Surely k-calculus kT is 'mixing units'

[pigeonrandle]

Hi,
This is my first post, so hello everyone!

I just have a quick question about the k-calculus, and am using the 2nd diagram on

http://www.geocities.com/ResearchTriangle/System/8956/Bondi/intro.htm

as a reference since it is the same one as in the book i am reading from.

In the diagram kT seems to be measured along B, but as time is measured along the y-axis, how on Earth can kT represent time - surely kT must be some sort of time/distance unit. Surely kT would be the vertical distance between O and 'the top of the kT bit of B' if you know what i mean? X0)

Thanks in advance,
James.

 

[robphy]

k is dimensionless

(It is the ratio of two proper-time intervals... on different inertial worldlines.
Eventually it can be written in terms of the relative velocity between the two worldlines.)

 

[pigeonrandle]

Thanks for your reply :0)

k is dimensionless

(It is the ratio of two proper-time intervals... on different inertial worldlines.
Eventually it can be written in terms of the relative velocity between the two worldlines.)

I see what you are saying, but it doesn't seem to make sense from the diagram...

if kT is measured along B (which is at an angle to the y axis) then surely kT is not just measured in time? And i suppose(!) as T is measured in time then k is not dimensionless?!

Please forgive my ignorance... and eventhough it might seem that i am just being awkward, i promise i am not. Thanks again,
James.

 

[robphy]

I think you are relying too much on the Newtonian notion of time.

The diagram shows axes labelled TIME and SPACE for the observer A.
These axes help A assign coordinates to events in spacetime... but you can ignore them for now.

The proper-time along a worldline is the time ticked off by a wristwatch carried by that worldline. T and kT are these elapsed times on A's and B's wristwatch's, respectively.

k is the ratio (elapsed wristwatch time of B)/(elapsed wristwatch time of A), assuming [for convenience] that A and B met at event O and, after a short time, A sent a light signal to B. (For example, the webpage says
"PROPOSITION - If A emits light for 1 second, B receives the light for k seconds. ") k is a pure number.At this stage, there is nothing particularly special about this construction.

Consider simple Euclidean geometry. Take two rulers that cross at point O, at some arbitrary angle.
Draw any line segment joining some point t_A on ruler A to a point t_B on ruler B (on the same side of the crossing point).
The length Ot_B is some number k times the length Ot_A. If I choose a different segment parallel to the first, drawn from a different point u_A to u_B... I'd get the same number k (by similar triangles).

 

[pigeonrandle]

I think you are relying too much on the Newtonian notion of time.

The diagram shows axes labelled TIME and SPACE for the observer A.
These axes help A assign coordinates to events in spacetime... but you can ignore them for now.

The proper-time along a worldline is the time ticked off by a wristwatch carried by that worldline. T and kT are these elapsed times on A's and B's wristwatch's, respectively.

k is the ratio (elapsed wristwatch time of B)/(elapsed wristwatch time of A), assuming [for convenience] that A and B met at event O and, after a short time, A sent a light signal to B.

Thanks for sticking with me!

So the axis' don't really apply to B? (in the same way that they apply to A and to the light beam).

And so is there a 'better diagram' that i could use to understand the relationship?

It just seems weird to me that this is a 'foundation of understanding relativity' eventhough the diagram doesn't exactly mean what it shows, and yet the simple geometery is taken as fact!

 

[robphy]

Thanks for sticking with me!

So the axis' don't really apply to B? (in the same way that they apply to A and to the light beam).

They are there, but aren't very convenient for B.
An example from PHY 101... if you have an inclined plane, you can set up axis that are horizontal and vertical... or you can set axes down the incline and perpendicular to the incline. The physics will be the same regardless of which you choose... but depending on your problem, one choice of axes might be more convenient than the other.

And so is there a 'better diagram' that i could use to understand the relationship?

It just seems weird to me that this is a 'foundation of understanding relativity' eventhough the diagram doesn't exactly mean what it shows!

The diagram is fine... if you don't bring along too much of your Newtonian or Euclidean intuition. If it helps, IGNORE the axes.
On a spacetime diagram, the important features are the worldlines [and those of the light rays] and events.
Axes can be set up for convenience [or can be a nuisance].
(Similarly on a free body diagram, the forces and the objects are important. Axes can be set up for convenience [or can be a nuisance].)

 

[pervect]

For an alternate approach, consider switching to geometric units. See the wikipedia article

http://en.wikipedia.org/w/index.php?title=Geometrized_unit_system&oldid=143483770

for a complete reference, but all you really need to do in this particular case is to chose units so that c=1.

 

[robphy]

For an alternate approach, consider switching to geometric units. See the wikipedia article

http://en.wikipedia.org/w/index.php?title=Geometrized_unit_system&oldid=143483770

for a complete reference, but all you really need to do in this particular case is to chose units so that c=1.

Even in geometric units, k is still dimensionless.

 

[pervect]

Even in geometric units, k is still dimensionless.

Sorry, I didn't mean to imply otherwise.

Where units might come into play is computing velocity as a function of k.

Also, there are other general situations where the units don't match up (expressing the schwarzschild radius r = 2M is a typical example), this problem is usually an indication that the author has used geometric units.