All Questions
37
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Proof of the invariance of $c$ using the Lorentz group
Apologies if this question was already asked a few times but i could only find proofs of the invariance of $ ds^2 $.
Is there any way of proving the 2nd postulate (that $c$ is invariant in all ...
- 309
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1
answer
100
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Proof of Invariance of Spacetime Interval?
I was going through Spacetime Physics by Taylor and Wheeler and came to a point where they showed a proof of Invariance of Spacetime Interval. You can find the proof Here and Here is the second part ...
- 121
5
votes
1
answer
398
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Linearity of Lorentz Transformation proof
I was reading this article and got to the part where the homogeneity of space and time leads to the linearity of the transformations between inertial frames.
In particular, the function $x^\prime=X(x,...
- 93
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1
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54
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From infinitesimal interval invariance to finite interval invariance in SR
In Landau and Lifshitz's The Classical Theory of Fields, on page 5 about interval invariance between different frames, it reads
Thus, $$ds^2=ds'^2,\tag{2.6}$$ and from the equality of the ...
- 613
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1
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66
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Is the invariance of the 4-dim scalar product the fundamental law behind time dilatation and length contraction?
The Lorentz Group is defined as the group of all transformations that leaves the 4-dim. scalar product invariant. An implication of this definition is that the absolute value of the first matrix ...
- 189
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54
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Lorentz transformation in extra-dimension
The Lorentz transformation is usually defined In 4D spacetime, how can we derive the generalized transformation in extra-dimensional space-times like 5D or more dimensions?
- 67
1
vote
2
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113
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Trying understand the argument on Special Theory of relativity on Goldstein book
I am reading Goldstein and trying to understand the special theory of relativity. I am not sure how did he make this following argument.
The books explain that $ds^2$ is invariant in spacetime. He ...
- 19
2
votes
2
answers
137
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Lorentz transformation and special relativity: Why the equations must be linear? [duplicate]
I'm reading Einstein's paper on special relativity (On the electrodynamics of moving bodies 1905), it gives a derivation of the lorentz transformation. In the derivation, he compares the stationary ...
- 93
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2
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107
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Light-like interval and simultaneity [closed]
Why if the interval between two events is light-like then there is no frame of reference where the events occur at either the same time? If I assume that the 2 events happen at the same time, I arrive ...
3
votes
5
answers
185
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Is it meaningful to draw $ct$ and $x$ axes by two lines intersecting at right angles?
In the Cartesian coordinate system, the x-axis is really perpendicular to the y-axis, by construction. Also, under a rotation of the coordinate system, the transformed coordinate axes $x',y'$ remain ...
- 11.8k
2
votes
3
answers
142
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What does it physically mean for the inverse of the metric tensor of inertial frames to be the metric tensor itself?
The metric tensor of inertial frames in S.R is given by $$g_{\alpha \beta}=diag(1,-1,-1,-1)$$
It's inverse $$(g_{\alpha \beta})^{-1}=g_{\alpha \beta}$$
I was wondering what this means geometrically. I ...
- 223
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2
answers
276
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Invariance of spacetime interval by Schutz
I am reading the book on General Relativity by Bernard Schutz. In it he proves the invariance of the interval in special relativity using the following argument.
$S^2=0$ for all light-like paths. This ...
6
votes
1
answer
467
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The metric of world lines in Newtonian and Galilean spacetimes
Consider a flat Newtonian or a flat Galilean 2+1 spacetime. So, mainly a flat 2D Euclidean space, evolving over time, where each time-slice is connected with the next one by a world line. Like in this ...
- 431
4
votes
0
answers
143
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Linearity of Lorentz Transformations from Principle of Relativity
Many derivations of the Lorentz transformations assume they must be linear maps on $\mathbb R^4$, where we identify the components of $\mathbb R^4$ with orthogonal coordinate systems associated to ...
- 3,407
-1
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1
answer
87
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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{...
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