Einstein theory

I am reading MTW page 407, Exercise 17.1. (a) Show that the most general second-rank, symmetric tensor constructable from Riemann and $g$, and linear in Riemann, is $$a R_{\alpha\beta} + b R ...
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1answer
56 views

Is metric tensor invariant under rotation?

It is said that metric tensor depend on the local coordinate system and therefore are not intrinsic to the surface of an 3d-object? How is it possible, kindly provide any proof or discussion. Also is ...
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145 views

Penrose diagram for spacetime which flows to $AdS_{2}$ at infinity

Consider I have the following 2 dimensional spacetime $(t,z)$: $$ds^2=\frac{4}{z^{2}}\left(1+\frac{1}{z}\right)^{-1}(-dt^{2}+dz^{2}).\tag{1}$$ When $z\rightarrow \infty$ we have ...
2
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1answer
57 views

Is this covariant derivative identity true?

Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$ If this is true, I'm ...
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50 views

Extracting something from Schwarzschild metric

In Papapertrou's lecture book on General Relativity he said in p 137 that from the metric $$ ds^2= e^\nu dt^2-e^\mu dr^2 -r^2(d\theta^2 +\sin^2\theta d\phi^2)$$ one deduces that $$\sqrt{-g} ...
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1answer
99 views

About Christoffel symbols in Riemann normal coordinates

According to the answer to this post, the Christoffel symbols in Riemann normal coordinates are approximated by $$\Gamma^{k}_{ij}(x)~\sim~\frac{1}{2} R^k{}_{ilj}(x_0) \xi^l \tag{5.10}$$ which came ...
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2answers
151 views

Can hyperbolic space be bounded?

There are many visualisations of hyperbolic geometry using Poincaré disks. What are their purpose? Can hyperbolic space be bounded? Can we endow the disk with the structure described by the FLRW ...
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1answer
55 views

Moving From Schwarzchild Geodesic Equations to Equations of Motion

So I am a student and decided (for some bizarre reason) to attempt to tackle general relativity for my final astrophysics and computational physics project this term. I have been doing a lot of ...
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49 views

Schwarzschild metric in Isotropic coordinates

As one wants to jump to Isotropic coordinates in order to write the Schwarzschild metric in terms of them, one does this coordinate transformation: $$r=r'(1+\frac{M}{2r'})^2$$ So we start with the ...
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0answers
43 views

Conditions for a diagonal induced metric?

Let $M$ be a manifold of dimension $n$ with a (say Lorentzian) metric $g$, that is diagonal in some choice of local coordinates. Let $S$ be manifold of dimension $k<n$ , embedded in $M$ by some ...
0
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1answer
121 views

Verifying a solution to Einstein's vacuum field equations

I need to verify a solution to Einsteins vacuum field equations. I have the solution as follows $$ds^2=a\,dt^2+b\,dr^2+\cdots$$ Is the following the right approach? Einsteins equation reduces to ...
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1answer
44 views

Circular orbit in Schwarzschild coordinates [closed]

This was an example in a general relativity textbook which I've been trying to work through myself. A spaceship uses its rocket engine to maintain a circular orbit around a Schwarzschild black hole ...
2
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0answers
51 views

Gravitational multi instantons

I was reading "GW Gibbons and SW Hawking, Gravitational multi-instantons, Physics Letters B 78 (1978), no. 4, 430–432." I had a few questions regarding the metric they define. I was wondering how ...
5
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1answer
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Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ ...
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1answer
80 views

What is Laplace operator of Schwarzschild-Spherical coordinates? [closed]

This is the Laplace operator of Spherical coordinates: What is the Laplace operator of Schwarzschild-Spherical coordinates? where the Differential displacement of Schwarzschild-Spherical ...
3
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1answer
122 views

Correct tetrad index notation

There seems to be some different conventions on the indexes of the tetrad. I am wondering which is the standard, which is correct, and which is an abuse of notation. In Sean Carroll's notes and in ...
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71 views

Metric with 5D signature: +---+

From a paper that a friend sent to me (on inflation theory which I am still in learner mode) a 5D signature +---+ was specified with the 5th dimension being a velocity dimension. I didn't know that ...
4
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1answer
113 views

Help understand article on thin shell formalism

I've been learning the Israel formalism (see original article here, although I prefer the exposition given by E. Poisson in his book A Relativist's Toolkit) for thin shells. I think I understand the ...
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1answer
31 views

Kaluza suggested metric

Is there a book or a paper that goes into the mathematical details of getting scalar curvature of the 5 dimensional metric that Kaluza wrote down? I am running into many mathematical issues for I am ...
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0answers
60 views

What's the meaning when Kerr-Newman metric's mass is zero?

Kerr-Newman metric represents the spacetime of a charged and rotating black hole. If the mass parameter is zero, this metric is still not the Minkowski spacetime. What's the meaning of a charged and ...
3
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0answers
48 views

How to derive the cigar soliton solution to the Ricci flow equation? [closed]

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
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3answers
149 views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: ...
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1answer
64 views

Does the metric define a Riemannian Manifold?

Does a Riemannian Manifold's metric tensor $g$ completely define the manifold, or are more parameters required?
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1answer
83 views

Ricci Scalar of the five-dimensiional Reissner–Nordström metric is different to zero?

The Ricci scalar of the four-dimensional Reissner–Nordström metric is equal to zero. In the case of the five-dimensional Reissner–Nordström metric, the Ricci scalar is different to zero?
6
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2answers
100 views

Why does the $L_2$ norm give the shortest path between 2 points?

Why not the $L_1$ or $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? Could we imagine a different universe where a different ...
1
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1answer
66 views

Timelike curves in Special Relativity

I have a question that probably might sound silly to most of you. We know that a natural Lorentz-invariant parametrization of a timelike curve is provided by: $$\tau$$ the Lorentz-invariant proper ...
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2answers
130 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
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1answer
317 views

How does the Hubble parameter change with the age of the universe?

How does the Hubble parameter change with the age of the universe? This question was posted recently, and I had almost finished writing an answer when the question was deleted. Since it's a shame to ...
2
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1answer
93 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with ...
2
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1answer
93 views

Is it okay to Wick rotate to give the negative of the Euclidean metric? Also, could we make the space-like coordinates imaginary instead?

There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + ...
3
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0answers
102 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
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75 views

Help with parametrization of surface if I'm given the metric [closed]

I've got a homework question. Consider a 2 dimensional space with metric $$ ds^{2} = \frac{dr^{2}}{1 -\frac{2}{r} } + r^{2}d\theta^{2} .$$ I need to show that this is the induced metric ...