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.
3,670 questions
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Divergence of canonical energy-momentum tensor in QFT [closed]
I have to show that the divergence of the canonical energy-momentum tensor is zero, i.e. $$\partial^{\mu}T_{\mu\nu} = 0.$$
The Lagrangian is $$\mathcal{L} = \frac{1}{2}\partial^{\mu} \phi \partial_{\...
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How do you obtain the coordinates of 3D space from the FLRW metric?
Under the assumptions of homogeneity and isotropy, it can be deduced that for a one-parameter family of spacelike surfaces folliating spacetime, the curvature within each such surface is constant. ...
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The variational derivative of the metric with respect to inverse metric
I have a somewhat naive question. However, consider the functional derivative
$$
\frac{\delta \partial_\mu g_{\alpha \beta}(x)}{\delta g^{\gamma \epsilon}(y)}.
$$
Is it should be
$$
\frac{\delta \...
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1+1D spacetime quantitative description
Suppose we try to imagine a 1+1D analogy of spacetime (1 dimension for space + 1 dimension for time). Lets say there is a point mass $m$ at $x = 5$. So the world line would be the line $x = 5$ in $x$-$...
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Spinor indices, raising indices of pauli matrices
How do we raise indices for a Pauli matrix. For example let $\left(\sigma^{y}\right)_{a\dot{a}} = \begin{pmatrix}
0 & -i \\
i & 0 \\
\end{pmatrix}$. How can I raise the index using 2D levi-...
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
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Does this tensor identity hold in any kind of generality?
Assuming Minkowski spacetime, I am given an antisymmetric tensor $F^{\alpha\beta}$, and am asked to prove the following identity:
$$ F_{\mu}{}^{\alpha}{}_{,\nu}F^{\nu}{}_{\alpha} = -F_{\mu\alpha,\...
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Would a degenerate coordinate system be acceptable?
Suppose I’ve got some spherically-symmetric metric akin to the Schwarzschild metric and I render it in spherical coordinates,
$$\text{d}s^2=f(r)\text{d}t^2-g(r)\text{d}r^2-r^2\text{d}\Omega^2$$
for ...
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The contraction of Christoffel symbols [duplicate]
I have a question regarding the the contracted Christoffel symbols from David Tongs PDF on general relativity.
He wants to prove that
$$\Gamma^{\mu}_{\mu v}=\frac{1}{\sqrt{g}}\partial_v\sqrt{g}$$
...
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3
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Deriving differential equation for the path of a particle in potential $U(r)$ using Maupertuis’ principle
I came across this Maupertuis' principle in Landau and Lifshitz, which, in it's final form looks like $$\delta\int\sqrt{2m(E-U)}dl=0.\tag{44.10}$$
They used this equation to show that path of a free ...
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Error in Di Francesco et al "Conformal Field theory" Eq 9.119?
In deriving equation 9.119 in their book the authors appear to claim that the metric variation of the Ricci tensor obeys
$$
g^{\mu\nu} \delta R_{\mu\nu}= (\frac 12 g_{\mu\nu}\nabla^2- \nabla_\mu\...
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Full model of gravitational waves?
Usually when we model gravitational waves use linearized gravity:
$$
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}
$$
when applying this to the EFE and making some simplifications, when we solve we get a ...
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1
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Wormhole metric by identification
In Matt Visser's book Lorentzian Wormholes the following metric is considered
$$ds^2 = -e^{2\phi(l)}dt^2 + dl^2 + r(l)^2 d\Omega^2 $$
The entry points of the wormhole exists at $l = -L/2, L/2$. Now, ...
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Lorentz scalar Lagrangian in curved spacetime
This question might be very simple but I guess I'm missing something. We know that Lagrangian has to be a Lorentz scalar. I can see why that should be the case when dealing with inertial frames of ...
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How to model a Black Hole inside a Black Hole?
If you put a Black Hole inside of a Black hole, then you get a spherically symmetric vacuum outside the inner Black Hole and the Schwarzschild metric is the only spherically symmetric vacuum in GR. ...
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How to find 4-acceleration scalar product in terms of $ds$ spacetime interval?
We know 4-velocity $$U^i =dx^i/ds$$ where $$ds=\sqrt{dx^idx_i}$$ so we have 4-acceleration $$A^i=dU^i/ds$$
Then we have $$A^iA_i=\dfrac{dU^i}{ds}\dfrac{dU_i}{ds}$$
How should I proceed to find this ...
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answer
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Interpreting a constraint on a simplified static spherically symmetric metric
Writing the following simplified metric
$$ ds^2 = - dt^2 + dr^2 + A(r)^2 d\Omega^2 $$
Under the most general static spherically symmetric matter:
$$ T^{\mu}_{\nu} = - \rho(r) \delta^{\mu}_0 \delta^0_{\...
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Obtaining Rindler metric from observer's coordinate basis
Example 5.7 in Hartle's Gravity: An Introduction To Einstein's General Relativity shows that a tetrad for an observer undergoing hyperbolic motion in flat spacetime is given by
$$
\begin{align*}
(\...
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Stretching of space as distance to massive body like black hole decreases, true or false?
This question is about the stretching/elongation/expansion of a measuring rod in a gravitational field according to general relativity. It's not about the spagettification of something falling into ...
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Do all the geodesic equation of Schwarzschild metric should have the same unit?
I am plotting the rotation of Earth around the sun by using Schwarzschild Geodesic equations considering that $c$ is different than 1.
My question is: Should the geodesic equations have the same unit? ...
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Questions For the Expansion of Gravitational Waves
In the appendix B of this paper for effective field theory of gravitational waves, they expand the metric perturbation:
$$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$$
here deviation $h$ is dimensionless. ...
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Strange coordinate transformation in this proof of Birkhoff's theorem?
In this paper a proof of Birkhoffs theorem is provided. However, there is one step that I am quite suspicious of, so I would like to know what you think about it. The relevant part of the paper for ...
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answers
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Well-defined metric on AdS boundary
I'm reading Lecture Notes on Holographic Renormalization, particularly section 3 on asymptotically anti-de Sitter spacetimes. I will summarize a part of the discussion.
The AdS metric in conformal ...
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1
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Finding Killing vectors for hyperbolic space [closed]
I want to find the Killing vectors for the hyperbolic space, which is described by the metric
\begin{equation}
ds^2 = \frac{dx^2 + dy^2}{y^2}.
\end{equation}
I have found the Killing equations, which ...
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1
answer
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Why can coefficient $a$ between spacetime intervals depend on absolute relative velocity between the systems?
I read Landau & Lifshitz' Classical Theory of Fields book (page 14-15) (see pic below) and I was confused when I saw in proof that coefficient $a$ between spacetime intervals $(ds)^2$ and $(ds')^2$...
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Understanding instantaneous rest frames in general relativity
In special relativity, to derive formulas for proper time and proper distance, one uses the concept of instantaneous rest frames.
In order to calculate proper time of a massive object moving in the ...
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Condition for quadratic correction to first-order perturbation of metric
In Wald's book on General Relativity, the linearized Einstein tensor $G^{(1)}_{ab}$ can be obtained by substituting $g_{ab} = \eta_{ab} + \gamma_{ab}$ in the Einstein equation and ignoring terms that ...
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Why is the Einstein Field Equation relevant in the area theorem?
I am studying Area theorem and the first assumption is as follows :
If Einstein equation holds satisfy null energy condition...
I don't understand in general, what does it mean to satisfy Einstein ...
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How do gravitational fields combine together in GR?
When we have 2 massive bodies coming close together say 2 black holes or 2 massive stars, how do their respective metrics/spacetime curvature combine in the space in between them?
Do we write
$$G_{\mu\...
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Extra factor in Fourier transform of $\bar{\gamma}_{\mu \nu}$
In the weak gravity limit (and with proper gauge transformations), the linearized Einstein equation is given by:
$$
\partial^c \partial_c \bar\gamma_{ab} = -16 \pi T_{ab}\tag{4.4.12}
$$
where $$g_{ab} ...
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What exactly is the role of the Lorentzian metric within spacetime? [closed]
I learned that twodimensional spacetime diagrams and fourdimensional spacetime manifolds are provided with Lorentzian (pseudo-Riemannian) metric. However, regarding a spacetime diagram with a light ...
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How can a theory of gravitons produce a metric that shortens spatial distances?
Consider the two principles:
A QFT of spin-2 particles (gravitons) cannot transmit information faster than the speed of light by special relativity. (Let's make an assumption that such a theory can ...
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Why in physics we work with metric without defining topology/smooth structure first? [closed]
I come from more mathematical background, so I always get confused how physicists
define spaces straight by writing metric tensor without specifying all point-set infromation on which this metric ...
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4d Riemannian Manifold as Part of 4d Lorentzian Manifold
If I'm understanding correctly, this paper talks about the union of an initial space-like hypersurface and a final one after a change in spatial topology as the boundary of a 4d manifold. But is a 4d ...
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Globally hyperbolic spacetimes and (1+3) decomposition
I am having trouble in understanding the relation between globally hyperbolic manifolds and the (1+3)-decomposition used in general relativity. Let me start with the following two preliminaries:
Let $...
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Angular velocity of an object in expanding universe
In flat FRW metric, the physical momentum of an object in free motion is inverse proportional to the scale factor, that is
\begin{align}
P_{phy} \propto 1/a
\end{align}
where $a$ is a scale factor.
...
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Can we use local proper time instead of global coordinate time in special relativity?
In special relativity we have:
$$dt^2-dx^2-dy^2-dz^2 = d\tau^2$$
Where $d\tau$ is the change in local proper time of the object (the change in the clock ticks).
I've always wondered why we can't just ...
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Timelike, spacelike, and lightlike normalization conditions
Let $u^\mu=\frac{dx^\mu}{d\lambda}$ be particle's "4-velocity" where $\lambda$ is affine parameter. If I am not mistaken, we have, for the different cases:
timelike (massive particles): $u_{...
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Metric tensor's and stress-energy tensor's interaction with a spin-1 field
The metric tensor $g_{\mu\nu}$ and stress-energy tensor $T_{\mu\nu}$ are both rank-2 fields. Is it conceivable for these fields to interact with a spin-1 field? In other words, could there be ...
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Time-like separated and Space-like separated events
I am trying to get an intuition about Time-like separated and Space-like separated events. I understand the definition of these terms, but I lack intuitive understanding of what these concepts ...
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Does an expanding universe naturally induce a Lorentzian metric in local nonexpanding coordinates?
When we consider the expansion of our spatial universe in time the standard method is through a conformal transformation of the spacetime metric as:
$$g_{\mu\nu}\rightarrow a(\tau)^{2}g_{\mu\nu}$$
...
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Green's function in General Relativity: when can we express the Ricci tensor as an operator?
One can express the Riemann tensor acting on a vector field $V^\mu$ as
$$R^\sigma_{\mu\rho\nu} V^\mu = \left(\nabla_\rho\nabla_\nu - \nabla_\nu\nabla_\rho\right)V^\sigma,$$
so we must be able to ...
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2
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Is proper time still maximized with different metric signature?
A free particle moving from event $a$ to event $b$, which is timelike connected to $a$, on spacetime follows geodesics. In most cases it is a path that minimizes this integral
$$S=\int_a^b ds=\int_a^b ...
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Can the Dirac trace of two gamma matrices always be made positive?
I'm doing a calculation where I have a number of traces of just two slashed matrices.
When Calculating a Dirac trace of two slashed matrices, we have the identity:
$$Tr[\displaystyle{\not} a \...
3
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2
answers
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Difference between these two Rindler metrics
I am uncertain about the difference between these two Rindler metrics:
$$ds^2 = -\left(1 + \frac{\alpha x}{c^2}\right)^2 c^2dt^2 + dx^2+dy^2+dz^2$$
$$ds^2 = -\left(\frac{\alpha x}{c^2}\right)^2 c^2dt^...
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Black hole radiating massive particles
Is it sensible to write the metric for a black hole that emits radiation that is composed of massive particles as
$$ds^2 = -\left(1-\frac{2M(t)}{r}\right)dt^2 + \frac{dr^2}{\left(1-\frac{2M(t)}{r}\...
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1
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How to obtain this relativistic limit for Rindler coordinates?
I have trouble following the calculation in this answer for Rindler coordinates. In particular, I am not sure how the second term is obtained in this equation:
$$f^x = m\frac{d^2x}{d\tau^2} + ma_0 \...
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votes
1
answer
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Is the quadrupole tensor a $(2,0)$ tensor?
In classical electromagnetism, the quadrupole moment is usually written as
$$Q_{ij}=\int d^3r \rho(\vec{r})(3r_ir_j-r^2\delta_{ij}). \tag{1}$$
However, this represents the quadrupole tensor as a ...
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0
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A confusing sign in Maxwell's equations with differential forms
I'm running into a basic issue with Maxwell's equations in terms of differential forms. Let's work in Minkowski space with mostly negative signature, and define the Hodge star in the usual way,
$$\...
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What happens to $g^{\alpha\beta}_{,\sigma}=-g^{\alpha\mu}g^{\beta\nu}g_{\mu\nu,\sigma}$ when $g_{\mu\nu}\rightarrow \eta_{\mu\nu}$ (weak field limit)?
The equation
$$g^{\alpha\mu}_{\,\,\,\, ,\sigma}\,g_{\mu\nu} + g^{\alpha\mu}\,g_{\mu\nu,\sigma} = (g^{\alpha\mu}g_{\mu\nu})_{,\sigma} = \delta^\alpha_{\nu,\sigma} = 0 $$
gives the useful relation
$$g^{\...
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