Why Is K-Calculus Confusing in Understanding Special Relativity?

[Tomer]

Hello everyone, and thanks for reading.

This will probably pretty long despite editing attempts, be forgiving :-)
I'm currently reading a book called "Introducing Einstein's relativity" by Ray d'Inverno. This is my hundredth attempt to try and fundamentally understand special relativity.
I have never heard of the "K-calculus" until I've read this book and was intrigued. However, I again fail to understand some really basic derivations in the book. I hope I will be able to portray my problems clearly enough.
Oh, by the way, I've done a bachelor in Maths and Physics (in Israel) 2 years ago, and planning on doing my master (in Germany) now. Just so you'd know I'm familiar with F=ma and so on :-)
However, I found myself encountering very basic questions. I'm either rusty (the questions might sound and even be very stupid), or I've developed a new (to me) form of critical thinking, but that's how it is.
The questions are sort of connected. That means, that by fully answering one I might be able to understand the rest. Might. :-)

Here goes -

1. When one draws a time-space diagram and world-lines in it, what is the significance of the coordinates on the different world lines and events? Are they the coordinates that I, the drawer, as an observer measure? Would it be correct saying, that every time I see such a diagram, I could draw my own world line simply as the y-axis?
I'm bothered, cause somehow, in the book, it's never mentioned who's doing the drawings. For example, the description sounds like "Observer B is moving with the velocity of v in respect to observer A", and the we see their world lines. Are "we" then observer A? Are the coordinates written on the diagram all in respect to observer A's clock and measuring equipment? What if there is no observer at rest? Are we a third observer?
--panic-- :-)

2. This is a hard one: Events. Events happen (ideally) at a point in space-time, (t,x).
What I don't really get, is how events are defined and what do they portray.

2.a When the event is "A is sending a light signal for a split of a second" I can "dig it". But if it's a moon, for example, that's just there (radiating constantly). Then A wants to "measure" the event, or more appropriately, to fit coordinates to it. He then sends a light signal to that moon, awaits it's reflection and writes the times down. (that's how it's been described in the book, and probably how it is in reality as well).
But then - what do get here? The coordinates of the moon? However, the moon has definitely moved 'till we got out signal back - so what do these coordinates describe? Where the moon was as the light signal arrived to him? What are we exactly measuring?
On the other hand - how can we measure an event happening in the split of a second? For example, if B smiles to me on the horizon for an instant, how am I to measure that? Are these then two prototypes of events?

2.b The book states that the distance coordinate is determined through the expression: 0.5(t2 - t1), where t1 and t2 are the times of the the sending and the acceptance times of the light signal respectively. That's clear (c being 1), if the space coordinate are meant to describe the location of the "event" when the light signal arrived to him. Same for the time coordinate (0.5(t1 + t2)). Is this then the goal of the coordinates? Why should we want to describe the location (time and spacewise) of objects by determining their positions when our signals hit them?

3. This is the hardest to explain without referring to the book, but I'll try.
This is in short a description in the book:
A and B are synchronized at the zero point, and B is moving with a velocity of v in respect to A. Accordingly there's a "k" factor defined, which I've pretty much understood, or thought I have. After a time interval of T A sends a signal to B, that is reflected back to him at event P. (P = reflection event)
B, in his clock, measures a time interval k*T until he gets the light. A then measures k*(k*T) as the time it took for the light to return (k^2 * T).
Then: t1 = T, t2 = k^2 * T.

What I don't get (I think that sums up my problem): how can we calculate coordinates measured by A while moving to B's clock and back. I mean, so B gets the light, in his clock, after k*T. Ok - what does it tell us about A?

Sigh... does this make any sense? Can someone maybe understand what's vague for me? It's not easy to explain.

I hope I didn't exhaust you to death, and I appreciate you reading!
That's it for now...

Tomer.

 

[robphy]

1. When one draws a time-space diagram and world-lines in it, what is the significance of the coordinates on the different world lines and events? Are they the coordinates that I, the drawer, as an observer measure? Would it be correct saying, that every time I see such a diagram, I could draw my own world line simply as the y-axis?
I'm bothered, cause somehow, in the book, it's never mentioned who's doing the drawings. For example, the description sounds like "Observer B is moving with the velocity of v in respect to observer A", and the we see their world lines. Are "we" then observer A? Are the coordinates written on the diagram all in respect to observer A's clock and measuring equipment? What if there is no observer at rest? Are we a third observer?
--panic-- :-)

Yes, we are a third observer... studying the motion of objects A and B.
(This situation is just like what you would do in plotting a position-vs-time graph in Galilean kinematics.)

2. This is a hard one: Events. Events happen (ideally) at a point in space-time, (t,x).
What I don't really get, is how events are defined and what do they portray.

2.a When the event is "A is sending a light signal for a split of a second" I can "dig it". But if it's a moon, for example, that's just there (radiating constantly). Then A wants to "measure" the event, or more appropriately, to fit coordinates to it. He then sends a light signal to that moon, awaits it's reflection and writes the times down. (that's how it's been described in the book, and probably how it is in reality as well).
But then - what do get here? The coordinates of the moon? However, the moon has definitely moved 'till we got out signal back - so what do these coordinates describe? Where the moon was as the light signal arrived to him? What are we exactly measuring?
On the other hand - how can we measure an event happening in the split of a second? For example, if B smiles to me on the horizon for an instant, how am I to measure that? Are these then two prototypes of events?

The coordinates of the event tell you (as you say)
"where the moon was as the light signal arrived [at his location, and when]"... but says nothing about how the moon is moving at that instant or where the moon was before or will be afterwards.
(Again, this situation is just like what you would do in plotting a position-vs-time graph in Galilean kinematics.)

2.b The book states that the distance coordinate is determined through the expression: 0.5(t2 - t1), where t1 and t2 are the times of the the sending and the acceptance times of the light signal respectively. That's clear (c being 1), if the space coordinate are meant to describe the location of the "event" when the light signal arrived to him. Same for the time coordinate (0.5(t1 + t2)). Is this then the goal of the coordinates? Why should we want to describe the location (time and spacewise) of objects by determining their positions when our signals hit them?

The radar method you describe is a practical way to measure distant events,
since it is usually difficult to extend a long ruler with a clock at the end of it to a distant event.

3. This is the hardest to explain without referring to the book, but I'll try.
This is in short a description in the book:
A and B are synchronized at the zero point, and B is moving with a velocity of v in respect to A. Accordingly there's a "k" factor defined, which I've pretty much understood, or thought I have. After a time interval of T A sends a signal to B, that is reflected back to him at event P. (P = reflection event)
B, in his clock, measures a time interval k*T until he gets the light. A then measures k*(k*T) as the time it took for the light to return (k^2 * T).
Then: t1 = T, t2 = k^2 * T.

What I don't get (I think that sums up my problem): how can we calculate coordinates measured by A while moving to B's clock and back. I mean, so B gets the light, in his clock, after k*T. Ok - what does it tell us about A?

The two numbers, t1 and t2, which are read off of A's clock,
provide the radar-coordinates of "when B gets the light" measured by A.

Since they met at a common origin (call it O), the product t1*t2 gives the square-interval between the events O and "when B gets the light". Another inertial observer through O (call her C) would generally measure another pair of times t1' and t2', but t1'*t2' would still equal the same square-interval between the events O and "when B gets the light".

Those radar coordinates can be expressed in rectangular form (as you noted above):
(delta t)=(t2+t1)/2 and (delta x)=(t2-t1)/2. Assuming A and B are inertial, (delta x)/(delta t) measures the relative velocity of B with respect to A (call it v).

If you express k in terms of v, you'll find that k is the doppler factor.
You can also find relations among gamma, v, and k... (which are secretly cosh, tanh, and exp functions). With a little more effort, you'll find the standard form of the Lorentz Transformations.

 

[ghwellsjr]

Prior to Einstein, a coordinate system included just three parameters for space (x, y, z) and you could call any "point" in that space a location (whether or not there was actually anything located at that location).

In Einstein's coordinate system, time has been added so the time/locations are not called points because that could exclude time. They could have been called "spacetime points" but that is rather cumbersome. So the term that is used is "event" which includes the four parameters defining a point in the spacetime coordinate system.

You should not restrict your idea of "event" to mean something happening at a particular place and time because it doesn't matter if any thing is actually located at that set of coordinates. Also, when you're talking about a scenario, like your example with the moon, it's not just one event, it's a whole series of continuous events and maybe even a set of arrays of events. A world-line on a space diagram is intended to show the series of events for a particle, ie, how it moves through space and time. The moon (or a spaceship) could be idealized as a particle, meaning its center of mass or some other point on it, if all you cared about was its trajectory through space as a function of time, but if you wanted to discuss the size of an object (like the front and back of a spaceship), you would need at least two sets of events.

"Event" is just the word in 4D spacetime that replaces "point" in a 3D coordinate system.

 

[Tomer]

Thanks for the answers.
I'll try to read it through again. In case things still don't add up I'll bug you again :-)

Tomer.