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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Confusion about 'minimally coupled' and 'massive' Scalar Tensor Theory
So if I write down a general action of Scalar Tensor Theory in a Jordan frame as
$$
S =
\frac{1}{16\pi G_0}\int \left( f_1(\varphi) R
- f_2(\varphi) g^{\alpha \beta} \partial_\alpha \varphi \partial_\...
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What is the gravitational field of an accelerated particle?
Could we simply change coordinates of the Schwarzschild metric in order to obtain the metric of a moving massive particle? Which would those coordinates be? Rindler coordinates? Maybe there is a ...
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Meaning of principal definitions of SR in relativistic quantum mechanics
I have just started with relativistic quantum mechanics in my advanced quantum theory class and we only had a very short intermezzo on special relativity. I feel like I don’t have enough knowledge on ...
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How are 'local Lorentz frames' defined?
I'm following Schutz's book to learn gr. We know that in an inertial frame, the metric tensor $g_{ab}$ has components $g_{ab} = \delta_{ab}$ (of course, $g_{00} = -1$). At the least the idea is that ...
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General relativity change of observer
I have a problem in calculating the module of the velocity of a particle measured by a static observer in a specific metric. This metric is
$$ds^2=(r^2-R^2)dt^2-\frac{dr^2}{r^2-R^2}-r^2d\varphi^2$$
...
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Resolving an apparent contradiction between Schwarzschild and ingoing Eddington-Finkelstein coordinates
I believe this is basic differential geometry issue. This may be obvious to many, but I was quite confused about it, and it took me quite a while to find the resolution. I'm going to ask and answer ...
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Do two coordinate systems cover the same patch of the de Sitter manifold
I am self studying general relativity and there is some especially hard problem (it is called bonus problem in book) I am currently working on it, but I am trully stuck, so I would appreaciate all the ...
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Proving that the Christoffel connection transforms like a connection
In Sean Carroll's intro to GR, he shows that a connection transforms as follows: $$\Gamma^{\upsilon'}_{\mu'\lambda'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\frac{\partial x^\lambda}{\partial x^{\...
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Christoffel symbol for time component
In Section 9.2, from "General Relativity: An Introduction for Physicists" by Hobson, I am trying to obtain the affine connections for the metric $$dS^2 = A(r)dt^2 - B(r)dr^2 - r^2(d\theta^2 +...
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Cannot understand this identity between kronecker and metric tensor [closed]
I'm working on Lorentz generators and I am really not able to understand this relation:
$$\omega_{\rho \sigma} \eta^{\rho\mu} \delta^{\alpha}_{\nu} = \frac{1}{2}\omega_{\rho \sigma} \left(\eta^{\rho\...
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Why cant a repulsive event horizon of negative mass be theoretically constructed?
An event horizon appears in the Schwarzschild metric when considering a positive point mass in General Relativity.
But for a negative point mass in the negative mass Schwarzschild metric, which ...
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Conformal flatness of $\rm dS$ spacetime
There exists a flat slicing in the lower triangle area of the Penrose diagram of $\rm dS$ spacetime.
To see this point, one introduces the planner coordinate, as defined in eq(13) of Les Houches ...
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Electron Muon Scattering
I have been doing studying in advance for Dirac Equation and stumbled upon the math on calculating the spin-averaged amplitude $\bar{M}^{2}$ for the electron muon scattering process. Below is an ...
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Killing vectors on the unit sphere
I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$.
I know how to solve this problem by considering ...
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Continuity equation as a four-divergence. Possible misunderstanding of contravariance, covariance, or metric signature
Context
I am studying electromagnetic quantities through the framework of special relativity. I find it notation heavy and the issue of tensors is something I am still not fully grasping. The ...
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Local Supersymmetry and Space-time Metric
So following from Simple Supergravity (arxiv:2212.10044), on page 5, it's written that
Only the spacetime metric can couple to the energy-momentum tensor...
Can anyone explain why it must be the ...
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Connection between pseudometric and Einstein elevator
I have a hard time understanding GR. I understand a lot (from a math point) about (pseudo)Riemannian manifolds, and I also learned about Einstein's elevator thought experiment. So let me elaborate:
...
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How is the Ricci scalar the trace of the Ricci tensor?
The Ricci scalar is the uncontracted version of the Ricci tensor $R=R^{\mu}_{\mu}=g^{\mu\nu}R_{\mu\nu}$. Carrol describes the Ricci scalar as being the trace of the Ricci tensor, but I do not ...
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Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
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Are there a more general metrics for the case of multiple masses? [duplicate]
I understand that for the case of one non-rotating mass with an electric charge the metric describing the spacetime around the mass is the Reissner-Nordström metric
$$-c^2d\tau^2=\left(1-\frac{2GM}{c^...
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Time-ordering and Minkowski metric's negative sign [closed]
I'm coming at the following question from a mostly lay perspective (i.e. barely-undergrad physics), so please bear with the weirdness of it if possible.
I've generally been uncomfortable with the ...
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Constant determinant of metric tensor of Schwarzschild solution in $(x, y, z, t)$ coordinates?
In spherical Schwarzschild coordinates, it's $A \cdot B$
= constant. Is there something similar in the Schwarzschild solution in $(x, y, z, t)$ coordinates? For example in Droste coordinates $g_{tt} \...
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From Klein-Gordon equation to Dirac equation: a wrong "derivation" [closed]
So let us start with the Klein-Gordon equation
$$\tag{KG}
(-p^\mu p_\mu + m^2)\phi = 0
$$
The idea is to "factorize" the operator $-p^\mu p_\mu + m^2$.
\begin{equation}\tag{1}
-p^\mu p_\mu + ...
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Is this a valid approach to think about the $N$-Body - Problem in General Relativity?
I would like to analyze the following problem:
Some point masses, between and around them just empty space ($N$-Body-problem). I would like to analyze the space between and around those point masses. ...
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Questions about Lorentz Matrices and Lorentz Metric
(I use the abstract index notation convention in this post)
In $\mathbb{R}^4$, denote the Lorentz Metric as $g_{\mu\nu}=$diag$(-1,1,1,1)$, then we can define the Lorentz Matrices to be all $4\times 4$ ...
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Schwarzschild and Kerr solution in $(x, y, z, t)$ coordinates?
The Schwarzschild solution ('simple' black holes) and the Kerr solution (rotating black holes) are very well known in General Relativity.
The coordinates which are used to describe them are mostly ...
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What consequences would it have to postulate zero shift vectors in GR?
The shift vector is a part of the metric tensor in General Relativity (GR). It's $g_{0i}$ with $i$ in $[1,3]$.
This post is related to this question. There, I ask whether a changing of coordinates is ...
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Can there be black holes in a 1+1D spacetime? [duplicate]
In 2D Einstein tensor is always zero, that means no mass (or cosmological constant or other stress energy tensor component) is allowed. Nevertheless, we can get nonzero Riemann, even nonzero Ricci ...
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Graphical representation of an arbitrary tensor
I am interested in finding graphical models for tensors. To begin with, let's think about graphical models of tensors in 2D and 3D. If possible, I would be interested in the graphical representations ...
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Why does a degree of freedom vanish from 3D to 2D in that tensor construction?
Let's assume an arbitrary tensor in 3D coordinates: $g_{ij} $ with $i, j$ in $[1,3]$.
It shall be arbitrary, meaning not symmetric.
It has 9 entries which equals 9 degrees of freedom (dof).
Now, I ...
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Is this rewriting of the metric tensor possible or not?
The metric tensor $g_{\mu\nu}$ is symmetric, meaning that $g_{\mu\nu}$ = $g_{\nu\mu} $ for every special $\mu$ and $\nu$ in [0,3].
That sums up to $g_{\mu\nu}$ having 10 degrees of freedom.
$g_{\mu\nu}...
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How would you reparametrize a worldline in terms of proper time in 2-dimensional Minkowski spacetime?
In a 2-dimensional Minkowski spacetime i.e. $x^\mu=(t,x)$, you can define the metric simply by the Minkowski metric, $ds^2=-dt^2+dx^2$, and the Christoffel symbols vanish. If you have a worldline ...
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GR: why "background-independent" and not only "background-interacting"?
My question is closely related to the answer of this question:
Why is general relativity background independent and electromagnetism is background dependent?
General Relativity is often stated to be &...
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$00$-component of the Ricci tensor for the Newtonian limit
For the past few days I've been busy with Relativity. I've been reading through the book "Spacetime and geometry" by Sean Caroll and currently I'm at chapter 4. In this chapter they make the ...
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How to solve nonlinear diff eq for a general Schwarzschild metric?
So I have a general form of a spherically symmetric metric:
$$ds^2 = -g(r)_t \, dt^2 + g(r)_r \, dr^2 + g(r)_s (d\theta^2 + \sin^2 \theta \, d\Phi^2)$$
$$ R_{\theta\theta} = \frac{-g'_s g'_t g_r + ...
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Confusion about choosing an Euclidean world sheet metric in String Theory path integral
When it comes to construct a well-defined path integral for the Polyakov action in Bosonic String Theory, most authors assume that the world sheet metric $g$ is Riemannian (i.e. it has Euclidean ...
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Zero Einstein Tensor in 4D
In 2D the Einstein tensor is always zero, and we can easily get solution with non-zero Ricci tensor but zero Einstein tensor. But is it possible in 4D? Can we get a space-time with zero Einstein ...
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Static spherically symmetric spacetimes
I would like to better understand a hypothesis that Wald uses to derive the general local formula of a static spherically symmetric spacetime.
A spacetime is said to be spherically symmetric if its ...
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In the Schwarzschild metric how is the inverse for the $g_{rr}$ component written?
In the Schwarzschild metric how is the inverse written for the component $g_{rr}$, which is $\frac{1}{1-\frac{2GM}{rc^2}}$?
Does $g^{rr}$ =$\left(\frac{2GM}{rc^2}-1\right)$, and therefore, $\left(\...
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Why does the Minkowski matrix appear in the free particle action?
It is usual to write the "kinetic" part of the SR action as the Minkowski space-time interval, here $(-,+,+,+)$, times $mc$
$$
S_{kin} = -\int_{\tau_1}^{\tau_2}mc\sqrt{-\eta_{\mu\nu}\dot{x}^{...
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Why is the density in GR equal to $\rho_0\dfrac{dx^0}{ds}\sqrt{-g}$?
In general relativity, the continuity equation says
$$
\partial_{\mu}\left(\rho_0c\dfrac{dx^{\mu}}{ds}\sqrt{-g}\right) = 0
$$
with $\rho_0$ being the proper density, as seen by an observer who is at ...
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Can contracted indices be flipped arbitrarily?
Let’s say we have a contraction of the form $a^{\mu}b_{\mu}$. Is it true that $a^{\mu}b_{\mu}= a^{\mu}b^{\alpha}g_{\alpha\mu} = a_{\alpha}b^{\alpha}$. If it is, is it also true that $T_{\alpha\beta}u^{...
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Trace of stress tensor in 2D average null energy condition
I was looking through Zamolodchikov's derivation of the $c$-theorem and stumbled across an equation which says the following -
$$\Theta = T^\mu_\mu = 4T_{z\bar{z}}.$$
As far as I understand, for two ...
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Similar to how there's field lines that make equations in Newtonian Gravity more intuitive, is there something that makes GR equations more intuitive?
One way I know to get intuition for the derivation of the force equation $$F=\frac{GM_1M_2}{r^2}$$ in Newtonian Mechanics is to imagine gravitational field lines, in combination with certain ...
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Time in the negative mass Schwarzschild solution
I have read that for the Schwarzschild metric solution with $M<0$, something odd happens with the time coordinate. For the constants of motion, $dt/d\tau=e(1 - 2GM/r)^{-1}$ with $M<0$ and $e$ a ...
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How to prove that vielbeins exist?
In Carroll's introductory book on General Relativity, he discusses the noncoordinate basis and how to construct the noncoordinate basis. When introducing this basis, he defines the vielbeins as the ...
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Is there a reason why $\det(g)$ is a scalar density of weight 2 and not 1, 3 or 4?
$\det(g)$ is not coordinate-independent - it is a scalar density of weight +2 (or −2 depending on convention) which generally changes across spacetime. In Minkowski space equipped with spherical polar ...
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Approximating curved spacetime with a grid of cartesian metric tensors?
Let's assume a universe with only some ($n$) single point masses $m_i$ in it. The point masses have initial positions in space-time, $x_{i0}$.
The spacetime between them is curved due to general ...
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Heisenberg's microscope and gravity [closed]
Is the Heisenberg's microscope gedanken experiment valid when considering spacetime kinematics?
That is, if we consider a small region of space and try to measure its curvature, then we may use ...
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Why can't the metric have more than one timelike coordinate? [duplicate]
In one of his lectures, L Susskind stated that he cannot make sense of a metric with more than one timelike dimension. I also have trouble imagining it, but is there a good mathematical or physical ...
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