Light wave behavior and frame velocity in Appendix 1: Simple Derivation

[RobikShrestha]

http://www.bartleby.com/173/a1.html
In this site, x=ct and x' = ct' are equations for light wave.

But at equation 6, it says, v= bc/a where v = x/t (for x'=0). It argues that v = velocity of x'. But how can x/t be velocity of k' when it is assumed to be velocity of light?? Please explain.

Even at the origin of k', should not the speed of light as seen in frame k still be c?

 

[HallsofIvy]

Where does it say anything about x'/t' being "the velocity k'"? x'/t' is the velocity of light in frame K' (which has speed k' with respect to frame K), not the speed of K' itself.

 

[harrylin]

http://www.bartleby.com/173/a1.html
In this site, x=ct and x' = ct' are equations for light wave.

But at equation 6, it says, v= bc/a where v = x/t (for x'=0). It argues that v = velocity of x'. But how can x/t be velocity of k' when it is assumed to be velocity of light?? Please explain.

- The equation of a light wave to the right, emitted from x=0 at t=0: x=ct
- The equation of the origin of k' for O'=O at t=0: x=vt

It can be confusing to write "x" without specification of what it is the coordinate.
So, you could write:

x_L = ct
x_k' = vt
Edit: or better, x_O' = vt

Harald

 

[RobikShrestha]

I still don't get it. First x = ct and x' = c t'. But finally x = vt ? How can we use same symbol 'x' for two different things: 1. for light and 2. for origin of k' ? I mean in his derivation, he first takes x' = ax - bct then takes x' = 0, and gets x = bct/a or v= bc/a. Now is this x = bct/a same as x = ct ? If not why do they have the same symbol?

 

[harrylin]

I still don't get it. First x = ct and x' = c t'. But finally x = vt ? How can we use same symbol 'x' for two different things: 1. for light and 2. for origin of k' ? I mean in his derivation, he first takes x' = ax - bct then takes x' = 0, and gets x = bct/a or v= bc/a. Now is this x = bct/a same as x = ct ? If not why do they have the same symbol?

Again: x is the symbol for position along the X-axis of K.
x can be the position of anything, a wave front or an origin or your dog.

For the light ray, the position x of the wave front as function of the time is equal to ct.
Similarly, the x position of O' is equal to vt.

Obviously v is not equal to c, simply because O' is not equal to the wave front (or your dog).
Therefore if you want to avoid confusion, you should distinguish them as I showed earlier:
instead of writing x and x, I suggest to write them as x_L and x_O'.

And note that in eq.5, x' refers to the transformation equation between any x and x': it should be valid for anything (thus, no subscript to add there).

So, although the derivation is simple, the understanding of what the different terms stand for is a little less simple!

 

[RobikShrestha]

I think I get it now. x = ct is the wave equation in our frame. x' = ct' is the wave equation in his frame. He travels (wrt to me) at 'v' so given I measure his origin to be at 'x' and 't'
after he crosses me at x=0, t=0 i would measure him traveling at v=x/t.
Another question though! I am new at relativity. It's not taught in my college. But I want to learn it. What are the topics I should learn? Is the site containing Appendix 1 enough or is it too vast for a newbie like me?

 

[harrylin]

I think I get it now. x = ct is the wave equation in our frame. x' = ct' is the wave equation in his frame. He travels (wrt to me) at 'v' so given I measure his origin to be at 'x' and 't'
after he crosses me at x=0, t=0 i would measure him traveling at v=x/t.
Another question though! I am new at relativity. It's not taught in my college. But I want to learn it. What are the topics I should learn? Is the site containing Appendix 1 enough or is it too vast for a newbie like me?

It's not bad as a start. But I would advice at first to only study special relativity.
You may carefully study the first part of it, as well as an introduction textbook on special relativity. There are a few online, but now I don't have a link ready to one of them, sorry.*

Also Einstein's 1905 paper can be useful: you could read sections 1 and 2, skip section 3 (replaced by the simple derivation), and continue with sections 4 and 5.
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

* With Google I now found http://science.howstuffworks.com/science-vs-myth/everyday-myths/relativity.htm which may be helpful.

 

[Neandethal00]

http://www.bartleby.com/173/a1.html
In this site, x=ct and x' = ct' are equations for light wave.

But at equation 6, it says, v= bc/a where v = x/t (for x'=0). It argues that v = velocity of x'. But how can x/t be velocity of k' when it is assumed to be velocity of light?? Please explain.

Even at the origin of k', should not the speed of light as seen in frame k still be c?

Anyone never raised his eyebrows about equation (3)?
Mathematically or physically it doesn't make any sense.
He is multiplying zero with a constant.
0 = constant * 0 ?

 

[harrylin]

Anyone never raised his eyebrows about equation (3)?
Mathematically or physically it doesn't make any sense.
He is multiplying zero with a constant.
0 = constant * 0 ?

That is correct of course. If you read the text, you will see:

"Obviously [relations 1 and 2] will be the case when the relation [3] is fulfilled in general".

What he forgot to mention is that x=ct is valid for a light ray emitted at x,t=0,0. It's not valid in general.
He is looking for a relationship that is valid for the position of the wavefront of light rays that are emitted at any time and from any x. For those we have x-ct=constant (but not generally zero).
For example, for a ray from x=5 at t=0 we get x-ct=5.
And of course, also x'-ct'=constant, but it may be a different constant. The ratio between those two constants is also a constant - the constant you cited.

 

[Neandethal00]

That is correct of course. For example, for a ray from x=5 at t=0 we get x-ct=5.

Ok, this makes sense. x is the distance from origin and ct is the distance from light-event.

Is there any reason they assumed observations by two frames would differ by a pure constant, independent of any variables, such as x or t? Doesn't this constant later turn out to be a function of velocity (x and t)?

Sorry, but derivation of this transform always brings in many 'hhhhmmm' questions to my mind.

 

[harrylin]

Ok, this makes sense. x is the distance from origin and ct is the distance from light-event.

Is there any reason they assumed observations by two frames would differ by a pure constant, independent of any variables, such as x or t? Doesn't this constant later turn out to be a function of velocity (x and t)?

Sorry, but derivation of this transform always brings in many 'hhhhmmm' questions to my mind.

At the bottom of that page you see that that constant is a function of velocity. The assumption was that it is constant for a constant relative velocity.

Here we have the combination of the light principle (light propagates with speed c) with the principle of relativity (the same law of light propagation for x' and t' as for x and t). The assumption of special relativity was that for any inertial frame, the speed of light (measured distance/duration) is constant and everywhere equal to c.

The only difference between such standard inertial coordinate systems can be that the length and time measurements differ, so that the constants differ; both must have for the wave front of any light ray (thus from any point at any time), in the direction of increasing coordinates:

[start coordinate] - [clock time] x [c] = constant.

That is required for a consistent measurement system, if indeed the light principle is correct.

Harald