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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.
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.