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How to prove $ g^{\mu\nu}\Lambda^{\rho}{}_{\mu}\Lambda^{\sigma}{}_{\nu}=g^{\rho\sigma} $ for the inverse metric?

Let me use condition $${\Lambda^{-1}}^{\rho}_{}{\nu}= \Lambda_{\nu}{}^{\rho}$$ We can raise $\nu$ index left side as well as right side, $${\Lambda^{-1}}^{\rho\nu}= \Lambda^{\nu\rho}$$ Now contract ...
Chauhan's user avatar
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1 vote

How to prove $ g^{\mu\nu}\Lambda^{\rho}{}_{\mu}\Lambda^{\sigma}{}_{\nu}=g^{\rho\sigma} $ for the inverse metric?

The $g_{ij}$ are the components of the metric tensor with respect to some local coordinate basis, which means they are scalars and can be imterchanged, i.e. $$g_{ij} g_{kl} = g_{kl} g_{ij}$$ This is ...
Octavius's user avatar
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3 votes
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In SR, why do we claim length contraction rather than faster than $c$ travel in the rocket frame?

The basic principle is that if you are holding a ruler and a stopwatch, and you measure the speed of some object with your ruler and your stopwatch at your location, then you will never measure a ...
John Rennie's user avatar
-1 votes

Inertial frames of reference

Question 1 - is the sheet of ice also therefore a good approximation to an inertial frame? The sheet of ice may or may not define an inertial reference frame - it depends on whether or not it is ...
gandalf61's user avatar
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Differential form of Lorentz equations

How do these differentials follow from the Lorentz transformations mathematically? And is there a chain rule involved? You simply take the total differentials of the Lorentz transform formula you ...
Dale's user avatar
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Length contraction in Michelson-Morley experiment

The transverse interferometer arm cannot logically experience length contraction. This arm is perpendicular to the direction of motion. The distance between mirrors cannot contract. When this arm is ...
AA ProperGander Morris's user avatar
1 vote

${}$Doppler shift

The form of the Doppler shift tells you a lot about your system. If the equations are symmetric in transmit (Tx) and receive (Rx), then all motion is relative and there is no preferred rest frame. If ...
JEB's user avatar
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3 votes

About Lorentz Invariant

It means $\phi(x^\mu) = \tilde{\phi}(\tilde{x}^\mu) = \tilde{\phi}((\Lambda^{-1})_\nu^\mu x^\nu)=\phi(x^\mu)\neq \tilde{\phi}(x^\mu)$
Wolphram jonny's user avatar
1 vote
Accepted

Lorentz Boosts and Hyperbolic Quaternions

I'd never heard of hyperbolic quaternions, and they seem very strange – not even an alternative algebra, which is worse than the octonions. The subspace $a+br$ is isomorphic to the split-complex ...
benrg's user avatar
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1 vote

I am basically having doubt about where Lorentz transformation is being applied?

Electric field is only one part of the electromagnetic field, that in SR is mathematically represented by a $2^{nd}$-order tensor, whose contravariant components are $F^{\mu\nu}$. The transformation ...
basics's user avatar
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1 vote

I am basically having doubt about where Lorentz transformation is being applied?

Now the solution uses the Lorentz transformation in 𝑦 and 𝑧 coordinate for 𝐸. Shouldn't it be only for 𝑥 but not 𝑦 and 𝑧 (they are perpendicular to 𝑥 direction, right?) You are thinking of ...
robphy's user avatar
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Invariance of Acceleration vs Invariance of Magnitude of Acceleration and help with proof

You are technically correct, but you are missing the forest for the trees. Yes, if we want to say what it is we want to say, in the rather rudimentary mathematical framework that we are using, then we ...
naturallyInconsistent's user avatar
0 votes

I am basically having doubt about where Lorentz transformation is being applied?

A charge moving in some direction will have its field lines contracted when they are at least partially perpendicular to the direction of motion! Think about an imaginary box or sphere that surrounds ...
mike1994's user avatar
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Can you tell who is moving through time?

By definition every observer is moving through time. In fact, in his own reference frame, am observer only moves through time, and not space. This time is called his Proper Time. If you are asking the ...
RC_23's user avatar
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0 votes

Can you tell who is moving through time?

Special relativity has two main postulates which are true for any relativistic problem In any reference frame, light travels at the velocity $c$. Every reference frame is equally valid and no ...
Ronny's user avatar
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2 votes

Can you tell who is moving through time?

However, can you know who is moving through time? In the sense that this phrase has meaning, every massive object is moving through time at $c$. Could you just turn on a flashlight while you are ...
Dale's user avatar
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0 votes

Weinberg QFT problem 2.1: transformation of quantum states

Note that $\Lambda$, $L(p)$ and $L(\Lambda(p)$ are all pure boosts in y-z plane so that any three vector along x-axis is unaffected by $W(\Lambda,p)=L^{-1}(\Lambda p)\Lambda L(p)$. Thus $W(\Lambda,p)$...
Damo's user avatar
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-1 votes

If velocity is relative, why does time dilate for some observers and not others?

If kinetic energy has been stored in Bob, then Bob's aging has decreased. How to decide if kinetic energy has been stored in Bob? Imagine a flywheel on which Bob is attached. If Bob's motion was such ...
stuffu's user avatar
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0 votes

Seemingly equivalent linear form of the Sagnac effect

However, while the round trip time $t_1'+t_2'=2\gamma=\gamma(t_1+t_2)$ is as expected from time dilation, the time differences $t_1'-t_2'\neq\gamma(t_1-t_2)=0$ where the opposite is true in the Sagnac ...
KDP's user avatar
  • 5,357
2 votes

If an observer was trapped in a closed box with no way to interact with the external surroundings how will he know if he is moving or at rest

This is the essence of Einstein's conception of special relativity: the laws of physics do not make any distinction between being at rest (relative to what?) and moving at constant velocity (again, ...
niels nielsen's user avatar
2 votes

If an observer was trapped in a closed box with no way to interact with the external surroundings how will he know if he is moving or at rest

how will he know if he is moving or at rest He can't. The two situations are indistinguishable. If the box is in uniform (constant velocity) motion, there is no way the person inside can sense how ...
joseph h's user avatar
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-4 votes

If an observer was trapped in a closed box with no way to interact with the external surroundings how will he know if he is moving or at rest

Your inner ear will tell you, it will pick up the motion, be it ever so slight, but one symptom that you may feel is nausea or get a sore head. I get a form of morning sickness from a genetic ...
user403769's user avatar
5 votes
Accepted

How is it that energy of matter yields gravity if the amount of energy in a system is frame dependent while the force caused by gravity is not?

The curvature of spacetime does not depend only on energy, but rather on a frame independent object called the stress energy tensor. This includes not only energy but also momentum and pressure. The ...
Eric Smith's user avatar
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3 votes

How is it that energy of matter yields gravity if the amount of energy in a system is frame dependent while the force caused by gravity is not?

I've been told that the gravitational field arises due to the energy density terms in the stress-energy tensor This is the misconception. Gravity is not sourced exclusively by the energy density ...
Níckolas Alves's user avatar
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Properties of inertial reference frames from different perspectives

a) the acceleration of the block with respect to someone inside the truck Lets label the angle between the slope and the horizontal as $\theta$. The acceleration of the block down the slope due to ...
KDP's user avatar
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1 vote

Seemingly equivalent linear form of the Sagnac effect

This is a derivative of the question regarding the Sagnac effect. Judging from the metric $ds^2=-dt^2+(rd\phi)^2$ for a constant $r$ for this question, it should be no different from that on a line ...
Dale's user avatar
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0 votes

Special relativity and accelerating twins

This is the usual twin paradox scenario, with A being the "stay at home" twin and B being the "traveling" twin, but viewed from the inertial reference frame co-moving with B on the ...
Eric Smith's user avatar
  • 9,256
-1 votes

Seemingly equivalent linear form of the Sagnac effect

As for $t_1-t_2$, the difference between the linear and circular cases is that in the linear case, you can arrange for two unprimed clocks that are a distance $2πr$ apart in the lab (primed) frame to ...
benrg's user avatar
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9 votes

Special relativity and accelerating twins

For any observer there is a property called the proper time that is equal to the time shown by the clock they are holding. So to find out how the elapsed times of the twins compare we just have to ...
John Rennie's user avatar

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