All Questions
5,969 questions
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Is there a known correspondence between curved supersymmetric ambitwistor space and curved supermanifolds in four dimensions?
The nonlinear graviton construction in twistor theory gives a correspondence between anti-self-dual conformal manifolds and certain three dimensional complex manifolds, which generalizes the twistor ...
- 310
0
votes
0
answers
40
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The second variational derivative of Ricci tensor with respect to metric
Is there a more efficient way to compute the second functional derivative of Ricci tensor,
\begin{equation}
\frac{\delta^2 R_{\mu \nu}(x)}{\delta g^{\alpha \beta}(y) \delta g^{\gamma \epsilon}(z)}
\...
- 85
1
vote
1
answer
28
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From material derivatives to partial derivatives in the wave equation
Consider the Cauchy momentum equation:
$$\rho \frac{d^2 \mathbf{u}}{d t^2} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$$
where $\rho(\mathbf{x},t)$ is the density, $\mathbf{u}(\mathbf{x},t)$ ...
- 225
2
votes
0
answers
32
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Getting the Double Field Theory action from the projectors
I am mainly focusing on the following paper by Olaf Hohm and Barton Zwiebach: On the Riemann Tensor in Double Field Theory, so I'll give broad brushstrokes as to what my qualms are.
The DFT (Double ...
- 125
-1
votes
2
answers
36
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Why must the total time derivative only be a linear function of velocity? [duplicate]
I'm hung up on page 7 of Landau & Lifshitz Course on Mechanics. They claim,
$$L(v'^2) = L(v^2)+\frac{\partial L}{\partial v^2}2\textbf v\cdot \epsilon \tag{p.7}$$
The second term on the right of ...
0
votes
0
answers
28
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How to do expansion of Lagrangians in terms of parametrized metric
For a parametrized metric $$g_{00}= -e^{2\varphi}, g_{0i}=e^{2\varphi}A_{i}, g_{i0}=e^{2\varphi}A_{j}, g_{ij}=e^{-2\varphi}(\delta_{ij}+\sigma_{ij})-A_{i}A_{j}.$$
How to expand $
\sqrt{-g} R(g_{\mu\nu}...
- 51
1
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0
answers
32
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How much does classical mechanics depend on the choice of symplectic form?
TlDr; a different choice of symplectic structure on a phase-space $\mathcal{M}$ affects the Hamiltonian mechanics insofar as it could affect what the canonical coordinates are, but is this the only ...
- 230
3
votes
1
answer
118
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Covariant derivative acting on Dirac delta function
Pardon my naive computational question. In my calculations, I encounter the following expression:
\begin{equation} \label{eq1}
\frac{\delta}{\delta g^{\gamma \epsilon}(z)} \left( g_{\mu \alpha}(x) \...
- 85
-3
votes
1
answer
90
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Origin of this equation attributed to Einstein
I came across this image of an equation that is apparently attributable to Einstein (his field equations). Does someone know where I can find the original equation? I'd like to dig a bit more into ...
0
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0
answers
45
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Ricci scalar of a dimensionally reduced theory (in D = 3)
I hope this question can be re-posed based on an answer. In a previous question, I asked about the dimensionally reduced Ricci scalar and got a very detailed answer. However, following the steps I got ...
- 55
2
votes
1
answer
113
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Jacobian and chain rule contradiction, Geodesic equation
Consider two components of a contravariant vector related by the jacobian $J^\bar{\mu}_\mu = \frac{\partial x^{\bar{\mu}}}{\partial x^\mu}$ , $x^{\bar{\mu}}= x^{\bar{\mu}}(x^\mu)$
$x^\bar{\mu} = J^\...
- 6,797
1
vote
0
answers
40
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
0
votes
0
answers
37
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Wald theorem 8.1.2 and the proceeding corollary
I am currently confused on the corollary of theorem 8.1.2 in Wald's book, specifically the paragraph separating the two. I've attached a screenshot below.
Why does Wald say that using theorem 8.1.2 ...
1
vote
1
answer
50
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Parallel transport of a vector on a $2d$ plane [closed]
Consider a 2d plane such that there is a curve $\gamma$ that traces a circle of radius $r=r_0$. Suppose a vector $A^\mu = (A_1, A_2)$ is attached on the circle as shown in the image below. I want to ...
- 512
2
votes
1
answer
81
views
Difference between an orthonormal frame and normal coordinates
I want to verify that my understanding of this topic is correct. Consider a manifold M, with metric tensor $\mathbf{g}$ and coordinates some spacetime coordinates $x^\mu$. The metric tensor in ...
- 4,737
-1
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0
answers
63
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
- 141
0
votes
1
answer
81
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Problem in deriving Killing equation
I am studying derivation of Killing equation by Wald (also reading some other literature) but having some problem in understanding the math.
Let $\chi ^a$ is killing vector on the horizon
$$\chi _{[a \...
- 185
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0
answers
27
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Visual proof of Stokes' theorem - how can we (for example) approximate a straight line by infinitesimally small aligned `v`'s? Non-standard analysis [migrated]
This webpage (unfortunately not in English) provides a visual proof of Stokes' theorem.
After showing at the local level - using some particular yellow contour (and particular green / red / blue ...
- 740
3
votes
0
answers
318
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Ricci scalar of a dimensionally reduced theory
I want to calculate the Ricci scalar using the Cartan equations in the context of reducing a spatial dimension of a 4-dimensional theory describing an empty spacetime with a metric of the form
\begin{...
- 55
1
vote
0
answers
25
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Reference request: an introduction to objective rates and constitutive equations
As a hobby, I am trying to better understand the theory around objective rates, frame invariance and objective constitutive equations in non-Newtonian fluid mechanics. Unfortunately, I have been ...
3
votes
0
answers
91
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Would a degenerate coordinate system be acceptable?
Suppose I’ve got some spherically-symmetric metric akin to the Schwarzschild metric and I render it in spherical coordinates,
$$\text{d}s^2=f(r)\text{d}t^2-g(r)\text{d}r^2-r^2\text{d}\Omega^2$$
for ...
- 3,534
0
votes
2
answers
84
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Light bends in what path? [closed]
I have heard light bends around black holes because they are very massive. That is because the shortest distance there is the displacement along a non planar curved surface where Euclidean Geometry ...
- 163
1
vote
1
answer
67
views
When a boundary value problem for geodesics equation has a unique solution (is well posed)?
By a boundary value problem (bvp) for geodesics, I mean that I know two points between which a geodesic is spanned but I have no information about initial and final velocities in these points.
What ...
- 175
1
vote
0
answers
37
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The contraction of Christoffel symbols [duplicate]
I have a question regarding the the contracted Christoffel symbols from David Tongs PDF on general relativity.
He wants to prove that
$$\Gamma^{\mu}_{\mu v}=\frac{1}{\sqrt{g}}\partial_v\sqrt{g}$$
...
- 19
2
votes
3
answers
84
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Deriving differential equation for the path of a particle in potential $U(r)$ using Maupertuis’ principle
I came across this Maupertuis' principle in Landau and Lifshitz, which, in it's final form looks like $$\delta\int\sqrt{2m(E-U)}dl=0.\tag{44.10}$$
They used this equation to show that path of a free ...
- 109
1
vote
1
answer
139
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Why can’t spacetime be projective?
I come from a pure mathematics background so I’m sure that there’s an obvious reason this cannot be the case, but it’s not obvious to me, however I haven’t found much discourse anywhere on the ...
- 21
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0
answers
35
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Derivation of spinor Newman-Penrose equation
I am trying to work through the original paper by Newman-Penrose, Newman, E., & Penrose, R. (1962). An approach to gravitational radiation by a method of spin coefficients. Journal of Mathematical ...
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-1
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1
answer
108
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What is stopping Einstein-Cartan theory (ECT) from being a bridge to a unifying theory? [closed]
I have recently read more about the Einstein-Cartan Theory and it seems to provide very interesting solutions (or hints thereof) that the General Relativity can't. Although it would not be the ...
- 323
4
votes
0
answers
71
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Error in Di Francesco et al "Conformal Field theory" Eq 9.119?
In deriving equation 9.119 in their book the authors appear to claim that the metric variation of the Ricci tensor obeys
$$
g^{\mu\nu} \delta R_{\mu\nu}= (\frac 12 g_{\mu\nu}\nabla^2- \nabla_\mu\...
- 56.6k
3
votes
2
answers
340
views
Understanding the definition of the covariant derivative
I'm currently working my way through the book "Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry" and I think I have a very fundamental ...
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1
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0
answers
74
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I am not able to derive strain tensor in different coordinate systems using Lie Derivatives [closed]
I've already seen a similar discussion in this forum (How to compute the strain rate tensor in non-Euclidean coordinates), but I still have problems.
The strain tensor is defined in Euclidean/...
-2
votes
1
answer
59
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
1
vote
1
answer
83
views
How curve is inextendible?
I am reading a literature arXiv:1901.03928v6 [hep-th] 12 Jul 2023 and it says :
In Minkowski space, the timelike geodesic $(t(s),x(s))=(s,0)$ is inextendible if $s$ is regarded as a real variable.
...
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1
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1
answer
84
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Why are curvatures interpreted as forces in Gauge Theory?
I've been learning differential geometry for a while, and am now reading up on Gauge Theories in Physics. I've come across the notion that curvatures on our Fiber Bundles correspond to forces a few ...
- 143
1
vote
1
answer
47
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Invariance under certain transformation in quantum mechanics and classical mechanics
I'm an undergraduate student in physics and have learned quantum mechanics (Griffiths) and classical mechanics (Marion). My question is bearing on the invariance under specific transformation.
In ...
- 31
3
votes
1
answer
67
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"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
0
votes
1
answer
53
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Derivative for the Maxwell field [closed]
I'm struggling with the following expression, which occurs in the derivation of the Maxwell Lagrangian in field theory.
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}...
2
votes
0
answers
131
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Understanding Bianchi identities in Newman-Penrose formalism
In the classical paper by Newman & Penrose, the authors introduce their formalism. However I am a bit confused over the Bianchi identities (4.5), in their paper. The first in particular reads:
$$D\...
- 1,187
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votes
1
answer
80
views
The definition of the Lie Derivative
I am aware that an answer to an almost identical question already exist, however, I found the already existing answer not helpful (at least to my current question).
Carroll defines, in his book, the ...
4
votes
1
answer
124
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Future inextendible curve
The definition of future inextendible curve is as follows:
A causal curve $\gamma$ is called future inextendible if it is impossible to find an event p $\in \mathcal{M}$ such that for all $U \subset \...
- 185
1
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1
answer
49
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Conceptual questions about Newman-Penrose Formalism
I'm taking an advanced undergraduate self-study course on tetrad methods and Newman-Penrose formalism. I'm following Chandrasekhar's masterpiece Mathematical Theory of Black Holes, I'm working through ...
0
votes
1
answer
70
views
Why are Weyl's Equations composed of only first-order derivatives?
I'm studying the Weyl's Equations from Section 1.5 of Perkins' Introduction to High Energy Physics.
The author says this:
Dirac set out to formulate a wave equation symmetric in space and time, ...
- 1,637
-2
votes
1
answer
58
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Problem solving geodesic equations numerically [closed]
I been having trouble solving the geodesic equations. The end goal is to plot them on a surface. I firstly calculated the Christoffel symbols and then inserted the differential equation in an ODE ...
- 1
2
votes
0
answers
64
views
Normal Vectors to a Hypersurface
I am reading the Hypersurfaces section on Carroll's GR book. He consider a hypersurface defined by the equation
$$f(x) = f^*$$
where $f$ is some function $f^*$ is a constant. Then he claims that the ...
9
votes
4
answers
4k
views
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant but acceleration is constant?
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant, but rate of change of velocity is constant?
Like speed is only the magnitude, so ...
0
votes
0
answers
68
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Mileage table argument in Weinberg's GR
The following sentences are from Weinberg's book on gravitation, where he says that we can use airline distances between four places on earth to tell earth surface is curved. The page number is 7.
...
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1
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1
answer
50
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Grassmann Numbers, anticommutation and derivative rules
If $\psi(t)$ is a complex Grassmann number and $\psi^*(t)$ is its complex conjugated. The following is true:
$$\frac{\partial (\psi^*\psi)}{\partial \psi}=-\psi^*\frac{\partial \psi}{\partial \psi}=-\...
- 1,628
-2
votes
0
answers
70
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
2
votes
1
answer
139
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Quantum mechanics in the language of differential geometry [closed]
So I am currently studying differential geometry and start recognizing a lot of concepts familiar from physics in the toolset of manifolds, tangential bundles and vector fields. In particular, we can ...
- 342
1
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1
answer
82
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Why the two methods give correct answer for the killing fields?
Consider the embedding of the $\mathbb{S}^2$ in $\mathbb{R}^3$,
\begin{align}
x & = r \sin \theta \cos \phi,\tag1 \\
y & = r \sin \theta \sin \phi,\tag2 \\
z & = r \cos \theta,\tag3
\end{...
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