Special theory of relativity  (Page 7/9)

u’ = u - v

where “u” and “u'” are the speeds of a particle or object as measured in two frames of reference which themselves move with a speed “v” with respect to each other.

Lorentz transformation

Like Galilean transform, Lorentz transform provides relation for space and time between inertial systems for all possible range of relative velocity. Importantly, it satisfies the postulate of special theory of relativity that speed of light in vacuum is a constant. The derivation of Lorentz transform has elaborate historical perspectives and is also the subject of insight into the relativistic space and time concepts. For this reason, we shall keep the derivation of this separate to be dealt later. Here, we shall restrict our consideration to the final form of Lorentz transform only. Let two inertial reference systems are denoted by unprimed and primed variables and their spatial origins coincide at t = t' = 0. Then, space (x',y',z') and time (t') co-ordinates of a "single arbitrary event" in primed inertial reference is related to space (x,y,z) and time (t) of primed inertial reference as :

Lorentz transformation

$$x\prime =\gamma \left(x-vt\right)$$ $$y\prime =y$$ $$z\prime =z$$ $$t\prime =\gamma \left(t-\frac{vx}{c^2}\right)$$

where,

$$\gamma =\frac{1}{\sqrt{\left(1-{\beta }^2\right)}}=\frac{1}{\sqrt{\left(1-\frac{v^2}{c^2}\right)}}$$

The dimensionless γ is called Lorentz factor and dimensionless β is called speed factor. For small relative velocity, v, the terms $v^2/c^2\to 0$ , $v/c^2\to 0$ and $\gamma \to 1$ . In this case, the Lorentz transform is reduced to Galilean transform as expected. Further, we can write transformation in the direction from primed to unprimed reference as :

$$x=\gamma \left(x\prime +vt\prime \right)$$ $$t=\gamma \left(t\prime +\frac{vx\prime }{c^2}\right)$$

Note the change of the sign between terms on right hand side.

Transformation involving two events

If two events, separated by a distance, occur along x axis at two instants, then we can write Lorentz transformations of space and time differences using following notations :

$$\text{Δ}x=x_2-x_1;\text{Δ}t=t_2-t_1;\text{Δ}x\prime =x_2\prime -x_1\prime ;\text{Δ}t\prime =t_2\prime -t_1\prime$$

The subscripts 1 and 2 denote two events respectively. The transformations in the direction from unprimed to primed references are :

$$\text{Δ}x\prime =\gamma \left(\text{Δ}x-v\text{Δ}t\right)$$ $$\text{Δ}t\prime =\gamma \left(\text{Δ}t-\frac{v\text{Δ}x}{c^2}\right)$$

The transformations in the direction from primed to unprimed references are :

$$\text{Δ}x=\gamma \left(\text{Δ}x\prime +v\text{Δ}t\prime \right)$$ $$\text{Δ}t\prime =\gamma \left(\text{Δ}t\prime +\frac{v\text{Δ}x\prime }{c^2}\right)$$

Constancy of speed of light in inertial references

We can test Lorentz transform against the basic assumption that speed of light in vacuum is constant. Let a light pulse moves along x-axis. Then, consideration in unprimed reference gives speed of light as :

$$c=\frac{x}{t}$$

If Lorentz transform satisfies special theory of relativity for constancy of speed of light, then the propagation of light as seen from the primed reference should also yield the ratio x’/t’ equal to c i.e. speed of light in vacuum. Now,

$$\frac{x\prime }{t\prime }=\frac{\gamma \left(x-vt\right)}{\gamma \left(t-\frac{vx}{c^2}\right)}=\frac{\left(x-vt\right)}{\left(t-\frac{vx}{c^2}\right)}$$

Dividing numerator and denominator by “t” and substituting x/t by c, we have :

$$\Rightarrow \frac{x\prime }{t\prime }=\frac{\left(\frac{x}{t}-v\right)}{\left(1-\frac{vx}{t{c}^2}\right)}$$ $$\Rightarrow \frac{x\prime }{t\prime }=\frac{\left(c-v\right)}{\left(1-\frac{vc}{c^2}\right)}=\frac{\left(c-v\right)}{\left(1-\frac{v}{c}\right)}$$ $$\Rightarrow \frac{x\prime }{t\prime }=\frac{c\left(c-v\right)}{\left(c-v\right)}=c$$

Clearly, Lorentz transform meets the requirement of special theory of relativity in so far as to guarantee that speed of light in vacuum is indeed a constant.

Lorentz factor

Lorentz factor ,γ, is the multiplicative factor in the transformation equations for x-coordinate and time. It is a dimensionless number whose value depends on the relative speed “v”. Note that the relativistic transformation for x-coordinate is just Lorentz factor times the non-relativistic or Galilean transformation.