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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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 $g_{\mu \nu}$ ...
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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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Curvy space in and around massive objects [closed]

If space curves around massive objects, what happens to the space within the massive objects?
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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 ...
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1answer
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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 = ...
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62 views

Conformal infinity in the Hawking-Hunter-Taylor-Robinson metric

I have been trying to follow some of the computations of this paper: http://arxiv.org/abs/hep-th/0408217 and particularly I couldn't derive the asymptotic form of the Kerr-AdS background (3.27) using ...
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63 views

Why are generators defined oppositely in Weinberg's vs. Maggiore's QFT books?

I've been confused about the sign conventions used in Weinberg's QFT book for a long time. Here's my question: The generators $J^{\mu\nu}$ are defined in this book as ...
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1answer
129 views

Is every solution of Einstein field equations unique?

Einstein's equation is $$8 \pi T_{ab} = G_{ab},$$ where the left side contains the stress-energy tensor and the right side contains the Einstein tensor. Is there exactly one unique stress-energy ...
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64 views

What is the metric at the center of a star? [duplicate]

If there is only one star in the universe then is the metric at the center of the star flat?
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80 views

What are world lines as opposed to arbitrary curves in spacetime?

In GR the spacetime manifold is equipped with a metric which makes it a Lorentzian manifold. It is the metric that is doing the separation of space and time (so that we end up with three dimensions of ...
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Could a 3+1 space be embedded in a 4+1 space and retain its 3+1 characteristics? [closed]

I'm confused because I can conceptualize this embedding scenario in two seemingly incompatible ways. Which of the following scenarios are possible?: 1) 4+1 space automatically enforces 4 dimensions ...
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2answers
45 views

Why is 90 degrees the standard for independence in vectors? [closed]

Why do so many laws and ideas in physics act separately if they are separated by 90 degrees? Say you have a force in one direction, x. You can't add a force within 0-90 degrees without changing the ...
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1answer
142 views

Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \begin{equation} \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = ...
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2answers
125 views

(Hyper)Surface of Simultaneity

How can I determine the surfaces of simultaneity if I know the metric? In particular, what are the surfaces of simultaneity for rotating disk with Langevin metric: $$ ...
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1answer
96 views

The Lie derivative of the metric $g_{ab}$ and index notation

I don't quite know where to start this question. I'm essentially not understanding how to compute the Lie derivative of a given metric and vector. So I have the following definition: $$ ...
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1answer
166 views

Jacobi equation in the book The Large scale structure of space-time

On pp. 79, it is obvious that equation (4.2) \begin{equation} \frac{D}{\partial s}Z^a = {V^a}_{;\ b}Z^b \end{equation} holds, where $Z$ is the deviation vector and $V$ is the unit tangent vector along ...
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1answer
51 views

Why is the spatial term for contravariant 4-gradient negative, whereas for other 4-vectors it is the covariant part that is negative spatially?

The contravariant 4-displacement is: $${x}^{\alpha} = (ct,\mathbf{r})$$ And the contravariant 4-gradient is: $${\partial}^{\alpha} = (\frac{1}{c}\frac{\partial}{\partial{t}},-\nabla)$$ From what I ...
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1answer
73 views

A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, \begin{equation} S=\int\sqrt{g}d^4x\ f(R) \end{equation} assuming there are no (or ...
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1answer
42 views

Ordering of Contravariant and Covariant spinors. Understanding the spinor space

I've been referring to Pg.36-Pg.38 in Introduction to Supersymmetry by Wiedamann. For understanding the precise origin of dotted, undotted indices on Spinors. He starts off my saying that $M$ acts on ...
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2answers
89 views

Standing wave on a circle [closed]

Suppose that we have a standing wave on a circle. I heard that by gradually increasing the radius of the circle, the wavelength will also increase to keep the standing wave. Is it right? If yes, ...
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2answers
86 views

How do you know what kind of space(time) you have when solving the Einstein Field Equations?

I'm experimenting with the EFE, and I ''invented'' a metric; a diagonal non-zero metric, and I discovered that the Riemann tensors are equal to zero which implies the Einstein tensor $G_{mn}$ equals ...
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3answers
67 views

What is the $ds^2$ notation in relativistic physics?

Could someone please explain me intuitively how $ds^2$ represents distance in relativistic physics?
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1answer
53 views

Using the Metric in Book Gravitation (MTW)

Here is the whole Box 2.2, at Page 55 The dot behind the second $-p^2$ seems to be a "planck mass" (sarcasm, flea egg) or just the book's style to use Dot behind the equations. So the Equation is ...
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4answers
937 views

Coordinates vs. Geometries: How can we know two coordinate systems describe the same geometry?

Specifically, I'm asking this because I'm taking a class on General Relativity, and in Hartle's book Gravity, in Ch. 12, after having spent some time using Schwarzschild coordinates, we are introduced ...
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70 views

Need some help understanding Relativistic Notation

My question originates from what is done in the book on Quantum Field Theory book by Mark Srednicki on page 21 (if anyone has it). So say you have an inertial frame that is represented in the ...
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52 views

Metric defining an sphere [closed]

I want to find for which cases this metric can define an sphere: $$\frac{1}{P^2}\left(\mathrm d\theta^2+\sin^2 \theta\; \mathrm d\phi^2\right)$$ where $P=\sin^2 \theta+K\cos^2 \theta$, with $K$ the ...
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127 views

Kerr Metric from rotated Schwarzschild?

Say we have got a system in GR that is described by the Schwazschild metric. Then we perform a coordinate transform that gives the metric in a rotating system. Why is the transformed metric not the ...
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1answer
63 views

Problem on parallel transport [closed]

I've been trying to go through an example for parallel transport but I cannot quite follow the solution. A surface (paraboloid) is given by the parametric equation $r(ρ, φ)$ = $ρ \cos(φ)\hat{i}$ + $ρ ...
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35 views

Electromagnetic tensor in cylindrical coordinates from scratch

I want to calculate the electromagnetic tensor components in cylindrical coordinates. Suppose I did not know that those components are given in Cartesian coordinates by $$(F^{\mu \nu})= ...
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1answer
114 views

What are gravitational waves relative to? [closed]

I have been having trouble picturing what kind of waves say Sun and Earth would make. Looking from top perspective Sun is in the middle and denting space while Earth is moving which is also denting ...
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1answer
44 views

Is there a parametrization for the shape of space?

I was thinking about how the space is curved. And how do we know that the shape of space arround a singularity is something like that: So I was trying to make a similar parametrization of this kind ...
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4answers
171 views

What is a metric? [closed]

I was taking a basic course in general relativity. They introduced a concept of a metric which I wasn't able to understand can somebody explain it to me why do we need a metric in curved spaces?
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1answer
65 views

Gravitational waves induce changes in the $h_{00}$ (time) component of the metric?

I'm rather stumped by a subtle point regarding metric perturbations of GW. I'm well aware the GW are able to produce changes in the flat space metric, They are transverse and have planes of ...
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1answer
45 views

Representing 1+1 Minkowski space as a surface in 3D Euclidean space

In 1+1 Minkowski space the distance between two points is given by$$ (x_1 -x_2)^2 -(t_1 - t_2)^2.$$ This is different from the Euclidean distance. But is it possible to come up with a 2D surface ...
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2answers
79 views

How does gravitational wave compress space time?

My question came from the talk of how gravitational wave stretches and compresses space time. Say there are two protons that are 1 centimeters apart, as a G-wave passes through them, would the ...
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Spherical metric multiply by a function

I know that if I want to get the metric for a two sphere I consider a Cartesian flat space, I change to spherical coordinates and then I consider that the radio is constant (so the space is not flat ...
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1answer
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Gravity with more than one metric tensor

As weird as it sounds, yes, there are gravity theories with more than one metric tensor. This is called bimetric gravity. My question to those who have encountered bimetric gravity before: a) ...
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1answer
145 views

How can I understand $\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $ in the simplest way?

How can I understand this equation $$\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $$ in the simplest way? I am a 13 year old boy who is totally ...
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49 views

Choice of the coordinate system in a spherically symmetric solution

For a static spherically symmetric solution, the metric can be written as $$ds^2=-A(r)^2 dt^2+\frac{dr^2}{B(r)^2}+C(r)^2d\Omega^2$$ In some cases, we can write $C(r)=R$ and interpret then $C$ (or ...
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525 views

Stokes theorem in Lorentzian manifolds

I've fallen accross the following curious property (in p.10 of these lectures): in order to be able to apply Stokes theorem in Lorentzian manifolds, we must take normals to the boundary of the volume ...
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1answer
53 views

Geometrically deriving Lorentz transformation from Minkowski diagram

How can we derive Lorentz transformation from a Minkowski diagram (like below image) by using only geometry theorems such as sines theorem and Pythagoras theorem?
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1answer
98 views

Working with indices of tensors in special relativity

I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector ...
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1answer
37 views

Three time dimensions and one spatial dimension degeneracy

Is there a sort of degeneracy in the space-time metric for our universe? What I mean is that it seems you would observe all the same properties of our universe if you simply placed a minus sign in ...
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1answer
67 views

Does Birkhoff's theorem hold inside the event horizon?

Can Birkhoff's theorem be used to say that the blackhole exterior and interior sections of Kruskal-Szekeres's solution (or coordinate transformations of it like Gullstrand–Painlevé coordinates, etc.) ...
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2answers
57 views

The Physical Basis of Our Assumptions about Physical Space

Let $\mathcal{S}$ represent the set of all points in physical space. Using measuring rods and assuming our use of them does not depend on time, we can establish a one-to-one correspondence between ...
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54 views

Schwarzschild manifold

I am given the following metric $$ds^2 = \frac{dr^2}{1-2m/r} + r^2dS,$$ where $dS$ is the standard metric on the unit sphere $S^2$. I am told that this is isometric to $\mathbb{R}^3$ or (taking its ...
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Exercise 18b in Schutz's First course in GR

The question is as follows: Show that a timelike vector and a non-zero null vector cannot be orthogonal. So we have a timelike vector $\vec{A}$, s.t $\vec{A}^2<0$; and a non-zero null vector, ...
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1answer
54 views

Does the Lorentz transformation necessary follow from the two postulates of relativity?

The two postulates of special relativity are: The choice of what inertial frame to use is arbitrary: all laws of physics are invariant. (the principle of relativity) The metric $$(\Delta s)^2 ...
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1answer
79 views

Meaning of $R=0$, $R_{ab}=0$. $R_{abcd}=0$

First let me state some definition The Einstein tensor is given by \begin{align} G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \end{align} and note that \begin{align} G^{\mu}_{\phantom{\mu} ...
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65 views

Divergence of inverse of metric tensor

I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?! I'm not so familiar with the divergence of the second ranked tensor. ...