Question

For this problem, assume that any two frames have the same origin.

Consider two events with coordinates (t1, 1) and (t2, 2) in an inertial frame of reference S. The coordinates of these two events in an inertial frame S'moving to the right with speed u relative to frame Sare (t) and (a).
If we assume that the y and z coordinates are constant in the two frames, then 2x is the space interval, more commonly known as distance, between the events as measured in frame S', and 21 is the space interval in the frame S. These two intervals are more commonly written ∆x' and ∆x, respectively. Similarly, t₂-t is called the time interval between the events in frame S', and tot is called the time interval in frame S. These two intervals are more commonly written ∆t' and ∆t, respectively
quantity is a mixture of space and time called the space-time interval s. To avoid possibly having to take the square root of a negative number, we usually talk about the square of the space-time interval, which is what you actually showed to be conserved in this part:
s² = (c∆t)²
A pair of events are observed to have coordinates (0 s, 0 m) and (50.0 s, 9.00 x 10⁹ m) in a frame S. What is the proper time interval AT between the two events?

Answer 1

The proper time interval Δτ between two events is calculated by setting spatial components to zero in the space-time interval formula and solving for Δτ, using the Lorentz transformation if necessary.

Step-by-step explanation:

When considering two events in a frame S, with coordinates (0 s, 0 m) and (50.0 s, 9.00 × 10⁹ m), we can calculate the proper time interval Δτ. The proper time interval is the time measured between two events that occur at the same spatial location in a particular inertial frame. The space-time interval (Δs) is defined as Δs² = Δx² + Δy² + Δz² - (cΔt)², which is Lorentz invariant. To find the proper time interval, we set Δx, Δy, and Δz to zero, since events occur at the same location in the proper frame, and solve for Δτ.

In a given frame of reference where the two events occur at the same location, the space-time interval's spatial components are zero (Δx = Δy = Δz = 0). Therefore, we use the relation c²Δτ² = -Δs² to find Δτ. However, because in frame S the events are separated by some distance, the space-time interval is positive, and we cannot directly calculate the proper time interval as the square root of a negative number. Instead, we must use the Lorentz transformation to find the proper time.

The Lorentz transformation relates the time interval Δt in the stationary frame S to the proper time interval Δτ in the moving frame S'. Given that the speed of light c is a constant, we can derive the proper time interval for the events in the proper frame as Δτ = Δt / √(1 - v²/c²), where v is the relative velocity of the moving frame. However, in this problem, since we are not given the relative velocity, we cannot calculate a numerical value for Δτ.