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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Expanding the Ricci tensor by summing over indices

I had an attempt at deriving the Schwarzschild metric. This is a 4-dimensional problem where the indices are being summed from 0 to 3. I got up to the part where I calculate the Ricci tensor which is ...
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30 views

How to derive the relation satisfied by “gravitational magnetic field” from an equation of the Weyl tensor?

Let us call the spacetime $M$ with a metric $g_{ab}$. There is a unit spacelike vector field $\eta^a$ orthogonal to a hypersurface. So that we can define the so-called gravitational electric and ...
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28 views

How many degrees of freedom to diagonalize the metric?

In A. Zee's Einstein Gravity in a Nutshell, he starts with the following expansion of the metric at some point $P$ of a Riemannian manifold, with coordinates $x^\mu$ that have the origin at $P$: $$ ...
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183 views

Why does the FLRW metric assume constant curvature?

So the FLRW metric takes the following form in reduced-circumference polar coordinates. $$ds^2 = -c^2 dt^2 + a^2(t) \left(\frac{dr^2}{1 - k\, r^2} + r^2 (d\theta^2+sin^2\theta\, d\phi^2)\right)$$ ...
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100 views

Does $g_{\mu\mu}$ in an expression follow the Einstein summation convention?

Assume that I have the expression for a Christoffel symbol: $$ \Gamma^\mu_{\alpha \beta}=\frac{1}{2}g^{\mu \lambda}(\partial_\alpha g_{\beta \lambda}+\partial_\beta g_{\alpha \lambda} - ...
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0answers
31 views

Solving the Stress-energy-momentum tensor with metric

Let's say I want to find an expression for $T_{00}$ in Einstein field equations given a particular metric. I need to find first $g_{00}$, which is not complicated to find, and $R_{00}$ which is ...
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3answers
111 views

GR: Pseudo Riemannian or Riemannian?

Is General Relativityy described by Pseudo-Riemannian manifold or Riemannian manifold? I cannot understand the vast difference between the two manifolds. In books, General Relativity is looked as a ...
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44 views

Explicit calculation using metric tensor

I'm trying to calculate the following quantity: $$S_{\bar{\alpha} \bar{\beta}} \equiv \left( \Lambda^{-1} \right)^\alpha_{ \ \bar{\alpha}} \left( \Lambda^{-1} \right)^\beta_{ \ \bar{\beta}} ...
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Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of dependence of ...
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242 views

How to eliminate this cross term?

Given $$ds^2=-A(r)dt^2+B(r)dr^2+2C(r)drdt+D(r)r^2(d\theta^2+\sin^2d\phi^2),\tag{23.1}$$ How can we eliminate the cross term?
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69 views

Killing vectors in General Relativity?

I'm looking to derive the surface area of the event horizon of a Schwarzschild black hole. I was just wondering if it were possible for someone to explain to me this: $$ ...
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38 views

Gravitational Time Dilation and Schwarzschild Coordinates

So I'm trying to use this equation for the time dilation of an object, but I don't know how to get the distance that I have (in meters) to a radial coordinate in terms of schwarzschild coordinates. ...
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2answers
82 views

Maintaining symmetry? [closed]

Minkowski metric is found to be $$ds^2=-dt^2+dr^2+r^2d\Omega^2$$ where $d\Omega^2$ is the metric on a unit two-sphere. Why should we keep track of the $d\Omega^2$ so that spherical symmetry holds ...
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628 views

Why do people put exponentials there

In his book, Sean Carroll, says p. 194 chapter 5: To impose spherical symmetry, we begin b writing the metric of Minkowski space in polar coordinates $x^{\mu}=(t,r, \theta, \phi)$: $$ ...
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Finding the Riemann tensor for the surface of a sphere with sympy.diffgeom

I have implemented a SymPy program that can calculate the Riemann curvature tensor for a given curve element. However, I am encountering problems solving for the case when the curve element is the ...
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3answers
82 views

Absolute direction in space [closed]

Rotation, from my understanding, is basically the "exchanging of different spatial dimensions with eachother", with $x^2+y^2=d^2$ being the "relationship" between any two spatial dimensions, aka. if ...
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1answer
68 views

Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
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691 views

How can we recover the Newtonian gravitational potential from the metric of general relativity?

The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration: ...
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2answers
90 views

How does a metric of the form $\mathrm{d}z \mathrm{d}\bar z$ work, if $z$ and $\bar z$ are not independent? [duplicate]

My question is motivated by 2D CFT where one works in "complex coordinates". The question is the following: Suppose I am in 2D flat Euclidean space, i.e. $$\mathrm{d}s^2 = \mathrm{d}x^2 + ...
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75 views

Minkowski metric and Null tetrad metric

I'm starting with the Newman-Penrose formalism and have a very basic question that I'm very confused about. The standard Minkoswki metric is $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$. Is then the null ...
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1answer
184 views

Tetrad formalism vs coordinate formalism example

Sources I have been reading Chapter 11 and 25 of Andrew Hamilton's amazing notes which has some material on tetrad formalism in general relativity (formulating GR in coordinate-free fashion). ...
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41 views

Is it true that GR waves can interfere if and only if the change to the metric can take negative values?

This appears to be a unique question compared to other topics about electromagnetic waves or quantum mechanics already on Stack Exchange. The topic on resonance doesn't really answer this question. ...
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1answer
36 views

Norm of Dilatation operator [closed]

The dilatation operator is given by $$D=x^{a}\frac{\partial}{\partial x^{a}}+z\frac{\partial}{\partial z}$$ How the norm can be $$D^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}x^{\mu}x^{\nu}+z^{2})$$ where ...
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101 views

Schwarzschild metric in lower-dimensional spaces

I was trying to explain the consequences of the Schwarzschild metric to someone last night and obviously it's pretty difficult to conceptualize in four-dimensional spacetime. Elementary googling has ...
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1answer
86 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
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3answers
296 views

Physically what does warping (of space-time) mean?

So there's general relativity and Einstein's field equations that tell us "mass(or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble ...
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1answer
88 views

Most general Ansatz for cylindrically symmetric metric in GR?

How would the most general Ansatz for a cylindrically symmetric metric in GR look like? To make this question more substantial, here is an example of what I have in mind. I ask this question in the ...
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1answer
65 views

Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ...
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2answers
63 views

Does decomposition of motion rely on Pythagorean theorem?

As an example, when analyzing a simple projectile motion with initial horizontal velocity in Newtonian mechanics, I'm enabled to decompose the projectile motion into the vertical and horizontal ...
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2answers
62 views

How to calculate the event horizon and the cosmological radius in a metric?

From reading about general relativity, the event horizon and the cosmological radius are the radius when $f(r)=0$, in the metric $$ ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2} $$ However, ...
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1answer
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How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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2answers
157 views

Why does the Lorentz transformation have to be a linear transformation? [duplicate]

In my textbook, they say the following statements before doing a proof for the Lorentz transformation: We know that the Galilean transformation $x' = x - vt$ is incorrect, but what is the ...
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2answers
71 views

Are orbits possible in de sitter space?

Since the de sitter space has constant positive curvature does that mean that objects can't orbit around other objects?
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51 views

Does Schwarzschild metric have cosmological horizon?

Since the space is not expanding in the Schwarzschild metric does that mean that there is no cosmological horizon? Also, what if the Schwarzschild metric was not asymptotically flat and we replace ...
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0answers
87 views

What is the degrees of freedom of metric tensor?

As $g_{\mu\nu}$ can be taken to be symmetric, it contains 10 functions of spacetime in 4 dimensions. But, why we call these 10 functions as the degrees of freedom of the metric while they are the ...
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How to invert this metric?

Reading this article i find a result that i am not sure how to obtain (page 3 eq 3). It is about the inversion of a metric of the type $$ g_{\mu\nu}=Al_{\mu\nu}+BH_{\mu\nu}. $$ In order to invert ...
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1answer
48 views

Do contractions with Dirac matrices involve a metric?

When figuring out where the spacetime metric enters an equation it is often useful to write all vector indices as covariant indices and write out the inverse metrics that are needed to contract them, ...
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1answer
64 views

Meaning of “physical” and “gravitational” metrics

I've recently been reading some notes (following a paper by J.D. Bekenstein, titled "The Relation between Physical and Gravitational Geometry": http://arxiv.org/abs/gr-qc/9211017) on alternative ...
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2answers
228 views

Variation of square root of determinant of metric, $\delta g$ [closed]

I am trying to calculate $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$ where $g = \text{det} g_{\mu \nu}$. We have $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 ...
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1answer
87 views

How to calculate spacetime intervals on a spacetime diagram?

In SR, the spacetime interval is given by the metric: $ds^2=-dt^2+dx^2$ (where I set $c=1$). To calculate $ds^2$ of a worldline on a spacetime diagram, I measure $dt$ and $dx$ of the line of ...
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2answers
98 views

Geometric definition of the Lorentz inner product

In Euclidean space one can define the dot product as projecting one vector to the other and multiply the length of the projected vector with the length of the other vector. This definition doesn't ...
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1answer
62 views

What is meant by “the Klein-Gordon equation is unsymmetrical between the temporal and spatial components”, and why is this a problem? [closed]

The Klein-Gordon equation explicitly reads $\left( \frac{\partial ^2}{c^2\partial t^2} - \nabla ^2+\left( \frac{m_0 c}{\hbar}\right)^2\right) \psi =0$ Now I read here on page 8 that: What is ...
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1answer
18 views

Black hole physics beyond the perturbation theory

Motivated by this question: Perturbation of a Schwarzschild Black Hole How would one deal with the situation where black hole experiences not only small perturbations but major changes to the metric? ...
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1answer
64 views

Event horizon from the metric

Let us suppose we have a metric of this form $$ds^2=-A(r)dt^2+\frac{dr^2}{B(r)}+r^2d\Omega^2$$ In all documents I can read, I've seen that the event horizon is defined by considering $A(r)=0$ But I ...
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117 views

What is the conformal mode of a metric?

I have a problem in terminology. This article talks about the conformal mode of a physical metric. I know what a conformal transformation is. But what is the conformal mode of a metric?
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104 views

How to define $\delta{g_{\mu\nu}}$?

In general relativity, when deriving the field equation using the variational principle we use $\hat{g}_{\mu\nu}=g_{\mu\nu}+\delta{g_{\mu\nu}}$. Does $\delta{g_{\mu\nu}}$ mean the measurement of how ...
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2answers
57 views

If the measurements of a clock above the earth depend on orientation, then what measurements are correct?

Take a clock in space above the earth (assuming a Schwarzchild spacetime) that works by relaying a light signal a small distance radially; ticking each time the light signal returns. Compare this to ...
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47 views

Is the scale factor Lorentz invariant?

Given that the Minkowski metric does not change under a Lorentz transformation, the scale factor does not change in the special case when it is equal to 1. Is this result true in general? i.e. is the ...
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1answer
79 views

What is the value of the variation stress energy tensor?

If we are living in a portion of space-time where the metric is very close to flat space and we know that the stress energy tensor is negligible at this portion of space-time is it ok to assume that ...
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1answer
77 views

Understanding the nature of metric tensor [closed]

The metric tensor for a flat spatial manifold gives us length on object, or separation between two space points. Similarly, $g_{\mu \lambda} dx{^\mu} dx{^\lambda}$ gives separation between two space ...