1. ## Relativity Discussion Thread

Ok, I'm starting this up because it seems that a lot of people (myself certainly included) want to know more about relativity and understand it better. This thread is absolutely NOT the place to debate relativity, it is only to discuss the theory as it currently stands and to ask questions to clarify any misconceptions or misunderstandings you may have. I think a lot of us learned new things in the "other threads," but we learned them in the wrong way and setting. So from now on anyone who has relativity questions, let's post them here and talk about them in a civilized and intellectual manner.

2. Oh please Thor! Not again!

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"When you are courting a nice girl an hour seems like a second. When you sit on a red-hot cinder a second seems like an hour. That's relativity."

Ok someone had too.

4. I'd be grateful if someone could explain Einstein's mathematics in §3 of his 1905 paper. He's deriving the Lorentz transformation equations, but goes about it in a very obtuse way. Then, in a footnote halfway through his explanation, he writes:
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply derived directly from the condition that [the speed of light should be the same in all inertial frames]
That, indeed, is how I derived them when I first studied relativity. But, still, I would like to know what Einstein is doing in this section.

On the Electrodynamics of Moving Bodies

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Originally Posted by Eroica
I'd be grateful if someone could explain Einstein's mathematics in §3 of his 1905 paper. He's deriving the Lorentz transformation equations, but goes about it in a very obtuse way. Then, in a footnote halfway through his explanation, he writes:
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply derived directly from the condition that [the speed of light should be the same in all inertial frames]
That, indeed, is how I derived them when I first studied relativity. But, still, I would like to know what Einstein is doing in this section.

On the Electrodynamics of Moving Bodies
"We can't solve problems by using the same kind of thinking we used when we created them."

Sorry again but man did he bank some good quotes for his own work!

6. Originally Posted by tuffel999
"We can't solve problems by using the same kind of thinking we used when we created them."
Einstein wasn't creating the problem then, though.
Sorry again but man did he bank some good quotes for his own work!
Is he the same guy who said, when all you have is a hammer, all your problems look like nails?

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Originally Posted by milli360
Originally Posted by tuffel999
"We can't solve problems by using the same kind of thinking we used when we created them."
Einstein wasn't creating the problem then, though.
Sorry again but man did he bank some good quotes for his own work!
Is he the same guy who said, when all you have is a hammer, all your problems look like nails?
I will look that up since I am not sure but I know he said:

"If A is a success in life, then A equals x plus y plus z. Work is x; y is play; and z is keeping your mouth shut."

and

"Two things are infinite: the universe and human stupidity; and I'm not sure about the the universe."

8. Originally Posted by Eroica
I'd be grateful if someone could explain Einstein's mathematics in §3 of his 1905 paper. He's deriving the Lorentz transformation equations, but goes about it in a very obtuse way. Then, in a footnote halfway through his explanation, he writes:
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply derived directly from the condition that [the speed of light should be the same in all inertial frames]
That, indeed, is how I derived them when I first studied relativity. But, still, I would like to know what Einstein is doing in this section.

On the Electrodynamics of Moving Bodies
That's kind of an open-ended question. How about this, you start going through §3, and as soon as you get to a step where you're not sure what he's doing (or why he's doing it), ask about it specifically.

9. Originally Posted by SeanF
Originally Posted by Eroica
I'd be grateful if someone could explain Einstein's mathematics in §3 of his 1905 paper. He's deriving the Lorentz transformation equations, but goes about it in a very obtuse way. Then, in a footnote halfway through his explanation, he writes:
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply derived directly from the condition that [the speed of light should be the same in all inertial frames]
That, indeed, is how I derived them when I first studied relativity. But, still, I would like to know what Einstein is doing in this section.

On the Electrodynamics of Moving Bodies
That's kind of an open-ended question. How about this, you start going through §3, and as soon as you get to a step where you're not sure what he's doing (or why he's doing it), ask about it specifically.
Eroica left part of the quote out.

Right before the part that Eroica paraphrased as "[the speed of light should be the same in all inertial frames]" should be the phrase " in virtue of those equations". I'll agree the rest of the footnote can be paraphrased that way, but it should be preceded by the "in virtue..." part.

That changes the meaning from what Eroica implied. Instead of proving that the Lorentz equations derive from the constancy of the speed of light, the footnote seems to be talking about assuming the Lorentz equations and deriving that the speed of light is constant.

10. Originally Posted by SeanF
you start going through §3, and as soon as you get to a step where you're not sure what he's doing (or why he's doing it), ask about it specifically.
OK. I'm referring to the bit of mathematics that follows these words (sorry I can't post the equations themselves):
Originally Posted by Einstein
or, by inserting the arguments of the function T and applying the principle of the constancy of the velocity of light in the stationary system:−
Thanks for the response. 8)

11. Originally Posted by milli360
Right before the part that Eroica paraphrased as "[the speed of light should be the same in all inertial frames]" should be the phrase " in virtue of those equations". I'll agree the rest of the footnote can be paraphrased that way, but it should be preceded by the "in virtue..." part.

That changes the meaning from what Eroica implied. Instead of proving that the Lorentz equations derive from the constancy of the speed of light, the footnote seems to be talking about assuming the Lorentz equations and deriving that the speed of light is constant. [emphasis added]
:-k Hmmm... I'm not so sure.
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply deduced directly from the condition that in virtue of those equations the relation x²+y²+z²=c²t² shall have as its consequence the second relation ξ²+η²+ζ²=c²t².
deduced, not assumed!

12. Originally Posted by Eroica
Hmmm... I'm not so sure.
You did leave something out.
Originally Posted by Einstein
The equations of the Lorentz transformation may be more simply deduced directly from the condition that in virtue of those equations the relation x²+y²+z²=c²t² shall have as its consequence the second relation ξ²+η²+ζ²=c²t².
deduced, not assumed!
Right. Let's break it down.

A=The equations of the Lorentz transformation
B=The condition

So, he's saying A may be deduced from B

What is B?

B=that in virtue of those equations the relation x²+y²+z²=c²t² shall have as its consequence the second relation ξ²+η²+ζ²=c²t²

What does that mean? "in wirtue of" means "On the grounds or basis of; by reason of" so if

C=those equations (clearly referring to the Lorentz equations)
D=the relation x²+y²+z²=c²t²
E=the second relation ξ²+η²+ζ²=c²t²

Then
B= C -> (D -> E)

Or since C=A,
(A -> (D -> E)) -> A

Physicists, what do they know about logic? What was meant in the original German, I'm not sure.

13. Originally Posted by milli360
What was meant in the original German, I'm not sure.
Are you sure the footnote is Einstein's, and not an editorial comment by the translator? It's not in the original German.
Zur Elektrodynamik bewegter Körper

14. Originally Posted by milli360
(A -> (D -> E)) -> A
You're not seriously suggesting that the above constitutes a valid deduction of A? :-?

15. Originally Posted by Eroica
Originally Posted by SeanF
you start going through §3, and as soon as you get to a step where you're not sure what he's doing (or why he's doing it), ask about it specifically.
OK. I'm referring to the bit of mathematics that follows these words (sorry I can't post the equations themselves):
Originally Posted by Einstein
or, by inserting the arguments of the function T and applying the principle of the constancy of the velocity of light in the stationary system:−
Okay, this section is kind of confusing. Let's start with this sentence:

Originally Posted by Einstein
If we place x'=x-vt, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time.
Okay, k is the moving system, with a velocity of v "in the direction of the increasing x of the other" system (K). A point in space that is stationary in k will be moving along the x-axis in K, right? In a duration of time t (as measured in K), that k-stationary point will move a distance of vt along K's x-axis. So, if we set x'=x-vt, then putting in the x and t coordinates of that moving point will give us a constant x'. Boy, I don't know if I'm explaining that very well. :-?

Anyhoo, next we have:

Originally Posted by Einstein
We first define τ as a function of x', y, z, and t. To do this we have to express in equations that τ is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1.
So, he says that given the coordinates of the event in K (and converting the variable x coordinate to a constant x'), we can calculate the time τ of the event in k. We don't know yet what the actual equation is to calculate τ from those coordinates, but there is one.

Originally Posted by Einstein
From the origin of system k let a ray be emitted at the time τ0 along the X-axis to x', and at the time τ1 be reflected thence to the origin of the co-ordinates, arriving there at the time τ2; we then must have ½(τ0+τ2)=τ1...
This comes from the definition of simultaneity earlier in the paper. Since these times are in the k system, where the emission/reception occurs at a stationary point, then the time out and the time back should be equal. Thus, the reflection time (τ1) should be exactly halfway between the emission time (τ0) and the reception time (τ2).

Originally Posted by Einstein
...or, by inserting the arguments of the function τ and applying the principle of the constancy of the velocity of light in the stationary system:-- ½[τ(0,0,0,t)+τ(0,0,0,t+x'/(c-v)+x'/(c+v))=τ(x',0,0,t+x'/(c-v))
This is just the same previous equation ½(τ0+τ2)=τ1, but with the variables τ0, τ1, and τ2 replaced with the "function(s) of x', y, z, and t" that we decided must exist at the beginning.

The coordinates for τ0 would be (0,0,0,t) {t is the time the signal was emitted as measured in K}
The coordinates for τ1 would be (x',0,0,t+x'/(c-v)) {x' is the converted x coordinate where the reflection occurs. x'/(c-v) is the duration of time the trip would take - adding it to t gives the time of reflection}
The coordinates for τ2 would be (0,0,0,t+x'/(c-v)+x'/(c+v))
{x'/(c-v) is the duration of the outbound trip and x'/(c+v) is the duration of the inbound trip - adding them both to t gives the time of reception}

Originally Posted by Eroica
Thanks for the response. 8)
You're welcome! I hope I'm helping!

16. Originally Posted by Eroica
Originally Posted by milli360
What was meant in the original German, I'm not sure.
Are you sure the footnote is Einstein's, and not an editorial comment by the translator? It's not in the original German.
Zur Elektrodynamik bewegter Körper
That version says, at the bottom "Numbered footnotes are as they appeared in the 1923 edition"

That is, it was taken from the English translation in the volume The Principle of Relativity, which was published in 1923.
Originally Posted by Eroica
Originally Posted by milli360
(A -> (D -> E)) -> A
You're not seriously suggesting that the above constitutes a valid deduction of A? :-?
Hence the reason I followed it with "Physicists, what do they know about logic? "

Deductions don't have to be logical.

17. Originally Posted by SeanF
I hope I'm helping!
So far, so good. I appreciate the help. (Actually, I already understood this bit. It's the next few steps that have me stumped!)

18. Originally Posted by Eroica
Originally Posted by SeanF
I hope I'm helping!
So far, so good. I appreciate the help. (Actually, I already understood this bit. It's the next few steps that have me stumped!)
I need a little more clarification. What did you mean, "That, indeed, is how I derived them when I first studied relativity. ", now that we've parsed it?

19. Originally Posted by milli360
What did you mean, "That, indeed, is how I derived them when I first studied relativity. ", now that we've parsed it?
Consider a thought experiment identical to the one Einstein describes in §3 of the 1905 paper, but in which the "mirror" which reflects observer k's light signal is at spatial co-ordinates (0,d,0) - that is, it's d units above the moving observer, at right angles to the direction in which he's moving.

From his perspective the light signal moves vertically up to the mirror, is reflected, and moves vertically back down to him - a round trip of 2d units. The time for the round trip by his clock we will call Δť. Since the speed of light is c:

cΔť = 2d ..... Equation 1

Now let's look at the same events from the point of view of the stationary K observer.

During the light signal's round trip, his clock registers a time of Δt. In this interval, the k observer moves a distance of vΔt. This means that the light signal does not move vertically up and down, but diagonally along the arms of a lambda-shape: Λ

By simple geometry, l, the length of one of these arms, is:

l = √[d²+(½uΔt)²]

Since the speed of light is the same for both observers (the condition that footnote is referring to):

cΔt = 2l

=> cΔt = 2√[d²+(½uΔt)²] ..... Equation 2

Solving Equation 1 for d, we get:

d = ½cΔť

Substituting this in Equation 2, we get:

cΔt = 2√[(½cΔť)² +(½uΔt)²]

Solving this for Δt, we get:

Δt = Δť/√[1-(u²/c²)]

This is the time-dilation factor. A similar thought experiment (which I won't rehearse), in which the light pulse is emitted horizontally in the same direction as k's motion, allows us to derive the Fitzgerald-Lorentz-contraction equation:

l = ľ√[1 - (u²/c²)]

It's fairly easy to derive the Lorentz transformation equations once you are equipped with these two relations.

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Originally Posted by Eroica
Originally Posted by milli360
What did you mean, "That, indeed, is how I derived them when I first studied relativity. ", now that we've parsed it?
Consider a thought experiment identical to the one Einstein describes in §3 of the 1905 paper, but in which the "mirror" which reflects observer k's light signal is at spatial co-ordinates (0,d,0) - that is, it's d units above the moving observer, at right angles to the direction in which he's moving.

From his perspective the light signal moves vertically up to the mirror, is reflected, and moves vertically back down to him - a round trip of 2d units. The time for the round trip by his clock we will call Δť. Since the speed of light is c:

cΔť = 2d ..... Equation 1

Now let's look at the same events from the point of view of the stationary K observer.

During the light signal's round trip, his clock registers a time of Δt. In this interval, the k observer moves a distance of vΔt. This means that the light signal does not move vertically up and down, but diagonally along the arms of a lambda-shape: Λ

By simple geometry, l, the length of one of these arms, is:

l = √[d²+(½uΔt)²]

Since the speed of light is the same for both observers (the condition that footnote is referring to):

cΔt = 2l

=> cΔt = 2√[d²+(½uΔt)²] ..... Equation 2

Solving Equation 1 for d, we get:

d = ½cΔť

Substituting this in Equation 2, we get:

cΔt = 2√[(½cΔť)² +(½uΔt)²]

Solving this for Δt, we get:

Δt = Δť/√[1-(u²/c²)]

This is the time-dilation factor. A similar thought experiment (which I won't rehearse), in which the light pulse is emitted horizontally in the same direction as k's motion, allows us to derive the Fitzgerald-Lorentz-contraction equation:

l = ľ√[1 - (u²/c²)]

It's fairly easy to derive the Lorentz transformation equations once you are equipped with these two relations.
=D>

A simple explanation that everyone can see! *The crowd goes wild*

21. I always liked that derivation. It's fairly simple and elegant. Well done!!

22. Originally Posted by Eroica
Originally Posted by milli360
What did you mean, "That, indeed, is how I derived them when I first studied relativity. ", now that we've parsed it?
I meant, which method are you referring to, the method of Einstein's paper, or the method of the footnote, or the method you thought that the footnote referred to.
]that is, it's d units above the moving observer, at right angles to the direction in which he's moving.
So, d is a measurement in the moving observer's frame.
By simple geometry, l, the length of one of these arms, is:

l = √[d²+(½uΔt)²]

Since the speed of light is the same for both observers (the condition that footnote is referring to):

cΔt = 2l
Did you use d to compute l, a measurement in the non-moving frame?

23. Originally Posted by milli360
I meant, which method are you referring to, the method of Einstein's paper, or the method of the footnote, or the method you thought that the footnote referred to.
Obviously the latter, since I don't understand the method Einstein uses.

Incidentally, you still haven't convinced me that there is any difference between the other two (the method of the footnote and the method I think the footnote is referring to). 8)

24. Originally Posted by milli360
Did you use d to compute l, a measurement in the non-moving frame?
Yes. d is at right angles to the moving observer's motion. Both observer's velocity in that direction is zero, so why should lengths in that direction be different in the two frames?

25. Originally Posted by Eroica
Incidentally, you still haven't convinced me that there is any difference between the other two (the method of the footnote and the method I think the footnote is referring to).
I'm still not exactly sure which method you think the footnote is referring to.

But it appears (to me ) to be that one assumes the constancy of the speed of light, and derives the Lorentz equations from that.
Originally Posted by Eroica
Originally Posted by milli360
Did you use d to compute l, a measurement in the non-moving frame?
Yes. d is at right angles to the moving observer's motion. Both observer's velocity in that direction is zero, so why should lengths in that direction be different in the two frames?
So, you're assuming the contraction is zero?

26. Originally Posted by milli360
So, you're assuming the contraction is zero?
Are you suggesting that the standard physics textbooks are mistaken? The derivation I gave is an accepted one. :-k

27. Originally Posted by milli360
I'm still not exactly sure which method you think the footnote is referring to.

But it appears (to me ) to be that one assumes the constancy of the speed of light, and derives the Lorentz equations from that.
That's my reading of the footnote, too.

28. Einstein has yet another method of deriving the Lorentz Transforms in an appendix to his 1920 Book, Relativity: The Special and General Theory.

Simple Derivation of the Lorentz Transformation

In this, he considers a light signal which is travelling along the x-axis only, so he avoids the problem you raise.

29. Originally Posted by Eroica
Originally Posted by milli360
I'm still not exactly sure which method you think the footnote is referring to.

But it appears (to me ) to be that one assumes the constancy of the speed of light, and derives the Lorentz equations from that.
That's my reading of the footnote, too.
No, I was asking if that was your reading. I disagree about that, for the reasons given above. It's possible--I'll have to look into it--that Einstein's paper is avoiding the problem that I brought up, and that's what makes it more complicated. He was a very careful guy.

PS: I looked at that link Simple Derivation of the Lorentz Transformation, and the first part (most of the page) is concerned with movement along the x-axis, with the light propagating along the x-axis as well. Isn't that different from your method, where the light goes off axis?

30. Originally Posted by milli360
I looked at that link Simple Derivation of the Lorentz Transformation, and the first part (most of the page) is concerned with movement along the x-axis, with the light propagating along the x-axis as well. Isn't that different from your method, where the light goes off axis?
Yes. That was why I brought it up - as an alternative (and still fairly easy) way to derive the transforms that obviates your objection.

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