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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How can I use Einstein's field equations?

Every time I try to find the answer to this question I get redirected to different pages that ultimately do not end up answering my question. I have some understanding of Riemannian geometry but have ...
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2answers
285 views

Finding the metric tensor from the Einstein field equation?

I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 ...
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3answers
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Why doesn't $ds^2 = 0$ imply two distinct points $p$ and $p'$ on a manifold are the same point?

Let's suppose I have a spacetime manifold $M$. Let $p$ be a point on my manifold. Now I move from $p$ to some other point $p'$. Presumably I should have moved some "distance" right? How can I speak of ...
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1answer
50 views

Does anyone recognize the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$?

I've stumbled upon the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$. Obviously the corresponding metric tensor has components: $\begin{bmatrix} g_{tt} & g_{tr} \\ g_{rt} & g_{rr} ...
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1answer
463 views

Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence. E.g. how can one derive ...
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3answers
353 views

Einstein tensor in Friedmann equations : where is the missing $c^2$?

I would like to demonstrate the several forms of the Friedmann equations WITH the $c^2$ factors. Everything is fine ... apart that I have a missing $c^2$ factor somewhere. In all the following $\rho$ ...
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1answer
64 views

How can I use Einstein's field equations to find the metric tensor? [duplicate]

I have watched and read a lot on the topic of General Relativity and the geometry behind it. I am confident that I can derive an approximation of the the stress-energy-momentum tensor with just the ...
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1answer
81 views

What is the covariant basis around a Schwarzschild black hole?

First of all, I'm not interested in time for this question. So lets consider a 3-manifold whose metric is the spatial part of the Schwarzschild metric, so it has the event horizon and the singularity ...
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3answers
150 views

Finding the appropriate coordinate transformation given two metrics

Given the two-dimensional metric $$ds^2=-r^2dt^2+dr^2$$ How can I find a coordinate transformation such that this metric reduces to the two-dimensional Minkowski metric? I know that ...
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0answers
20 views

Decomposition into symmetric and antisymmetric form [on hold]

(a) Given a second-rank tensor Tμν, often viewed as an $N \times N$ matrix (for a space of dimension $N$), show by explicit construction that one can always decompose $T_{\mu\nu}$ into a symmetric ...
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1answer
90 views

Is any spacetime metric physically realizable?

Given a spacetime metric, you could work out a stress-energy tensor for each position that would result in that metric. I know building a wormhole requires negative energy densities, which are ...
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1answer
47 views

Will a stress-energy tensor have the same identities as it's metric?

Say I have a metric tensor where $$g_{00} = -c^{2}\ and $$ $$g_{01}=g_{02}=g_{03}=0$$ and $$g_{12}=g_{13}=g_{23}$$ and $$g_{11}=g_{22}=g_{33}$$ My question is straightforward: would the same or ...
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2answers
70 views

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric?

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric? Consider the metric $g_{\mu\nu} = \begin{pmatrix} 1 & 0 &0 \\ 0 & r^2 & 0\\ 0 & 0 & r^2\sin^2\theta \end{pmatrix}$
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3answers
431 views

Why is the scalar product of four-velocity with itself -1?

My GR book Hartle says the scalar product of four-velocity with itself $-1$? Consider the definition of four velocity $\mathbf{u} = \frac{dx^{\alpha}}{d\tau}$. Suppose I take the scalar product of ...
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1answer
98 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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1answer
81 views

How to find a metric of a general observer?

Yes, that's it. How to find a particular metric of an observer in general relativity? Let's say we have a static metric: ...
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1answer
52 views

Computation of $T^{\mu\nu}$ from Lagrangian density $\mathscr{L} $

I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as ...
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4answers
158 views

What is Minkowski spacetime?

I was browsing through an article on spacetime when I caught the words Minkowski Spacetime. A Wikipedia search brought me an article too complex for me to totally understand. So what is Minkowski ...
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2answers
1k views

Why is spacetime not Riemannian?

I apologize if this is a naïve question. I'm a mathematician with, essentially, no upper-level physics knowledge. From the little I've read, it seems that spacetime is Lorentzian. Unfortunately, the ...
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1answer
36 views

Energy-Momentum Tensor with mixed indices

I know that $T_{\mu\nu}$ is the Energy-Momentum Tensor and $T=g^{\mu\nu}T_{\mu\nu}$, but does anyone know what $T^{\nu}_{\mu}$ is and how its calculated?
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0answers
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Boyer–Lindquist coordinates

In the Kerr solution to the vacuum Einstein Equation written in Boyer–Lindquist coordinates. Because it is not spherical polar coordinates, $r$ ranges from 0 to infinity does not cover all the space, ...
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2answers
508 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
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1answer
69 views

Inverse Metric Tensor

First the setup: Let $\mathcal M$ be a $2$-dimensional manifold. Let $U_P$ be some open neighbourhood of a point $P \in \mathcal M$. Let $\mathcal F : U_P \rightarrow \mathbb R \times \mathbb R$ be ...
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1answer
56 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
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3answers
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Why does Minkowski space provide an accurate description of flat spacetime?

What is the chain of reasoning (beginning, of course, from observations about the universe) that leads one to predict that Minkowski space provides an accurate description of space-time in the ...
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1answer
36 views

Given any metric, how to find the straight line path between two points? [closed]

Say we are given a two-dimensional metric $$ds^2=f_1(x)dx^2+f_2(x)dy^2,$$ for any kind of function. How do we calculate the distance along a straight line path (not the shortest possibly) between, ...
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1answer
64 views

Four-vectors and metric tensor

I think it's safe to say that if $x^\mu=(x^0,x^1,x^2,x^3)$, then $x_\mu=(x^0,-x^1,-x^2,-x^3)$. But I don't really understand why one follows from the other. Could someone explain? Also, I've been ...
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53 views

Eddington-Finkelstein coordinates: Why $\ln(r-2m)$ instead of $\ln|r-2m|$?

If one considers the Schwarzschild metric $$ \text d s^2 = -V(r)\text d t^2 + \frac{1}{V(r)}\text d r^2 + r^2 \text d \Omega^2\;,\qquad V(r) = 1-\frac{2m}{r}\;, $$ and introduces the ...
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1answer
136 views

Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical ...
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2answers
49 views

Hermitian Metric and Geodesics

Why isn't general relativity developed with a Hermitian metric and a theory of complex valued paths and geodesics? The concept of arc length and geodesic suffers under a pseudo-Riemannian metric. My ...
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0answers
31 views

Symmetric energy-momentum tensor using derivative wrt. metric

I can find the Noether current for space time translation symmetry by demanding that the first order correction to the Lagrangian vanishes upon infinitesimal translations of coordinates. But in cases ...
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539 views

Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric?

Why is it that all problems I encountered until now have metrics that when represented in a matrix form turn out to be symmetric? Aren't there asymmetric matrices representing some metrics?
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1answer
41 views

Schwarzschild Solution Convention

In looking at the components of the Schwarzschild Metric, one finds $ g_{00} = (1 - \frac{r_s}{r})c^2 $. Wikipedia states that $r$ is measured as the circumference, divided by $2π$, of a sphere ...
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5answers
144 views

Is $ds^2$ just a number or is it actually a quantity squared?

I originally thought $ds^2$ was the square of some number we call the spacetime interval. I thought this because Taylor and Wheeler treat it like the square of a quantity in their book Spacetime ...
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2answers
64 views

Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric

My calculus has 30+ years of rust on it and I am stuck on the integration of the interval in General Relativity... I wish to calculate the spatial coordinate at time t of an object moving with ...
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2answers
138 views

Why do things slow down when you move faster, rather than speed up?

I've been trying to get to grips with SpaceTime. As I understand it, we move at a set rate through spacetime. Any increase in our rate of travel through space results in a decrease in our rate of ...
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1answer
61 views

Geodesic deviation

In S. Carroll Lecture Notes on General Relativity, chapter 6, pages 152-153 we have equation (6.62) $$\tag{6.62} \frac{\partial^2}{\partial t^2} S^\mu=\frac{1}{2} S^\sigma \frac{\partial^2}{\partial ...
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2answers
146 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
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1answer
139 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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0answers
24 views

Norm of Killing vector field

Let us suppose we have a Killing vector field with $X^a = 1/2$ and $X^b = 1/3$ and $g_{ab}=1$ where the other $c$ and $d$ components are zero. Now we want to find its norm: The formula for finding ...
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1answer
79 views

Naturalness of tensor fields in general relativity?

In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ...
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1answer
33 views

isotropy of 3-space and spacetime metric

The most general spacetime metric is given by $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=c^2dt^2+g_{0i}dtdx^i-g_{ij}dx^i dx^j$$ Why does the second term said to violate isotropy of 3-space? It is true that, ...
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1answer
73 views

Why does $\frac{d\tau}{d\sigma} = L$?

I am given a (3+1)-dimensional spacetime that has the line element \begin{equation} ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 + r^2 d\phi^2 \end{equation} ...
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1answer
41 views

Acceleration of stationary observers in their own reference frame?

In the beginning of this link: https://www.math.ku.edu/~lerner/GR/Schwarzschild.pdf they calculate the acceleration of a stationary observer. As I understand, this accleration is seen by an ...
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3answers
354 views

How to connect Einstein's Special Relativity (SR) with General Relativity (GR)?

How Einstein's SR becomes GR? $$ds^2=dr^2-c^2dt^2,$$ $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.$$ When the $s$ is constant $ds^2=0$, isn't it true? How to connect Einstein's SR with GR? What is the ...
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1answer
70 views

Little problem with indexes

Suppose I have a diagonal matrix metric, like $$b_{\mu\nu} = \mbox{diag}(1, -1, -1, -1)$$ namely there are nonzero values only for $\mu = \nu$. My problem is this (please be quiet to explain me ...
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1answer
60 views

Can I simply find the Christoffel symbols by dividing by $g$?

Given the following equation \begin{equation} g_{\alpha\delta} \Gamma^{\delta}_{\beta\gamma} = \frac{1}{2} \left(\partial_\gamma g_{\alpha\beta} + \partial_\beta g_{\alpha\gamma} - \partial_\alpha ...
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3answers
134 views

What besides the metric do you need to set up the EFEs and the geodesic equation?

One of my professors wrote on the board (1) Mass tells spacetime how to curve $\to$ Metric/Einstein Field Equations (2) Spacetime tells mass how to move $\to$ Geodesic equation Suppose I am given ...
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1answer
47 views

Metric Tensor Identity Manipulation

If I take the metric identity in 4d minkowski spacetime $$ g_{uv}\frac{dx^u}{d\tau}\frac{dx^v}{d\tau}=1, $$ where $\tau$ is proper time parameterisation. Can I conclude that ...
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On GR with perturbation

Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation? The paper is http://arxiv.org/pdf/0704.0299v1.pdf In equation 25 for $R_{00}$, ...