Basic Lorentz transformation derivation

[Ibix]

We're talking about A's perception of the event, and not his reasoning as to what it would have been if he had not been where he was but somewhere else.

This, IMO, is the basic problem. You are talking about this but also saying you want to derive the Lorentz transforms, which have nothing to do with the perceptions of an observer. So you are talking about two different things as if they were the same thing.

You can, of course, derive transforms between any non-degenerate coordinate systems. But note that your basic transforms are non-linear (if you do them correctly, there's a $|x/c|$ in there), so it's probably rather messy.

 

[jeremyfiennes]

That doesn't make any sense. A only sees the photon when it enters his eye.

An observer doesn't see the photons. They are part of his 'seeing'. Photons from the event arrive at his retinas, send neural impulses to the visual cortex of his brain, and he subjectively experiences seeing the event. There is one event and two inertial observers views of it, (x,t) and (x',t').

 

[stevendaryl]

An observer doesn't see the photons. They are part of his 'seeing'. Photons from the event arrive at his retinas, send neural impulses to the visual cortex of his brain, and he subjectively experiences seeing the event. There is one event and two inertial observers views of it, (x,t) and (x',t').

Whatever. But what does it mean to say that A sees the event occurring at $(x, t_x + x/c)$? That is nonsensical. Or at least, I don't know what it means. It seems that it is mixing up two different things: The first coordinate, $x$ is where the event happened, and the second coordinate, $t_x + x/c$ is when A received the image of event. Why are you pairing those two things, and what does pairing them have to do with the derivation of the Lorentz transformations?

 

[Ibix]

Whatever. But what does it mean to say that A sees the event occurring at $(x, t_x + x/c)$? That is nonsensical.

And it should be $(x,t_x+|x/c|)$, just for extra fun.

 

[stevendaryl]

An observer doesn't see the photons. They are part of his 'seeing'. Photons from the event arrive at his retinas, send neural impulses to the visual cortex of his brain, and he subjectively experiences seeing the event. There is one event and two inertial observers views of it, (x,t) and (x',t').

I'm thinking that you don't understand that when people talk about the coordinates of event being $(x,t)$, they don't mean that $t$ is the time they saw the light from the event. $t$ is the time that the event took place.

The time that somebody sees an event is different for different observers, even if they are at rest relative to each other.

 

[stevendaryl]

I'm thinking that you don't understand that when people talk about the coordinates of event being $(x,t)$, they don't mean that $t$ is the time they saw the light from the event. $t$ is the time that the event took place.

The time that somebody sees an event is different for different observers, even when those observers are at rest relative to each other.

If I say that an event happened at longitude 21 degrees East, latitude 11 degrees North, at 5:21 pm Greenwich time, I don't mean that that's when I saw the light from the event. I can't see the light of something on the other side of the world.

 

[weirdoguy]

You are talking about this but also saying you want to derive the Lorentz transforms, which have nothing to do with the perceptions of an observer.

And that has been stated in this thread quite a few times @jeremyfiennes and it seems that you saw it, so I don't know why you are unwilling to accept this fact.

 

[jeremyfiennes]

So we at least agree that the Wikipedia x=vt+x'/γ is wrong, and that the γ should be
somewhere in the numerator?

 

[jeremyfiennes]

"t is the time that the event took place."

This implies an absolute time, which you have rightly observed is a meaningless idea. There is the time t on observer A's clock, and time t' on observer B's clock, and the Lorentz transform that gives the relationship between them: t' = ??(x,t).

 

[stevendaryl]

So we at least agree that the Wikipedia x=vt+x'/γ is wrong, and that the γ should be
somewhere in the numerator?

No. You have two equations that are both true:

1. $x = x'/\gamma + vt$, which implies $x' = \gamma (x - v t)$
2. $x' = x/\gamma - vt'$, which implies $x = \gamma (x' + v t')$

The equations with $\gamma$ in the denominator are just rearrangements of the equations with $\gamma$ in the numerator.

 

[Nugatory]

So we at least agree that the Wikipedia x=vt+x'/γ is wrong, and that the γ should be
somewhere in the numerator?

Are you talking about the $x=vt+x'/\gamma$ in the section "Time dilation and length contraction"? Wikipedia is correct and the $\gamma$ belongs where it is, in the denominator.

See posts #10, #12, #23, #24 above... And I strongly recommend that you try the exercise I suggested in #24. Of course using the Lorentz transform to derive the Lorentz transform is unacceptably circular, but that's not what we're asking you to do. The point is that once you look at the transformed coordinates it will become clear how they relate to one another and you'll see why the $\gamma$ belongs where it does.

[Edit: You might also find it helpful to draw a Minkowski diagram showing the event M and the worldline of the origins of the two coordinate systems. Pay particular attention to the distance between the origin of F' and M in the two frame - this is where the relativity of simultaneity is misleading you].

 

[stevendaryl]

This implies an absolute time,

$t$ is the time in a particular COORDINATE system. It's not the time on any particular clock. A coordinate system is not a single clock.

 

[stevendaryl]

There is the time t on observer A's clock, and time t' on observer B's clock, and the Lorentz transform that gives the relationship between them: t' = ??(x,t).

That is absolutely NOT what the Lorentz transformation tells you. The Lorentz transformations don't tell you how A's clock relates to B's clock. They tell you how A's COORDINATE SYSTEM relates to B's coordinate system. A coordinate system is a way of assigning (in the case of one spatial dimension) two numbers, $x$ and $t$ to every event. So here's one event:

• A's clock shows $t = 12$

The coordinates for this event in A's coordinate system are: $(x=0, t=12)$. The coordinates for this event in B's coordinate system is:
$x' = - \gamma v \cdot 12$ (that is, $x' = \gamma (x - v t)$ with $x=0$ and $t=12$) and $t' = \gamma \cdot 12$ ($t' = \gamma (t - \frac{vx}{c^2})$).

Suppose that $v = 0.866 c$ so that $\gamma = 2$. Then that means that B's coordinates for that event are $(x' = -20.784, t' = 24)$

This is NOT relating what's on B's clock to what's on A's clock. It is NOT saying that "When A's clock shows time 12, B's clock shows time 24". That would be contradictory, because we can also look at a different event:

• B's clock shows $t' = 12$

B's coordinates for this event are $(x'=0, t'=12)$. A's coordinates for this event are $(x=+20.784, t=24)$.

The Lorentz transformations transform between coordinates for an event in one frame and coordinates for an event in another frame. It is NOT a transformation between the time on one clock and the time on another clock.

 

[PeterDonis]

At this point this thread seems to be going nowhere, so it is now closed.

 

[Nugatory]

This thread is closed because it is clearly going nowhere.
If the answers already supplied are not enough for the original poster to work out for themself why $x=vt+x'/\gamma$ and not $x=vt+\gamma{x'}$ is the correct expression all the way back at post #8, we can have another thread devoted to only that question.