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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Renormalisation and the Fisher-Rao metric
The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be thought of as a flux in a space of possible Hamiltonians for a ...
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Holonomy group of Schwarzschild spacetime, other interesting examples?
I'm teaching myself a little about holonomy groups in the context of general relativity. This paper by Hall and Lonie classifies a lot of the possibilities for simply connected spacetimes in 3+1 ...
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How to prove that Weyl tensor is invariant under conformal transformations?
I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
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Solving Maxwell equations on curved spacetime
I have difficulties to understand how to solve the Maxwell equations on curved spacetime. I want to solve the equations in the weak regime $g_{\mu\nu}=\eta_{\mu\nu}+h{\mu\nu},~ h_{\mu\nu}\ll 1$ ...
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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
$$U(1+\omega)=1+\frac{i}{2}\...
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Why is the Taub-NUT instanton singular at $\theta=\pi$?
Consider the following metric
$$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$
where $$V=1+\frac{4m}{r}.$$
That is the Taub-NUT instanton. I have been ...
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General relativity 2 particle problem with negative mass
I am looking at a problem of 2-particle system of which one has negative mass. I have situation described on Wiki under section "Runaway motion". Prticulary, if we assume, that negative mass ...
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Correct statement of Birkhoff's theorem (spherically symmetric does not imply static?)
If I understand correctly, the appropriate statement of Birkhoff's theorem in general relativity is that
The Schwarzschild metric is the unique spherically symmetric vacuum
solution.
(Or we might ...
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Propagator in Brans-Dicke Gravity
Consider an action of the form
$$
S = -\frac{2}{\kappa^2}\int d^4x\sqrt{-g}~\left(\phi R + \phi\mathcal{L}_{matter}\right).
$$
Expanding this to second order in $h_{\mu\nu}$ and including a harmonic ...
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Energy-Momentum Tensor of a Gravitational Wave
In radiation gauge ($\gamma=0$), the Einstein field equation in vacuum for a perturbation $\gamma_{\mu\nu}:=g_{\mu\nu}-\eta_{\mu\nu}$ is given by $$ \boxed{ \partial^\alpha\partial_\alpha \gamma_{\mu\...
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In the context of condensed matter physics, what does it mean for time to have two dimensions?
In an online article that describes condensed matter physics for laypersons, the author describes various so-called "designer materials" that have exotic properties, including one in which ...
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Can some components of metric be Finslerian while the others be Riemannian?
A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
user65852
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Pseudo-Riemannian 2D manifold (visualize time curvature)
My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
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Is it possible to create a Nil geometry in real spacetime according to general relativity? (What metrics are possible in the real world?)
Background
I've heard that it is possible to construct a Penrose triangle in the 3D geometry Nil. And I wondered: Can we build a Penrose triangle in the real world if spacetime is appropriately ...
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Can one build Wilson lines in general relativity?
This question has two parts:
Firstly, I am curious if one can build Wilson lines as a 'parallel transport operator' in general relativity in direct analogy with what is done in gauge theory. For a ...
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Can two points always be joined by timelike curve?
I have asked this question on MSE but now I think it is better suited for the Physics Stack Exchange. Suppose $p$ and $q$ are connected by a causal (i.e. its tangent vectors have non-positive norm) ...
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What is the physical motivation behind the mathematical definition of an inertial system?
In this German Classical Mechanics lecture by Frederic Schuller, it is given that a Newtonian spacetime with an absolute inertial frame is one in which
$$ \nabla_{v} G=0$$
Where $\nabla_v$ is the ...
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Spin-2 operators from metric fluctuations in the AdS/CFT
This is a computational question. I am pretty sure that there is a simple explanation, and something obvious that I am missing but I cannot figure it out.
I want to add that this is not meant to be a &...
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On the embedding of the Schwarzschild metric in six dimensions
At every point of the 4-D space-time, it's metric, being a symmetric 2-tensor, has $\frac{D(D+1)}{2}=10$ independent components. From this we can subtract four degrees of freedom according to the four ...
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Scalar Curvature of a Conformally Flat Metric
Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
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Question about derivation of tensor in Di Francesco's CFT
This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the $2$-point Schwinger function in ...
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How to derive connection Lie algebra valued one-form on the frame bundle if given the pulled back of it on the physical space?
I am following this YouTube lecture by Schuller where he finds the appropriate formalism for the quantum mechanics in the physical curved space.
Everything makes sense to me but at the very end I see ...
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Electromagnetism in spacetime with 2-times split-signature $(+,+,-,-)$ metric
Maxwell's equations can easily be generalized to any $(m,n)$-spacetime. Is there any material analyzing what such a theory will look like?
Note that a particle still moves in spacetime forming a line. ...
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Why is AdS spacetime like a "saddle"?
When the shape of the universe is discussed, the three cases are flat, closed and open. Where AdS spacetime with a negative cosmological constant describes the open spacetime, as in the middle in the ...
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Linearity of Lorentz Transformations from Principle of Relativity
Many derivations of the Lorentz transformations assume they must be linear maps on $\mathbb R^4$, where we identify the components of $\mathbb R^4$ with orthogonal coordinate systems associated to ...
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Is there a coordinate system for the Schwarzchild spacetime where $r$ is proper length/time?
I don't see any when I search online. The way I would try to construct them would be as follows. Starting from the standard metric
$$ds^2 = -\left(1 - \frac{r_s}{r'}\right)dt^2 + \left(1-\frac{r_s}{...
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Linearised diffeomorphisms on an arbitrary gravitational background Part 2
This question is a follow on from my recent post here, in the sense that I will use the notation introduced there. In that post, I considered infinitesimal diffeomorphisms of a metric $g_{\mu\nu}$ ...
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Linearised diffeomorphisms on arbitrary gravitational background Part 1
Consider some spacetime $\big(\mathcal{M},g_{\mu\nu}\big)$ parameterised by local coordinates $x^{\mu}$ ($\mathcal{M}$ is a smooth differentiable manifold equipped with a Lorentzian metric $g_{\mu\nu}$...
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Error of $-i$ factor in light cone indices in conformal field theory in Becker's book
In Becker's book of String theory Ch-$3$ I'm getting an error of factor $-i$ in the definition of lightcone indicies after Wick rotation. The convention of the book is following $\sigma_{\pm}=\tau\pm\...
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Equivalence between path integral formalism and operator formalism on curved spacetimes and radial quantization for a 2D boson field
We know that the path integral formalism and operator formalism are equivalent on flat spacetime. I am wondering whether we can also make it explicit on a curved spacetime.
Let us consider a concrete ...
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First order quantum string action
Considering this post: Quantum String action the action given is of the lowest order but the effective action, for low energies, is given by:
$$ S_{ef.}= -\frac{1}{2k^2} \left( S^{(0)}+ \alpha S^{(1)} ...
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Angle-preserving linear transformations in 2D space for relativity
I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the $ct$ axis and the worldline of an ...
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Does every curved spacetime have non-commuting generators of translations?
If we define the generators of translations in a general spacetime to be $P_\mu$, is it true that in every curved spacetime we have $[P_\mu,P_\nu]\neq0$? Is it also true that for every spacetime where ...
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Solving scalar quantum field in 1+1D Milne space
So our line element is
\begin{equation}
ds^2=dt^2-a^2t^2dx^2
\end{equation}
doing following coordinate transformation
\begin{equation}
y^0=t\hspace{2pt}\cosh ax, \hspace{2pt}y^1=t\hspace{2pt}\sinh ...
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Quantum corrections to metric on non-linear sigma model target space
I am trying to make sense of what physicists mean when they talk of quantum corrections to the metric on the target spaces of nonlinear sigma models, for example [GHL99].
First some quick notation. ...
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How does the Equivalence Principle imply that derivatives of the metric vanish in a freely falling frame?
Why do the first derivatives of $g_{\mu\nu}$ vanish in a freely falling coordinate system?
I would like to start from the Equivalence Principle that for any point in spacetime there exists a locally ...
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Robertson-Walker metric and cosmic homogeneity
The Robertson-Walker metric is of the form
$$\tag{1} ds^2 = dt^2 - a(t)^2 \Big(\frac{dr^2}{1 - kr^2} + r^2 d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \Big).$$
My question is related to the $a^2(t)$ ...
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Role of the observer in Gödel's universe
I am here to clarify myself about the role of the observer in Gödel's solution (1949) of Einstein's field equations.
The Universe we are dealing with is anisotropic, since the axis of rotation ...
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Connection between contra-/covariant vectors in SR and complex numbers?
If we take a spacetime with one spatial dimension, we can write a vector as $A^\mu=(t, x)$. This is a contravariant vector, and we can calculate the covariant vector by multiplying it with the ...
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What is the geometry of light cones if space is curved/non-Euclidean?
In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as ...
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Constructing the Kruskal diagram for a 2-dim metric of the following form
We are given $ds^2 =- \frac{du\,dv}{M - uv},$
where $$v=t+x\, \qquad u= t - x\qquad -\infty < t, x < \infty$$ and $M$ is a positive constant.
The Riemann curvature tensor is proportional to $\...
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How many Killing spinors exist on $S^5$?
So, I know that on $S^n$, a spinor of the form
$$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$
where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
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Dirac equation in curved spacetime - found second derivatives of the metric, violation of the principle of equivalence?
I am working on the Dirac equation on curved spacetime. A Foldy-Wouthuysen transformation was applied to obtain the semiclassical limit of the equation to study the dynamics of the spin of the ...
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Squashed 3-sphere?
What is a squashed 3-sphere? In context of quantum gravity. I stumbled upon a term 'squashed 7 sphere' but that's concerning supersymmetry.
Is it just normal 3-sphere metric, that is just 'squashed' ...
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About Dirac equation in curved spacetime (spherical)
I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta d\...
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Linearized gravity: When do we let the metric be $\eta_{\mu \nu} + h_{\mu \nu}$ and when does it reduce to $\eta_{\mu \nu}$?
I am following a standard text on GR. In the chapter on linearized gravity, the metric $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$ reduces to $\eta_{\mu \nu}$ when the metric act on tensor components ...
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How to relate Riemannian and Lorentzian tetrad fields on the same manifold/spacetime?
Consider Gibbons and Hawkings paper wherein a Riemannian metric $\overset{\mathcal{R}}{g}_{\mu\nu}$ and everywhere well defined normalized line field $l_{\mu}$ on spacetime $M$ may be used to ...
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Symmetry at spatial infinity
One definition for asymptotic flatness is
$$\lim_{r \rightarrow \infty}g_{\mu\nu}=\eta_{\mu \nu}+\mathcal{O}(r^{-1}),$$
where $\eta_{\mu \nu}$ is the Minkowski metric.
The asymptotic symmetry group is ...
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Physical meaning of the Riemann curvature tensor with all 4 lower indexes
Since Riemann with 1 upper and 3 lower index mean the changes of a vector after being parallel transported around a closed loop, does Riemann with all 4 lower indexes mean the changes of a co-vector ...
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Are there non-smooth metrics for spacetime?
I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics:
Lorentz invariance holds locally in GR, but you're right that it no longer applies ...
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