An event has spacetime coordinates (x,t) = (1200 m, 2.0 μs) in reference frame S. What are the event's spacetime coordinates (a) in reference frame S' that moves in the positive x-direction at 0.80c and (b) in reference frame S'' that moves in the negative x-direction at 0.80c?

Verified step by step guidance

1. Step 1: Understand the problem. The event's spacetime coordinates are given in reference frame S as (x, t) = (1200 m, 2.0 μs). We need to find the transformed spacetime coordinates in two different reference frames, S' and S'', which are moving relative to S at velocities of 0.80c in the positive and negative x-directions, respectively. This requires the use of the Lorentz transformation equations.
2. Step 2: Write down the Lorentz transformation equations. For a reference frame moving at velocity v relative to another frame, the transformations for the coordinates are: $x_{\text{'}}=\gamma (x-\mathrm{vt})$ and $t_{\text{'}}=\gamma (t-\mathrm{vx}/c^2)$, where γ is the Lorentz factor given by $\gamma =(1/\sqrt{1-{v/c}^2})$.
3. Step 3: Calculate the Lorentz factor γ for v = 0.80c. Substitute v = 0.80c into the formula for γ: $\gamma =(1/\sqrt{1-{0.80}^2})$. Simplify this expression to find γ.
4. Step 4: Apply the Lorentz transformation equations for S'. Substitute x = 1200 m, t = 2.0 μs (convert to seconds: t = 2.0 × 10⁻⁶ s), v = 0.80c, and γ (calculated in Step 3) into the equations for $x_{\text{'}}$ and $t_{\text{'}}$. Perform the substitutions and simplify the expressions to find the transformed coordinates in S'.
5. Step 5: Repeat the process for S''. For the reference frame S'' moving in the negative x-direction at v = -0.80c, use the same Lorentz transformation equations but substitute v = -0.80c instead of 0.80c. Follow the same steps as in Step 4 to find the transformed coordinates in S''.

Key Concepts

Spacetime Coordinates

Spacetime coordinates combine space and time into a single four-dimensional continuum, represented as (x, t). In this framework, an event is defined by its position in space (x) and its occurrence in time (t). Understanding spacetime is crucial for analyzing how different observers perceive the same event, especially in the context of special relativity.

Lorentz Transformation

The Lorentz transformation equations relate the spacetime coordinates of an event as observed in different inertial reference frames moving relative to each other at a constant velocity. These transformations account for the effects of time dilation and length contraction, ensuring that the speed of light remains constant for all observers. They are essential for converting coordinates between frames S, S', and S'' in the given problem.

Relative Velocity and Time Dilation

Relative velocity refers to the velocity of one object as observed from another moving object. In special relativity, time dilation occurs when an observer measures time intervals differently due to relative motion, with moving clocks ticking slower compared to stationary ones. This concept is vital for understanding how time and space are perceived differently in the frames S', and S'' as they move at significant fractions of the speed of light.