Velocity addition via k-calculus

In summary, Bondi's k-calculus approach can be used to derive the special relativity velocity addition rule.
  • #1
pervect
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We had a thread a while ago where a poster was particularly interested in the SR rule of velocity addition. And in that thread, I suggested a better foundation was the k-calculus approach, with a reference to Bondi's treatment in "Relativity and Common Sense".

Here I would like to show how to derive the special relativity velocity addition rule using some results from the k-calculus approach.

K calculus basically says that if a light signal is transmitted via some observer O and received by some observer A, there will be a doppler shift such that the received frequency at A is some multiple k of the transmitted frequency from O. This factor k is only dependent on the relative velocity v between two observers, not the distance between them.

Using the results from Bondi, one can use a simple radar setup to determine the relationship between k and v given the fact that the speed of light is constant for all observers.

This is not particularly hard, but somewhat lengthly, and would require diagrams to illustrate clearly. I will summarize the important results from Bondi as the following relations between k and v.

$$k = \sqrt{ \frac{1-\beta}{1+\beta } } $$

$$v = c \, \frac{1-k^2}{1+k^2} $$

Here c is the speed of light, and ##\beta = v/c##.

The part that I wish to show is how to compute the velocity addition formula from these results. Suppose we now have 3 observers, O, A, and B. And there is some velocity ##v_1## between the pair of observers (O,A), and some velocity ##v_2## between the pair of observers (A,B). We wish to find the velocity between the pair of observers (O,B), which we will denote as ##v_t##. We will also use the notation that ##\beta_1 = v_1/c## and ##\beta_2 = v_2/c##.

When we reformulate this in terms of doppler shift, we will note that there is some k-factor ##k_1## between observers (O,A) and some k-factor ##k_2## between observers (A,B). And we can conclude that the k factor between (O,B) is just the product of k_1 and k_2, namely.

$$k_t = k_1 * k_2$$

To see why this is true, take an example. Observer A emits light, whose frequency is multipled by some factor k_1, which we will take as 1/2, so that the frequency received by A is 1/2 the frequency emitted by O. Similarly, when B emits light, there will be some factor k_2, which we will take as 1/3, so that in the example observer B receives light at 1/3 the frequency emitted by A. We can then conclude that observer B in this example receives light at 1/6 the frequency as emitted by O, and more generally that the k-factor between O and A is just k_1 * k_2

The rest is algebra, which is perhaps slightly messy, but it's still just algebra. We write

$$k_t = \sqrt{ \frac{1-\beta_1} {1+\beta_1} \cdot \frac{1-\beta_2} {1+\beta_2} }$$
$$v_t = \frac{1-k_t{}^2}{1+k_t{}^2}$$

And when we simplify this we find

$$v_t = c \, \frac{\beta_1 + \beta_2} {1+\beta_1 \beta_2}$$

the expected velocity addition formula of special relativity.
 
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  • #2
You should post this as an Insight. In the forums it will just be lost among 200 threads misunderstanding the relativity of simultaneity.
 
  • #3
For what it's worth, there is a Wikipedia article Bondi k-calculus with lots of diagrams. (Disclosure: I wrote most of it.)

 
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Related to Velocity addition via k-calculus

1. What is velocity addition via k-calculus?

Velocity addition via k-calculus is a mathematical method used to combine velocities in special relativity. It takes into account the effects of time dilation and length contraction to accurately calculate the combined velocity of two objects moving at different velocities.

2. Why is velocity addition via k-calculus important?

Velocity addition via k-calculus is important because it allows us to accurately calculate the velocity of objects in special relativity, where the traditional method of adding velocities using the Galilean transformation does not apply. It is necessary for understanding the behavior of objects moving at high speeds.

3. How does velocity addition via k-calculus work?

Velocity addition via k-calculus uses the Lorentz transformation equations to calculate the velocity of an object in a different reference frame. It involves converting the velocities of the objects into a common frame of reference and then applying the k-calculus formula to calculate the combined velocity.

4. What is the k-factor in velocity addition via k-calculus?

The k-factor in velocity addition via k-calculus is a constant value that takes into account the effects of time dilation and length contraction. It is used in the k-calculus formula to accurately calculate the combined velocity of two objects in special relativity.

5. What are the limitations of velocity addition via k-calculus?

Velocity addition via k-calculus is limited to the special case of objects moving in the same direction. It does not take into account the effects of acceleration or objects moving in different directions. It is also only applicable in the context of special relativity and does not apply to objects moving at non-relativistic speeds.

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