Question

Demonstrate that equations (2 - 12) and (2 - 13) become the classical transformation equations (2 - 1) when v << c, except when applied to events very far away, in which case time is still not absolute.

Short answer

At low speeds, the Lorentz transformation equations will change into the well-known classical Galilean transformation equations, with the exception that time will start to take up a component that relies on location, making it no longer be an absolute. The influence must be considered even though we won't recognise it until we are very far away from reality.

Step-by-step solution

Write the given data from the question.

Consider that equations (2 - 12) and (2 - 13) become the classical transformation equations (2 - 1) when v << c.

Determine the formula of Lorentz transformation equations at low speeds.

Write the formula of Lorentz transformation equations at low speeds.

$\mathit{x}\mathbf{\text{'}}\mathbf{=}{\mathit{\gamma }}_{\mathbf{v}}\mathbf{(}\mathit{x}\mathbf{-}\mathit{v}\mathit{t}\mathbf{)}$ …… (1)

Here, ${\mathit{\gamma }}_{\mathbf{v}}$ is relativistic velocity transformation, t is time, v is speed, x is transformation position and c speed of light.

Determine the value of Lorentz transformation equations at low speeds.

The Galilean transformation, which is ultimately expressed in Newton's equations, should be the limit of the Lorentz transformation. Let's see what a Lorentz transformation will appear to be at different speeds.

Determine the Lorentz transformation equations at low speeds (v << c).

Substitute $\frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$for ${\gamma }_v$into equation (1).

$x\text{'}=\frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\left(x-vt\right)$

Since,

$v<<c\Rightarrow \sqrt{1-\frac{v^2}{c^2}}\approx 1$

Then will be:

$x\text{'}=x-vt$ …… (2)

In essence, you will do the same for the remaining equations: put the factor of equal to one, and then rewrite the equations as follows:

$t\text{'}=t-\frac{vx}{c^2}$ …… (3)

Then, Galilean transformation equations are:

$x=x\text{'}+vt\text{'}$ …… (4)

And time equations are:

$t=t\text{'}+\frac{vx\text{'}}{c^2}$ …… (5)

The Galilean transformation is perfectly represented by equations (3) and (5), but if you look at the time equations (Eqs 4 and 6), you'll see that it wasn't entirely recovered. We kind of adopted some of the concepts from special relativity, which holds that time is relative and dependent on one's position in space rather than being an absolute. At low velocities, this effect will, however, be insignificant and imperceptible due to the speed of light squared in the denominator of the second term. Large distances, or particularly big distances, require study of this issue since it becomes relevant in certain cases.