6
votes
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 ...
- 75.6k
5
votes
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 $X^a-X^bu_bu^a = (\delta^a_b - u_bu^a)X^b={P^a}_bX^b$
- 1,897
3
votes
Accepted
Chose coordinates where $g_{01}=g_{02}=g_{03}=0$ to disentangle space and time?
Yes of course you can do that. You can even go beyond to impose $g_{00}=-1$. This is thank to the coordinate freedom which lets you fix 4 out of 10 metric components. Then you are remained with ...
- 1,120
2
votes
Accepted
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 ...
- 1,809
2
votes
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 ...
- 36.7k
1
vote
Functional derivative and exterior derivative
If the 3-manifold does not have a boundary, then functional derivatives and exterior derivative commute, i.e.
$$\begin{align} \left[\frac{\delta}{\delta A(y)},\mathrm{d}^{(x)}\right]A(x)~=~&0, \cr ...
- 209k
1
vote
Lorentz scalar from the second derivative of the metric tensor
For the fluctuations $h^{\mu \nu}$ around a flat background, $\partial_\mu \partial_\nu h^{\mu \nu}$ is a Lorentz scalar. For highly curved backgrounds, it is not sensible to classify quantities by ...
- 8,431
1
vote
Derivation or origin of projection tensor
This is very simple linear algebra, not even differential geometry. You first need to understand things at the level of one vector space; for the differential geometry case, you just apply this ...
- 6,880
1
vote
Accepted
Understand the Lorentz transformation in QFT
A Lorentz transformation from coordinates $x^i$ to $y^i$ has components $\Lambda^i_{\ j} = \frac{\partial y^i}{\partial x^j}$. Its pullback on the volume form $\mathrm d^4x \equiv \mathrm dx^{0}\wedge\...
- 1,351
1
vote
Is it possible to use topology arguments to find analogies in thermodynamic systems?
I use the answer here rather than commenting since I (still) don't have enough reputation to add comments.
I assume that Charateodory first and then others, e.g. Truesdell and Noll, have tried a ...
1
vote
Definition of global supersymmetry on curved spacetimes and use of constant spinor fields
The covariantly constant or parallel spinors arise as follows:
Suppose we start with a superalgebra with supercharge $Q$. Then an infinitesimal transformation by that supercharge is generated by $\bar\...
- 127k
1
vote
How to derive the commutation coefficient from coordinate basis (GR)?
An anholonomic basis is basically just the linear combination of holonomic bases where the expansion coefficients $f$ are functions of coordinate variables -
$$\begin{equation}\tag{1}\mathbf{e}_{i}=f^{...
- 23
1
vote
Accepted
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}{\...
- 26
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