All Questions
Tagged with metric-field or metric-tensor
3,670 questions
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Question in prove infinitesimal Lorentz transformation is antisymmetric
I know we need prove this property by:
\begin{align*}g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*}
and
$$\Lambda^\mu_{\ \ \ \nu}=\delta^\mu_{\ \ \ \nu} + \...
-1
votes
1
answer
87
views
Is the 4D Minkowski spacetime a physical and/or mathematical necessity?
I want to know the physical and mathematical justification for formulating Special Relativity in terms of a four-dimensional pseudo-Euclidean space with metric $\mathrm{d}s^2=c^2\mathrm{d}t^2-\mathrm{...
- 1,112
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votes
1
answer
352
views
What would happen if a black hole disappeared? [closed]
Imagine if a black hole disappeared. Would spacetime act like a rubber band and propel objects that used to be caught in its gravitational field outwards - i.e. some kind of space time explosion? How ...
- 131
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votes
1
answer
313
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Criterion for a black hole in Anti-de Sitter background
Consider a Schwarzschild-Anti de Sitter (SAdS) metric
$$ds^2=-(1-\frac{2M}{r}+ k\, r^2 )\, dt^2+\frac{dr^2}{1-\frac{2M}{r}+k \,r^2}+r^2 d\Omega_2^2, $$
with $M,k>0$. This solution has only ...
user56224
-1
votes
2
answers
391
views
Antisymmetric Lorentz parameters [closed]
In the expression for infinitesimal Lorentz transformation parameters omega appeare with all indices down and this forms antisymmetric matrix. What I think is that if we raise one of the indices up it ...
- 1,813
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votes
1
answer
81
views
Basic Index Raising and Lowering Question [duplicate]
I am trying to understand the order of the indices when raising or lowering tensors.
For example, the electromagnetic tensor:
$$F^{\alpha \beta} =
\begin{bmatrix}
0 & -\frac{E_{x}}{c} &...
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-1
votes
1
answer
1k
views
What is the difference between space-like four-momentum and time-like four-momentum? [closed]
I was reading a wiki page on Tachyon, came across these terms? What i need is bit of a mathematical description to understand these terms?
- 175
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votes
2
answers
660
views
What is a light cone? [closed]
What is a light cone? Why we can't escape the light cone? Why the speed of light being the limit for us to escape the cone the future and the past events of the light cone is that governs the future? ...
-1
votes
2
answers
73
views
The central point is singular or not?
Let take the metric
\begin{eqnarray}
\mathrm ds^2 = -f^2(r) dt^2 + g^2(r)dr^2 + r^2~\mathrm d\Omega^2
\end{eqnarray}
with $f^2(r) = 1 -ar^3$ and $g^2(r) =\frac{b}{1-ar^3}$.
Is the central point $r=...
- 471
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votes
2
answers
231
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Indices in energy-momentum tensor
Let's assume next Lagrangian:
$$\mathcal{L}=\partial_0\phi + \partial_0\psi^*+\nabla \phi \nabla \phi^*$$
and try to derive energy-momentum tensor:
$$T^{\mu \nu}=\frac{\partial \mathcal{L}}{\...
- 31
-1
votes
1
answer
61
views
Is it clear that $\epsilon_{abcd}F^{ab}(\delta^{\mu}_e\delta^{\nu}_f-\delta^{\mu}_f\delta^{\nu}_e)\eta^{ce}\eta^{df}$ vanishes? [closed]
Is it clear that $$\epsilon_{abcd}F^{ab}(\delta^{\mu}_e\delta^{\nu}_f-\delta^{\mu}_f\delta^{\nu}_e)\eta^{ce}\eta^{df}$$ vanishes without computing explicitly?
Here $\epsilon_{abcd}$ is the totally ...
- 2,022
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votes
2
answers
640
views
Problem in Metric Tensors
In the second book, Field Theory, of popular series of Theoretical Physics by Landau-Lifschitz are obtained following equations. After making linear coordinate transformation (prime=old+ipsilon) a new ...
- 638
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1
answer
446
views
Why isn't the Schwarzschild radius defined as the distance from the event horizon to the "place" where all mass is concentrated?
The Schwarzschild radius is defined as the radius of the sphere in 3-dimensional space surrounding a black hole. Let's switch to 2-dimensional space to make things easier to visualize. Why is the ...
-1
votes
1
answer
257
views
Can Spacetime interval be positive between two objects or events? [closed]
We know that spacetime interval can be negative like if I am watching a star far away or it could be 0 like if it I an watching a guy sitting next to me but can it be Positive?
-1
votes
2
answers
78
views
Curvy space in and around massive objects [closed]
If space curves around massive objects, what happens to the space within the massive objects?
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1
answer
429
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Distance of closest approach
When deriving the gravitational bending angle of light, In this paper, the author introduced $R$ (the distance of closest approach), in equation ($7$), to solve the problem.
My question: How is $R$ ...
- 789
-1
votes
1
answer
459
views
Metric tensor in General Relativity or otherwise [closed]
What is the metric tensor?
How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices?
How it relates to distance function (metric) and angles?
How does ...
- 19
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votes
0
answers
63
views
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}{\...
- 141
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votes
2
answers
99
views
Is it possible for a Lorentz scalar to NOT be invariant under another linear transformation?
Lorentz scalars are invariant under Lorentz transformations, which are a subset of linear transformations. I wanted to know if it is possible, for a Lorentz scalar, to NOT be invariant with respect to ...
- 454
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2
answers
349
views
Construction of infinitesimal Lorentz transformation
I'm following the book from Greiner on relativistic QM and I got two questions here:
In (3.36b) where does the last expression come from? From which we get delta = delta + omega_nu^sigma + omega^...
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votes
2
answers
309
views
Isotropy and Homogeneity of the internal Schwarzschild Metric
I've been working on a paper involving the internal Schwarzschild metric and something I've been told multiple times is that the internal metric is not isotropic. I understand that the external ...
- 2,525
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1
answer
247
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Is the Euclidean metric a necessary accessory of Lorentzian spacetime?
Spacetime manifolds, as well as their simplified form, twodimensional spacetime diagrams, are always Lorentzian (Lorentzian metric $ds^2 = dt^2 - dx^2$). Normally, the Lorentzian metric is hidden ...
- 3,151
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votes
1
answer
109
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Contraction of a second-rank tensor with the metric tensor
Is it legitimate to perform the following tensor contraction?
$ g_{\alpha\beta} G_{\mu\nu} \partial^\mu \phi \partial^\nu \phi = g_{\mu\nu} \delta^\mu_\alpha \delta^\nu_\beta G_{\mu\nu} \partial^\mu ...
- 69
-1
votes
1
answer
422
views
Line element to polar coordinates [closed]
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
\begin{equation}
\mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta}
\end{equation}
So:...
-1
votes
2
answers
312
views
Clearing up a discrepancy when deriving the Lorentz transformation from length contraction
I've been working through the Feynman Lectures on Physics. I'm currently on lecture 15: The Special Theory of Relativity, specifically 15-5, the section on the deriving the Lorentz Transformation ...
-1
votes
1
answer
300
views
Can an a distance in Minkowski space, based on a Euclidean plane, be time-like?
In a Minkowski space diagram of ct vs x, with 2 stationary events in the same plane of spacetime (1 event located along a worldline, 1 not on a worldline) is this still considered euclidean space?
...
- 201
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1
answer
1k
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Lower/Raising Riemann curvature tensor
I want this :$R^{a}_{bcd}=R_{abcd}$
I tried this:$R^{a}_{bcd}=g^{ae}R_{ebcd}$ but it is not $R_{abcd}$.
How do you lower $a$ without changing it with another dummy index ?
Edit: This is what i ...
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1
answer
100
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What is $\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$?
Title says it all, is there a closed expression for
$$\frac{\delta (\partial_\kappa \sqrt{g})}{\delta g^{\mu\nu}}$$
where $g = \det g_{\mu\nu}$?
- 1,323
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1
answer
423
views
Is the Pythagorean theorem one of the assumptions needed in order to make special relativity consistent?
In all the formulations of special relativity I have seen and I am no expert mind you and in fact a beginner I have noticed that the Pythagorean theorem is always used in one form or fashion to ...
user86411
-1
votes
1
answer
678
views
Conformal Transformation: Minkowski sheet to cylinder
What conformal transformation can I make to 2d Minkowski with metric $ds^2=-dt^2+dx^2$ to show that it is conformal to a cylinder?
- 759
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2
answers
1k
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Off-diagonal terms in metric for 4D space-time [closed]
Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
- 81
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1
answer
86
views
How do you calculate the time-time component of a tangent vector in spacetime?
The norm of a tangent vector of a spacetime manifold is the sum of the square of its components:$$\lVert \boldsymbol {V_t}\rVert^2=\left(\frac{\partial t}{\partial\tau}\right)^2+\left(\frac{\partial x}...
- 584
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3
answers
382
views
What is the evidence that gravitational fields don't sum up as a superposition?
Einstein's field equations are non-linear. Gravity gravitates (self-interacts). It's very complicated to solve Einstein's field equations for more than one central object. That are keystones in ...
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1
answer
85
views
Confusion about Einstein's field equations
There is an issue I'm having regarding the Einstein field equation I would like clarification on... Given that:
$$R_{\mu v} -\frac{1}{2}g_{\mu v}R=kT_{\mu v}$$
Why can't the $g_{\mu v}R$ term be ...
- 613
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votes
4
answers
227
views
How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it
Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
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votes
1
answer
333
views
Calculating speed in four dimensions [closed]
If you are moving at $c$ in 3D space and $c$ in time axis too, What would be your total speed?
Edit: Since question has been voted to be closed, I shall make an Edit.
In 4D world all objects move ...
- 2,111
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votes
2
answers
77
views
Background spacetime in general relativity [closed]
When calculations of spacetime curvature are rendered for a given situation, the resultant curvatures are obtained with respect to Minkowski spacetime.... Flat. What evidence exists that the ...
- 470
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votes
1
answer
472
views
Spacetime stretching [duplicate]
When spacetime stretches, does both space and time stretch proportionally. That is, if you stretch Planck-length doesn't Planck-time also stretch proportionally? I find this to be a conundrum.
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3
answers
1k
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Why did Einstein put a negative sign in the Pythagorean theorem? [duplicate]
In 4-dimensional spacetime, when we study the spacetime interval, why did Einstein put a negative sign in it?
$$x_1=x$$
$$x_2=y$$
$$x_3=z$$
$$x_4=ct$$
$$ds^2=dx^2+dy^2+dz^2-(cdt)^2$$
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votes
2
answers
129
views
Angular coordinates are unaffected?
Why then is there no change in the angular coordinates in Schwarzschild metric? Why arent they unaffected by the mass?
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1
answer
142
views
How do I compute the trace and the inverse of this tensor?
$$[X^{\mu\nu}] = [X] = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 1 & 1 & 3 \\ 1 & 2 & 3 & 0 \end{pmatrix}$$
How to compute the trace of $X^{\...
- 383
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votes
3
answers
255
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Is this a valid approach to think about the $N$-Body - Problem in General Relativity?
I would like to analyze the following problem:
Some point masses, between and around them just empty space ($N$-Body-problem). I would like to analyze the space between and around those point masses. ...
- 93
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votes
1
answer
100
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Klein-Gordon in Schwarzschild Curved Spacetime [closed]
Given such a Schwarzschild metric, the covariant Klein-Gordon equation for a mass $m$ takes the form
$$\left[\frac{1}{g_{00}} \frac{\partial^2}{\partial t^2 }-\frac{1}{r^2} \frac{\partial}{\partial r}...
- 1
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votes
1
answer
146
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Finding equation of motion for given Lagrangian with respect to metric
Given the following action in $d$ dimensional $(0,1,...,d-1)$ curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[...
-2
votes
1
answer
69
views
Time difference between all particles and waves [closed]
Since all elementary particles and waves were created simultaneously in the big-bang (t0) would there be any time difference between any interacting elementary particles and/or waves after t0? I'm ...
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votes
1
answer
144
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Norm of summation of vectors
If we have a vector $\partial_v$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$ g^{vv}\partial_v\partial_v$.
My question what if we have ...
- 167
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votes
1
answer
67
views
Divergence of canonical energy-momentum tensor in QFT [closed]
I have to show that the divergence of the canonical energy-momentum tensor is zero, i.e. $$\partial^{\mu}T_{\mu\nu} = 0.$$
The Lagrangian is $$\mathcal{L} = \frac{1}{2}\partial^{\mu} \phi \partial_{\...
- 141
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votes
1
answer
307
views
Is this posible in GR $g_{ab}g^{ab}=1$? [duplicate]
Metric tensor multiplied by its inverse. I always see this with different indices.
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votes
1
answer
107
views
Raising and lowering indices: how to compute $\sigma_{\mu}$ when we know $\sigma^{\mu}$ [closed]
I have a question about raising and lowering indices. How to compute $\sigma_{\mu}$ when we know $\sigma^{\mu}$? Here
\begin{align}
\sigma^{\mu} = \left( \begin{matrix}
\left( \begin{matrix} 1 & ...
- 257
-2
votes
1
answer
304
views
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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