Question

How can the crews of two spaceships, which are observing that their clocks are ticking differently (for instance with the help of a Pulsar) to apply Lorentz's transformation for time dilation to calculate the relative speed between them?

Answer 1

If the crews of two spaceships are observing that their clocks are ticking differently due to time dilation, they can indeed use Lorentz's transformation to calculate the relative speed between them. Lorentz's transformation is a mathematical formula that describes how time, space, and other physical quantities appear to change when observed from different inertial reference frames.

To calculate the relative speed between the two spaceships, the crews would follow these steps:

1. Measure the Time Difference: Each spaceship's crew would measure the time difference between their onboard clocks and establish how much time has elapsed according to their own reference frame. For example, if Spaceship A observes its clock ticking slower than Spaceship B's clock, they note the time difference.

2. Exchange Information: The crews need to exchange information about their respective measurements. They can use a communication method such as radio signals or laser transmissions to transmit the observed time differences between the two spaceships.

3. Apply Lorentz's Transformation: Using Lorentz's transformation equations, the crews can calculate the relative speed between the spaceships. The transformation equations account for time dilation effects caused by relative motion.

   The equation for time dilation (Δt') as observed from a moving reference frame is given by:

   Δt' = Δt / sqrt(1 - (v^2 / c^2))

   where:

   • Δt is the time difference measured by the stationary spaceship (as observed from its own reference frame).
   • v is the relative velocity between the spaceships.
   • c is the speed of light in a vacuum.

   By rearranging the equation, the relative velocity (v) can be calculated as:

   v = c * sqrt(1 - (Δt / Δt')^2)

   The crews can plug in the measured values of Δt and Δt' to compute the relative velocity (v) between their spaceships.

It's important to note that both spaceships must be in relative motion to experience time dilation effects. If both spaceships are at rest relative to each other, there will be no time dilation, and Lorentz's transformation will not be applicable for calculating their relative speed.