New answers tagged differential-geometry
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How to derive the Kerr killing vector?
I believe you were almost there :)
If you consider that without loss of generality you can set a=1 (or, in alternative, you wanna ask yourself what is b going to look like with a set to 1) you then ...
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Lorentz scalar from the second derivative of the metric tensor
For the fluctuations $h^{\mu \nu}$ around a flat background, $\partial_\mu \partial_\nu h^{\mu \nu}$ is a Lorentz scalar. For highly curved backgrounds, it is not sensible to classify quantities by ...
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How does an electromagnetic wave warp spacetime?
Yes, there are plane wave solutions of the Einstein-Maxwell equations. Indeed there are various parallel plane wave solutions to the Einstein equations, the pp-wave spacetimes. They correspond to ...
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How to combine Vierbein fields?
On the first place Vierbein's transform coordinate components $V^\mu$ of vectors in components $V^a$ which transform consistently under Lorentz transformations:
$$V^a(x) = e^a_\mu(x) V^\mu(x)$$
If the ...
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Derivation or origin of projection tensor
This is very simple linear algebra, not even differential geometry. You first need to understand things at the level of one vector space; for the differential geometry case, you just apply this ...
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How does an electromagnetic wave warp spacetime?
This is a good question. The answer is tricky, because it requires that you look at the electromagnetic wave as part of a system. Considered in isolation, a light pulse can be made to have arbitrarily ...
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Derivation or origin of projection tensor
You can approach this from elementary linear algebra. For the first part we will deal with Euclidean vectors so there will be no distinguished upper and lower indices.
With a unit vector $\vec e$ the ...
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Derivation or origin of projection tensor
Take an arbitrary vector $X^a$. Its projection along $u^a$ direction will be $X^bu_b u^a$. Then the component orthogonal to $u^a$ will be simply $X^a-X^bu_bu^a = (\delta^a_b - u_bu^a)X^b={P^a}_bX^b$
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On the Background Independence condition
This could sound counterintuitive but the background independence condition means that the dynamics of a field depend on the background. I am gonna explain myself:
Classical Field Theory
Assume we ...
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Tetrad formalism vs coordinate formalism example
Tetrads offer a way to locally interpret tensor (physical) quantities in GR in terms of what a normal (physical) observer would measure. They are particularly powerful in connecting GR with SR, ...
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Physical intuition of spin connection and spinor bundles?
A spinor bundle is defined via the associated bundle construction. Lets first make a couple of preliminary definitions:
A rotation bundle is defined as a principal bundle whose gauge structure group ...
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Showing that a generator exponentiates to a $\mathbb{R}$ group
From now on $\{\cdot,\cdot\}$ denotes the Lie commutator of smooth vector fields on a smooth manifold.
As far as I understand, you have a vector field $X$ on $TM$, where $M$ is the Schwarzschild ...
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Understanding and Expressing the Definition of Inertia Tensor in the Language of Differential Geometry
$\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathcal{E}}
\renewcommand{\O}{\mathrm{O}}
\newcommand{\SO}{\mathrm{SO}}
\newcommand{\so}{\mathfrak{so}}
\renewcommand{\d}{\mathrm{d}}
\renewcommand{\r}{\...
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Why is the covariant derivative of a one-form $\nabla_{i}v_j=\frac{\partial v_j}{\partial u^{i}}-\Gamma^k_{~ij}v_k$?
Let me add a more intuitive explanation that follows Ferrari, Gualtieri, Pani argument to introduce covariant derivatives.
Let's differentiate a 1-form along the coordinates, writing it as a ...
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How to derive the commutation coefficient from coordinate basis (GR)?
An anholonomic basis is basically just the linear combination of holonomic bases where the expansion coefficients $f$ are functions of coordinate variables -
$$\begin{equation}\tag{1}\mathbf{e}_{i}=f^{...
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Is it possible to use topology arguments to find analogies in thermodynamic systems?
I use the answer here rather than commenting since I (still) don't have enough reputation to add comments.
I assume that Charateodory first and then others, e.g. Truesdell and Noll, have tried a ...
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Understand the Lorentz transformation in QFT
I want to propose a different approach, as I believe the OP might benefit from comparing different viewpoints on distribution theory.
The question is really about the transformation of distributions ...
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When is a classical field theory on curved spacetime supersymmetric?
Assume a given field theory with fields $\Phi^I$, where $I$ spans all the fields. $\Phi^I$ can be either a boson or a fermion and have any spin consistent with its statistics. Let the flat-space ...
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Understand the Lorentz transformation in QFT
A Lorentz transformation from coordinates $x^i$ to $y^i$ has components $\Lambda^i_{\ j} = \frac{\partial y^i}{\partial x^j}$. Its pullback on the volume form $\mathrm d^4x \equiv \mathrm dx^{0}\wedge\...
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Definition of global supersymmetry on curved spacetimes and use of constant spinor fields
The covariantly constant or parallel spinors arise as follows:
Suppose we start with a superalgebra with supercharge $Q$. Then an infinitesimal transformation by that supercharge is generated by $\bar\...
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Can one encode topology into the position operator in quantum mechanics?
Mathematically the data of the topology is encoded in the structure of the operators themselves. I will focus on the simple case of the punctured plane $M=\mathbb{R}^2\setminus \{0\}$.
Suppose we are ...
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In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?
It's not clear that GR generalizes to 1+1D. In two dimensions the stress-energy tensor has more components than the curvature ($3>1$), so it's not obvious what the GR field equation should look ...
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