Question

Question: A friend says, "It makes no sense that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another-she decides to do it, and its done." By considering the tractable, if somewhat unrealistic, situation of Anna's thought being communicated to her hands by light signals, answer this objection.

Short answer

Answer

It is proved that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another.

Step-by-step solution

Lorentz transformation.

The Lorentz transformation defines the relationship between the two frames that move at the constant velocity and relative to each other.The frame which has a motion with a constant velocity is known as the inertial frame, and the frame that has a rotational motion with constant angular velocity in a curved path is known as the non-inertial frame.

Determine the formulas to prove that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another.

The expression to convert the time from the frame to frame is given as follows.

$\mathit{t}\mathbf{\text{'}}\mathbf{=}{\mathit{\gamma }}_{\mathbf{v}}\left(t-\frac{v}{c^2}x\right)$ (1)

Here,t^' is the time in the frame S^', is the time in the frame S,is the Lorentz factor for the velocity, C is the speed of the light.

Prove that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another.

Consider the figure which shows the position of two candles in Anna’s hand.

Calculate the time measured by the observer in the moving frame when the candle is at its position .

Substitute t₁ for t and x₁ for x into equation (1).

$t{\text{'}}_1={\gamma }_v\left(1-\frac{v{x}_1}{c^2}\right)$

Calculate the time measured by the observer in the moving frame when the candle is at its position .

Substitute$t{\text{'}}_1$ for t and x₂ for x into equation (1).

$t{\text{'}}_2={\gamma }_v\left(1-\frac{v{x}_2}{c^2}\right)$

Calculate the time difference between the two observations,

$\Delta t\text{'}=t{\text{'}}_1-t{\text{'}}_2$

Substitute${\gamma }_v\left(1-\frac{v{x}_1}{c^2}\right)$ for$t{\text{'}}_1$ and${\gamma }_v\left(1-\frac{v{x}_2}{c^2}\right)$ for$t{\text{'}}_2$ into the above equation.

$$\begin{gathered} \Delta t\text{'}={\gamma }_v\left(1-\frac{v{x}_1}{c^2}\right)-{\gamma }_v\left(1-\frac{v{x}_2}{c^2}\right) \\ \Delta t\text{'}={\gamma }_v\left[\left(1-\frac{v{x}_1}{c^2}\right)-\left(1-\frac{v{x}_2}{c^2}\right)\right] \\ \Delta t\text{'}={\gamma }_v\left[1-\frac{v{x}_1}{c^2}-1+\frac{v{x}_2}{c^2}\right] \\ \Delta t\text{'}={\gamma }_{v}v\left(\frac{x_2-x_1}{c^2}\right) \end{gathered}$$

Since the time difference$\Delta t\text{'}\ne 0$ , therefore it is proved that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another.