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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Why are there no covariant particles?

Why is physics described by scalar, spinor, vector, and tensor bundles, but not convector bundles or tensors of other ranks? In tensor bundle notation, these are respectively $T_0^0, S, T_0^1, T_0^2$. ...
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Cosmology - Confusion About Visualising the Universe as the Surface of a 3-Sphere

Consider the FRW metric for the Universe in the form found in many standard cosmology textbooks: $$ds^2 = -dt^2 + a(t)^2\left(\frac{dr^2}{1-Kr^2}+r^2(d\theta^2 + \sin^2\theta d\phi^2)\right)$$ I am ...
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Linearised diffeomorphisms on 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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Modulus of four acceleration

The four acceleration is defined as $$\alpha^\mu = \gamma_V ^4 \left(\frac{\vec{v} \cdot \vec{a}}{c},\frac{\vec{v} \cdot \vec{a}}{c^2} \vec{v} + \frac{1}{\gamma_V ^2} \vec{a} \right)$$ where $\vec{v}$ ...
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Ricci tensor for FRW Metric [closed]

I am attempting to prove that the FLRW metric given by $$ds^2 = -dt^2 + g_{ij}dx^idx^j = -dt^2 + a^2(t)\left(d\vec{x}^2+K\frac{(\vec{x}\cdot d\vec{x})^2}{1-K\vec{x}^2}\right)$$ has $$R_{ij} = \left[\...
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Equivalence principle and different metrics

If I do not interpret equivalence principle wrongly, it says that we could always free fall ourselves to achieve local inertial frame. The metric would be $ds^2=-dt^2+dx^2+dy^2+dz^2$. Suppose an ...
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Levi-Civita tensor in tetrad/vielbein basis

I am reading about tetrads in a GR textbook and a question occured to me. It seems natural to assume that the Levi Civita tensor in tetrad/vielbein basis is the Levi Civita symbol in the flat indices, ...
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Proper time invariance and conformal symmetry group

Special relativity is based on the fact that the proper time is always the same in any inertial frame: $$ds^2=(cdt)^2-dx^2=ds'^2=(cdt')^2 -dx'^2 $$ If I understand it correctly this is based on the ...
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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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Christoffel Symbols and Metric Tensor

I know that the Christoffel Symbols are made out of the first derivative of Metric Tensor. Is there any relation between the number of metric components and Christoffel symbols? Is there any general ...
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Why is stressed in Kruskal metric that the part in $u$ and $v$ is conformally flat?

Many books show how Kruskal metric, derived putting together both Eddington-Finkelstein results, is conformal flat (if just the first 2 coordinates are considered): $ds^2 = \left(1-\frac{2GM}{r} \...
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How to handle a lightlike metric with cross terms, and its corresponding proper time?

I understand that the metric of null-separated events is $\mathrm{d}s^2 = 0$. This would mean that the time component is equal to the spatial components, $\mathrm{d}t^2 = dl^2$. If there are cross ...
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Missing sign in Dirac equation

This is very trivial, but it's really bugging me. The ansatz for the Dirac equation in terms of $\boldsymbol\alpha$ and $\beta$ matrices is $$ [i(\partial_t+\boldsymbol\alpha\cdot\boldsymbol\nabla)-\...
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Raising and lowering indices in quantum field theory

Is raising and lowering indices in quantum field theory works the same as in the general theory of relativity? By means of this metric tensor? $g^{μν}= \begin{pmatrix} 1 & 0 & 0 & 0\\ ...
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Different versions of Schwinger parameterization

One common used trick when calculating loop integral is Schwinger parameterization. And I have seen two versions among wiki, arxiv and lecture notes. $$\frac{1}{A}=\int_0^{\infty} dte^{-tA}$$ or, $$\...
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How do 'locally Euclidean' and 'Lorentzian' requirements in manifolds reconcile?

In GR, we define our manifolds to be locally Euclidean. However, we also demand that our metric tensor have a Lorentzian signature. Since the metric tensor is a measure of curvature, doesn't the first ...
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Interpretation of black hole metric with fractional $\kappa$ instead of the usual $\kappa\in\{-1,0,1\}$

The metric for a black hole can be written: $$d s^{2}=-\left(\kappa-\frac{2 M}{r}\right) d t^{2}+\left(\kappa-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Sigma_{2, k}^{2}$$ where $\kappa=-1,0,1$ ...
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Clock synchronization and definition of simultaneity along a curve or contour in general relativity

What I understand so far Fix a point $A$ with coordinates $x^\alpha$ with respect to a frame $K$ with metric tensor $g_{\alpha\beta}$. Then take an arbitrary point $B$ infinitesimally close to it with ...
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Is the distance between all points on the event horizon zero?

These are the Kruskal-Szekeres coordinates of Schwarzschild spacetime: It is isomorphic to the split-complex plane. But on the split-complex plane the distance between all points on the null ...
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Which tensor should the metric raising or lowering?

For something like $g^{ij} n_i h_{kj}$, how do I know which one should the metric operate on? $n_i$ or $h_{kj}$? The results could be $n^j h_{kj}$ or $n_i h^i_{k}$, which are different. The question ...
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Spacetimes where $R_{ij}\neq 0$ but $R_{ij}V^iV^j=0$ on a timelike and/or null geodesic?

Do there exist spacetimes with a timelike and/or null geodesic $\gamma$ with tangent vector $V$ for which $R_{ij}\neq 0$ on the geodesic, but $R_{ij}V^iV^j=0$ on it? If so, are there any general ...
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Decay of a pion using four momentum vector

I was stuck on the following question and I can't see what the relation of this is, The question in regard was Determine the energy E of the muon in terms of $m0_µ$, $m0_π$ and $c$ when the resting ...
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Raising and lowering indices in line elements - why do we raise and lower them in line elements?

My question refers to Piattella's lecture notes on cosmology. On page 15, the Euclidean line element is defined as $$ ds^2 = \vert d\mathbf{x}\vert^2 = \delta_{ij}dx^idx^j. $$ My first question is ...
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The tensor product in the Hamiltonian of graphene

I have the Hamiltonian of pristine graphene \begin{equation} H=v_{F}.\boldsymbol{\gamma}.\boldsymbol{p} \end{equation} with $\boldsymbol{p}=(p_{x},p_{y})$ is the momentum operator, $v_{F}$ is the ...
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Kerr metric in Eddington–Finkelstein form

I am searching for a reference in which I can find out the Kerr metric in Eddington–Finkelstein form. I have computed it by hand and I have obtained the following form but I am not sure above all of ...
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Understanding 4-momentum and the Lorentz invariant product

What I am trying to do is show that if $\textbf{P}_1$, $\textbf{P}_2$ are time-like, future point 4-vectors, then $\textbf{P}_1 .\textbf{P}_2\geq0$. My understanding of the meaning of timelike is that ...
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Killing equation and normal coordinates

I have already seen a similar question but I have not sure to have understand completely so I hope you can help me. If I write the killing equation $\cal{L}_X g=0$ as $X_{\alpha;\beta}+X_{\beta;\alpha}...
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Visualisation of space-time curvature for a black hole

I want to visualise the curvature of a black hole for different coordinates by using the metric. The general approach I (have to) use is as follows: Let the Schwarzschild metric for the coordinates $(...
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Can flat spacetime have curvature?

Hi I am new and I hope this can make sense. Is it true that flat spacetime can curve? For example, you have a flat piece of spacetime that curves at a constant rate of pi/2. the flat spacetime would ...
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Hyperbolic isometries in the context of General Relativity

In the context of hyperbolic geometry, it is possible to create a classification for isometries. I would like to know if these isometries have any particular meaning in the context of general ...
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The contravariant derivative of a substitution for the de Sitter metric

Consider the de Sitter metric: $$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$ I know that we rewrite the metric as $(u,r,\theta \phi)$ using the substitution $$u = t-...
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Physics models using (non-lorentzian) Indefinite metrics [duplicate]

I would like to understand better the role of indefinite metrics in physics. As far as I know, Lorentzian metric is the natural setting for Einstein's Relativity Theory. Somewhere I read something ...
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Raising and lowering indexes

I'd like to check if my understanding of the following is correct: consider a contravariant object $x^\mu\in V$, where $V$ is a vector space with a metric $g^{\mu\nu}$. From linear algebra, we know ...
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Equation of a conformal Killing vector

Question 10(a) on pages 469-470 in the book "Spacetime and Geometry" by Sean Carroll asks: Suppose that two metrics are related by an overall conformal transformation of the form: $$ \tilde{...
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Conformal coordinate change: gauge symmetry or global symmetry

While reading the fourth chapter "Introducing Conformal Field Theory" of D. Tong's string theory notes, I read that A transformation of the form $\sigma^\alpha\to\tilde{\sigma}^\alpha(\...
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Scaling of charged metrics in General Relativity

Lets say we have a metric g. This metric is created by a combination of electrical charges, fields, masses, and momenta. In other words, the metric corresponds to the stress-energy tensor given by all ...
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Proper time of a timelike geodesic

In the contest of the newtonian limit in general relativity, if I consider a timelike geodesic that can represent the motion of a free falling particle under the influence of the gravitational force ...
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Static spacetime and Schwarzschild solution

The Schwarzschild spacetime $(\cal{M},g)$, for which the metric is a solution for the Einstein field equation in vacuum, $$g=-\Big(1-\frac{2m}{r}\Big)dt^2+\Big(1-\frac{2m}{r}\Big)^{-1}dr^2+r^2d\Omega^...
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Understanding the general metric of a stationary spacetime

Wikipedia says the metric for any stationary spacetime, when made independent of $t$, has the form $ds^2 = \lambda(dt - \omega_i\ dy^i)^2 - \lambda^{-1}\ h_{ij}\ dy^i\ dy^j$ with the Killing vector $\...
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Killing equation in coordinates

In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$ I have read that the key observation, to write the equation in ...
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$SU(2)$ infinitesimal transformation of pion triplet lagrangian raising and lowering indices

How can I show that $$\epsilon^{ijk} (\partial^\mu\pi^i)\pi^j\partial_\mu\alpha^k(x)=\epsilon^{ijk}(\partial_\mu\pi^i)\pi^j\partial^\mu\alpha^k(x)~?$$ can I do the following? $\pi$ is the pion triplet ...
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Does the invariance of interval hold only for events that occur in inertial reference frames?

For example, if we have two frames that have constant relative velocity, would the interval be the same between any two events? Or should we impose the condition that the particle must not accelerate ...
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Spacetime translation in QFT

I have a question about the field under the spacetime translation. For example, in page 26 of Peskin's textbook, they give the translation properties of the field. So consider the space translation, ...
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A problem with parallelism of tangent vectors

In 2D-space with metric $ds^2=du^2+2\lambda dudv+ dv^2$, where $\lambda= \lambda (u,v)$, how can we show that the tangent vectors to the curves $u$ = constant form a field of parallel vectors along ...
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Homomorphism between $\mathrm{SL}(2, \mathbb{C})$ and $\mathrm{O}(1, 3)$

By definition if $L$ is an element of the Lorentz group we have $(x,y)=(Lx,Ly)$. Now $S$ be an element of $\mathrm{SL}(2, \mathbb{C})$. Usually people defines the homomorphism $\lambda:\mathrm{SL}(2, \...
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Chasing down minus sign in the expression $p_\mu - e A_\mu$ for the momentum

In his Quantum Field Theory book, Folland says that the prescription for incorporating a 4-potential $A$ into the Dirac equation $(i \gamma^{\mu} \partial_\mu - m)\psi = 0$ is to replace $i\partial_\...
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How to express the fact that $\delta_\mu{}^\nu$ is symmetric?

In relativity, given a metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$, by definition of the inverse, we have the relation $g_{\mu\rho}g^{\rho\nu} = \delta_\mu{}^\nu$. In matrix form, $\delta_\...
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Curve is geodesic iff $\nabla_k g(V,V)$ vanishes

Let $V$ be a Killing vector field and let $s \longmapsto x^i(s)$ be a curve such that $$\dot{x} \enspace \equiv \enspace \frac{dx^i}{dx}(s) \enspace = \enspace V^i\big(x(s)\big)$$ Show that $x^i(s)$ ...
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Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed]

I've shown that $\nabla_{\lambda} g_{\mu\nu} = 0 $ rigorously by the following method: $ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...

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