All Questions
20
questions
-1
votes
1
answer
133
views
Geometric Construction of Minkowski Space and Proper Time
In Minkowski spacetime, we define a tangent space at each point and using the metric, we calculate a real number that is the infinitesimal invariant interval between two points. This real number, ...
- 489
0
votes
0
answers
57
views
Commutation of exterior product in spacetime
I am following these notes (A Practical Introduction to
Differential Forms, by Alexia E. Schulz and William C. Schulz 2013) and on page 67 (pdf page number 71) there is an expression for the ...
- 147
1
vote
0
answers
106
views
Energy is the time component of 4-momentum in SR: Proof as per R. Wald's book
This is an excerpt fom R. Wald's book on General Relativity (page 61). I'm not able to understand how he deduces that $E$ must be the time component of $p^a$ with only the assertions made before this ...
- 107
6
votes
2
answers
433
views
Confusion regarding bundle structure of Galilean spacetime in Penrose's The Road to Reality
I am reading Roger Penrose's The Road to Reality. In section 17.3, I encounter the following passage. To give a context, Penrose was explaining that even though an Aristotelian spacetime can be ...
- 682
3
votes
1
answer
231
views
Meaning of the metric tensor
I took a relativity class as an undergraduate but lost contact with the theory many years ago. Recently I took some old notes to revisit some concepts. I am a layman in the subject, so I apologize in ...
- 585
4
votes
3
answers
853
views
Is there a physically meaningful example of a spacetime scalar potential?
From Misner, Thorne and Wheeler, page 115.
0-Form or Scalar, $f$
An example in the context of 3-space and Newtonian physics is temperature $T\left(x,y,z\right),$ and in the context of spacetime, a ...
- 2,015
0
votes
1
answer
89
views
What are physical evidence that we are living in affine space?
An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping
$\phi:E \times E \rightarrow V,\...
- 445
1
vote
1
answer
129
views
Is there a more satisfactory answer than just saying that the manifold of special relativity is the $\mathbb R^4$/some set of "events"?
I'm an undergraduate who visited a course on differential manifolds and now I have the task to reformulate the maxwell equations in terms of differential forms. The most obvious question that arises ...
- 1,801
1
vote
2
answers
160
views
Doubt about the ways to define Spacetimes
My Question:
The most general way to define a Spacetime is by the aid of Differentiable Manifolds; therefore the underlying structure is of a topological manifold. Furthermore, we can talk about the ...
- 3,085
4
votes
1
answer
454
views
Notation of Mixed Tensors: Risk of Confusing Index Positions?
The convention for notating indices of a tensor is to write a contravariant index superscript and a covariant index subscript. If one has a pure contravariant or a pure covariant tensor of $2$nd order,...
- 43
0
votes
1
answer
2k
views
How to find a normal to an hypersurface?
I have to apply the Israel junction conditions in a region in which a hypersurface with O(3) symmetry separates two spacetime with Schwarzschild metric (with masses $M_+$, the exterior one, and $M_-$, ...
- 41
2
votes
0
answers
66
views
Is the linearity of transformation of intervals postulate in SR? [duplicate]
Let $O$ and $\overline{O}$ be inertial frames, $A$ and $B$ be events, the interval between $A$ and $B$ be $(\Delta t, \Delta x, \Delta y, \Delta z)$ in $O$ and $(\overline{\Delta t}, \overline{\Delta ...
- 153
0
votes
0
answers
82
views
Isometry is equivalence relation on Lorentzian manifold?
Proofwiki says isometry is an equivalence relation on metric spaces, and e.g. Minkowski space is a metric space - but as a metric space its metric is not the Minkowski metric (see this S.E. question).
...
- 772
2
votes
1
answer
514
views
Area in Minkowskian Spacetime
If, in a $d$ dimensional space with Euclidean metric, we parametrize a two-dimensional surface with parameters $\xi^1$ and $\xi^2$ then the area can be written as
$$A = \int ~\mathrm d\xi^1 \mathrm d\...
user87745
2
votes
0
answers
288
views
Active and passive transformations in coordinate-free relativity
Consider a coordinate-free metric tensor $g$ which can take different coordinate forms, say $g_{ij}(x)$ and $g_{i'j'}(x')$. Also consider a metric tensor $h$ which is related to $g$ by an active ...
- 514