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Questions tagged [metric-tensor]

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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2answers
65 views

What is the way to represent a Lorentz tensor field?

For a vector field one can represent this with an array of arrows. There is a standard sort of way to represent tensors in Euclidean space as small ellipses. Is there any standard way of ...
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1answer
65 views

Is spacetime defined mathematically without using $c$ speed?

Is there a mathematical definition of spacetime that does not use $c$ speed as a conversion factor or involve the spacetime interval? If not why?
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1answer
34 views

Distinguishing between matrix forms when reordering indices of tensors

I'm studying general relativity and tensors. It seems that in the cooridnate independent form of the tensor, the order of indices matters even between an upper and lower index. For example, in general,...
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4answers
766 views

Twin paradox in curved space time [duplicate]

In a flat space, where special relativity works, a travelling body can only return to the same point if we apply some kind of acceleration to the body. So twin paradox is not a paradox because a ...
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1answer
62 views

Spatial part of Robertson-Walker metric

The spatial part of the FRW metric can be written as $$d\Sigma^2=d\rho^2+f^2(\rho)(d\theta^2+{sin}^2\theta d\phi^2)$$ where $f(\rho)$ satisfies $$\frac{df}{d\rho}=\frac{f(2\rho)}{2f(\rho)}.$$ I am ...
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0answers
29 views

Notion of 'functional degrees of freedom' for the metric function in GR?

I have read through the numerous questions on 'degrees of freedom' in the metric tensor, and won't list them all here. However none of them address my question on 'functional' degrees of freedom in ...
4
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1answer
83 views

How do the properties of a Lie group (represented as a manifold) manifest in the metric tensor of that manifold?

I know this is a math question; however, physicists are more likely to be familiar with what I'm asking (also, I'm directly trying to utilize it in the context of general relativity). I may have ...
2
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1answer
58 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...
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0answers
30 views

Contravariant metric in Newton-Cartan spacetime

I'm interested in the geometrized newtonian gravitation or Newton Cartan theory. In every reference that I have found begins saying that a Newton-Cartan spacetime is a manifold M with some structures. ...
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3answers
60 views

Confusion over use of contravariant notation in Noether's theorem and Lagrangian filed theory

The variational principle clearly gives $$\frac{\partial \rho}{\partial t} + \overrightarrow{\nabla}\cdot \mathbf{J} = 0.$$ So the sign is positive. However in my lecture notes it is claimed that ...
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1answer
58 views

What is the difference between space-like four-momentum and time-like four-momentum? [on hold]

I was reading a wiki page on Tachyon, came across these terms? What i need is bit of a mathematical description to understand these terms?
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0answers
62 views

Integrate isotropic coordinates [on hold]

When I take isotropic coordinates at the general metric for a static-spherically symmetric spacetime, finally you get the following integral: $$\dfrac{dr'^2}{r'^2}= \dfrac{dr^2}{r^2 (1-2M/r)}$$ i.e: ...
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1answer
55 views

Are there timelike 3D surfaces in special relativity

I am reading Scharf's 'finite QED' and I am puzzled at the beginning. He first introduces Minkowski space with $(+,-,-,-)$ signature, and here is a definition I find difficult: A three-...
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1answer
29 views

How to get the four-velocity components from a given metric tensor?

I’m a little bit confused about how to get the four-velocity components from a given metric tensor (or line element). For instance, which are the components of the four-velocity in the Schwarzschild ...
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1answer
45 views

Integral over an area of spacetime [on hold]

Is it possible to evaluate this integral in spacetime? $$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$ If it is ...
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2answers
67 views

Lorentz Factor is just the SINE function Opposite/Hypotenuse

Has anyone noticed that the Lorentz Factor used to calculate the relativistic length compression and time dilation for an object moving through space, can be written as the SINE function… Opposite ...
3
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1answer
60 views

Deriving Friedmann Equations without General Relativity

Can we derive the analytic Friedmann Equations without general relativity, starting from completely classical/nonrelativistic arguments? (If we consider sufficiently small volumes.)
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2answers
51 views

Divergenceless of energy momentum tensor for any metric $g_{\mu\nu}$

As suggested by @my2cts, from this post, I want to know if the divergenceless of energy-momentum energy tensor is valid for any metric $\eta_{\mu\nu}$ (i.e for example with $\eta_{\mu\nu}=g_{\mu\nu}$)?...
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3answers
94 views

Doubt about the vacua equations of General Relativity

I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical. So, in a elementary point of view, ...
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2answers
77 views

Determinant of the metric tensor

After a change of coordinate system on flat space from $x\rightarrow y$, we have the metric tensor: $$g_{\mu \nu} = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\...
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0answers
54 views

Way of splitting spacetimes into space and time

On several occasions, we would like to separate the components of the metric into three sets ($g_{00}$,$g_{0\alpha}$,$g_{\alpha \beta}$). Please refer to exercise $4.2$ of "Gravitation, Foundations ...
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2answers
72 views

Maxwell Equations in Friedman-Robertson-Walker metric

The Maxwell equations are relativistic. But what happens to them in an expanding space time? I assume that only the charge density $\rho$ is affected, i.e. only Gauss's law gets modified. Am I right ...
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1answer
146 views

The derivation of the Lorentz transformation: addition of distances

In the derivation of the Lorentz transformation, one has a reference frame, $S$, at rest and another, $S'$, moving away at constant speed $v$. At time $t$ there is an event at a point $x$ in $S$. The ...
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0answers
65 views

Difficult coordinate transformation

I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric $$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\...
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1answer
40 views

How to lower both indices on the metric tensor?

If I have the tensor matrix $g^{\mu \nu}$ and I want the tensor matrix too $g_{\mu \nu}$, what is the calculation? Inverse? Adjoint? Some rule?
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2answers
76 views

How can the $v$ coordinate be null if $g_{vv}\neq 0$?

I'm probably missing something very basic here. As far as I know, a coordinate is called null when its coordinate lines are null. This that if $(M,g)$ is spacetime and $x^\mu$ a coordinate chart, the ...
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1answer
74 views

Relation between curvature in orthonormal basis and in “standard” metric form

Im familiar with both formulations of GR - standard with metric and connection coefficients and that based on orthonormal frames and differential forms (Cartan's structure eqns) in solving Einstein's ...
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0answers
50 views

Showing that $4$-velocity and $4$-acceleration are mutually perpendicular in special relativity

The following is said in Landau and Lifshitz I'm trying to understand how we arrive at formula $7.4$. In my class a different approach was taken if we let $$u^a = \frac{dx^a}{dt'}$$ where $dt' = dt \...
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3answers
85 views

Question about inner products of tensors and Einstein summation convention

So I am studying Special Relativity and basic tensor calculus and got stuck at an exercise. $$F^{\mu \nu}: = \left[ \begin {array}{cccc} 0&-{\it E_x}&-{\it E_y}&-{\it E_z} \\ {\it E_x}&...
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0answers
53 views

Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{γ_μ,γ_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*...
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2answers
48 views

Dual of Kalb Ramond field [closed]

i've been studying string theory for 4 days, i have a Kalb Ramond $B_{(2)}$ of this kind (from a $5^2 _2$ solution [1]) and i want evaluate its dual but i don't obtain the right result: The variables ...
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2answers
88 views

Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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2answers
106 views

Is the geodesic equation independent of an initial condition?

The following argument is used to determine the unknown factors (e.g., $A(r)$ and $B(r)$) in the Schwarzschild metric. $$ \lim_{r \to ∞}A(r) = \lim_{r \to ∞}B(r) = 1 \space\space\space\space\space\...
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1answer
107 views

Using the Minkowski metric show that $u_au^a = -c^2$

Using the Minkowski metric show that $u_au^a = -c^2$. I am trying to solve the above problem. Firstly $4$-velocity can be defined as $$u_a = \frac{dx^a}{dt'}$$ where $dt' = dt\sqrt{1-\frac{v^2}{c^2}...
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1answer
67 views

Metric of Gravitational Field near Earth's surface

I am trying to do a calculation in which I am trying to work out how a scalar field behaves in the earth's gravitational field near the surface. I know that the Schwarzschild metric would describe the ...
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0answers
56 views

Mass-Density relation in General Relativity

Suppose one has a static and spherically symmetric spacetime with line element defined by: $$ds^{2}=-c^{2}e^{\nu(r)}dt^{2}+e^{-\nu(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$ where $\nu(r)...
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2answers
44 views

Covariant and contravariant coordinates - index notation

I am learning about electrodynamics and have recently been introduced to the four vector. I also come fresh to the idea of covariant four vectors and contravariant four vectors. My question concerns ...
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4answers
2k views

Are any of Euclid's 5 postulates false in Minkowski spacetime?

I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any),...
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0answers
73 views

Convention Special Relativity

I am a little confused about the factor $c$ in different conventions of special relativity I know that $$ds^2=c^2dt^2-dx^2-dy^2-dz^2.$$ This can be interpreted in two different ways: Either we define ...
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0answers
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Most used convention about Christoffel symbols

Just a simple question: what is the most used form for Christoffel symbols, (1) or (2), see below: (1) $$\Gamma_{ij}^{k} = g^{kl}\Gamma_{lij}$$ and then, we have: $$\Gamma_{lij}=\Gamma_{lji}$$ (2) $...
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1answer
81 views

Invariance of spacetime interval in special relativity: linearity

I'm trying to understand which assumption are necessary to prove the invariance of the spacetime interval $$\Delta s^2=c^2\Delta t^2-\Delta \mathbf{x}^2$$ in special relativity. The postulates of ...
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0answers
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Silly and unusual question about BH-BH binary

My question is quite simple. As we all know, we have detected gravitational waves. The first detection [https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves#References] was about ...
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1answer
32 views

Schwarschild Metric from Kepler law

Can we redrive Schwarschild metric from Kepler's law without assuming General Relativity?
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4answers
135 views

Symmetry of a tensor

This is from my notes, which I don't fully understand: It is straightforward to check that (anti)symmetry is a coordinate-independent notion, e.g., if the components of a tensor are symmetric in some ...
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4answers
147 views

Metric of the Universe

As some of you know, there is a fundamental flat space-time metric that describes our universe without any energy or matter in it. Correct me if I am wrong, but this metric and existance of the ...
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1answer
76 views

What is the difference between a quadrivector and a 4-vector? [closed]

What is the difference between a quadrivector and a 4-vector? Why is the square of a 4-vector equal to $t^2+x^2+y^2+z^2$ while the square of a quadrivector is equal to $t^2-x^2-y^2-z^2$? Aren't they ...
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0answers
35 views

Interpreting Kruskal for cosmological spacetime

I'm dealing with a metric of the form: $$ds^2 = -f(r)dt^2 + a(t)(1/f(r)dr^2 + a(t)r^2d\Omega^2)$$ Where $f$ is just $f = 1 - 2m/r$. The first thing I did was to introduce a conformal time coordinate $...
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1answer
63 views

How to compare the observation with the theoretically predicted result?

On the Wikipedia Article on “Geodesics in general relativity”, it says the following: “Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved 4-D ...
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2answers
75 views

How do I look for (possibly) all coordinate transformations with a given metric?

From what I learned in tensor calculus so far, coordinate transformations are supposed to preserve the metric of the space. (Here I used GR notation, but the metric doesn't have to be the spacetime ...
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0answers
12 views

Mistaken contra-variant base calculation in the famous Dvorkin-Bathe 1984 paper?

From all sources, I read that the covariant metric tensor matrix and the contravariant metric tensor matrix are the inverse of each other. For example, on this webpage: http://mathworld.wolfram.com/...