Chapter 36: Q. 61 (page 1061)

Derive the Lorentz transformations for $t$ and $t'$ .
Hint: See the comment following Equation $36.22$ .

Short Answer

Expert verified

$t^{1} = γ (t - \frac{v}{c^{2}} x)$

Step by step solution

Given Information

We have to derive the Lorentz transformations for $t$ and $t'$ .

Simplify

To derive the Lorentz transformations we have the below equation:

$γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{1}{\sqrt{1 - β^{2}}}$

As we know that

$x = c t$ and

$x' = c t'$

Then,

$x' = γ (x - v y) x = γ (x' + v t) x x' = γ^{2} (x x' + x v t' - v t x' - v^{2} t t' c^{2} t t' = γ^{2} (c^{2} t t' + c t v t' - c t v t' - v^{2} t t') c^{2} = γ^{2} (c^{2} - v^{2}) γ^{2} = \frac{c^{2}}{c^{2} - v^{2}} γ^{2} = \frac{1}{1 - \frac{v^{2}}{c^{2}}} γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Next,

$x = γ (x' - (- v) t') x = γ (x' + v t') \frac{x}{γ} = x' + v t' \frac{x}{γ} - x' = v t' \frac{x}{γ v} - \frac{x'}{v} = t' t' = \frac{x}{γ v} - \frac{γ (x - v t)}{v} t' = γ (\frac{x}{γ^{2} v} - \frac{x}{v} + t) (E q . 1)$

To get a number for t, we need to simplify the indicated component of the equation and then return it to the same equation.

$\frac{x}{γ^{2} v} - \frac{x}{v} = x (\frac{1}{γ^{2 v}} - \frac{γ^{2}}{γ^{2} v}) \frac{1 - γ^{2}}{γ^{2} v} = \frac{\frac{c^{2} - v^{2}}{c^{2} - v^{2}} - \frac{c^{2}}{c^{2} - v^{2}}}{\frac{c^{2} v}{c^{2} - v^{2}}} \frac{- v^{2}}{c^{2} - v^{2}} \cdot \frac{c^{2} - v^{2}}{c^{2} v} = \frac{v}{c^{2}}$

To find the value for t, we'll now substitute values in equation (1).

$t' = γ (t - \frac{v}{c^{2}} x)$

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Derive the Lorentz transformations for $t$ and $t'$ .
Hint: See the comment following Equation $36.22$ .

Short Answer

Step by step solution

Given Information

Simplify

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Derive the Lorentz transformations for tand t'. Hint: See the comment following Equation 36.22.

Short Answer

Step by step solution

Given Information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Medical Physics

Measurements

Physics of Motion

Quantum Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Derive the Lorentz transformations for $t$ and $t'$ .
Hint: See the comment following Equation $36.22$ .