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Special Relativity
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In classical physics, velocities simply add. If an object moves with speed u in one reference frame, which is itself moving at v with respect to a second frame, the object moves at speed u+v in that second frame.

This is inconsistant with relativity because it predicts that if the speed of light is c in the first frame it will be v+c in the second.

We need to find an alternative formula for combining velocities. We can do this with the Lorentz transform.

Because the factor v/c will keep recurring we shall call that ratio β.

We are considering three frames; frame O, frame O' which moves at speed u with respect to frame O, and frame O" which moves at speed v with respect to frame O'.

We want to know the speed of O" with respect to frame O,U which would classically be u+v.

The transforms from O to O' and O' to O" can be written as matrix equations,

$\begin{pmatrix} x' \\ ct' \end{pmatrix} = \gamma \begin{pmatrix} 1 & - \beta \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad \begin{pmatrix} x'' \\ t'' \end{pmatrix} = \gamma' \begin{pmatrix} 1 & - \beta' \\ -\beta' & 1 \end{pmatrix} \begin{pmatrix} x' \\ ct' \end{pmatrix}$

where we are defining the β's and γ's as

$\begin{matrix} \beta = \frac{u}{c} & \gamma = \frac{1}{\sqrt{1-\beta^2 }} \\ \beta^\prime = \frac{v}{c} & \gamma^\prime = \frac{1}{\sqrt{1-{\beta^\prime}^2 }} \end{matrix}$

We can combine these to get the relationship between the O and O" coordinates simply by multiplying the matrices, giving

$\begin{pmatrix} x'' \\ ct'' \end{pmatrix} = \gamma \gamma^\prime \begin{pmatrix} 1+\beta \beta' & - (\beta + \beta') \\ - (\beta + \beta') & 1+\beta \beta' \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad (1)$

This should be the same as the Lorentz transform between the two frames,

$\begin{pmatrix} x'' \\ ct'' \end{pmatrix} = \gamma'' \begin{pmatrix} 1 & - \beta'' \\ -\beta'' & 1 \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad (2) \mbox{ where } \begin{matrix} \beta'' & = & \frac{U}{c} \\ \gamma'' & = & \frac{1}{\sqrt{1-{\beta''}^2 }} \end{matrix}$

These two sets of equations do look similar. We can make them look more similar still by taking a factor of 1+ββ' out of the matrix in (1) giving#

$\begin{pmatrix} x'' \\ ct'' \end{pmatrix} = \gamma \gamma' (1+\beta \beta') \begin{pmatrix} 1 & - \frac{\beta + \beta'}{1+\beta \beta'} \\ - \frac{\beta + \beta'}{1+\beta \beta'} & 1+ \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix}$

This will be identical with equation 2 if

$\beta''=\frac{\beta + \beta'}{1+\beta \beta'} \mbox{ (3a) and } \gamma'' = \gamma \gamma' (1+\beta \beta') \mbox{ (3b)}$

Since the two equations must give identical results, we know these conditions must be true.

Writing the β's in terms of the velocities equation 3a becomes

$\frac{U}{c}=\frac{\frac{u}{c} + \frac{v}{c}}{1+\frac{uv}{c^2}}$

which tells us U in terms of u and v.

A little algebra shows that this implies equation 3b is also true

Multiplying by c we can finally write.

$U = \frac{u+v}{1+\frac{uv}{c^2}}$

Notice that if u or v is much smaller than c the denominator is approximately 1, and the velocities approximately add but if either u or v is c then so is U, just as we expected.