Final answer:
The proper-time interval (Δτ) between two events is the time interval measured in the frame where events occur at the same location. It's found using the spacetime interval (Δs^2), which incorporates the differences in space and time coordinates. After calculation, one can determine the proper time, which is invariant across all inertial frames.
Step-by-step explanation:
To determine the proper-time interval between two events, one must calculate it in the reference frame where both events occur at the same location. In the given scenario, we would convert the space-time coordinates to the same units, typically meters and seconds, and then apply the definition of proper time (Δτ). The formula for the time dilation effect in special relativity is Δt = γΔτ, where Δt is the time interval measured by an observer moving relative to the event, and γ is the Lorentz factor, which equals 1/√(1 - v^2/c^2). However, since the question states that events occur at different locations, we need to calculate the spacetime interval (Δs^2) first and then extract the proper time from that.
Calculating Δs^2 = c^2Δt^2 - Δx^2 - Δy^2 - Δz^2, where Δx, Δy, and Δz are the spatial differences between events (in meters) and Δt is the time difference (in seconds). In this case, Δx would be 1.06 km (or 1060 meters), and Δt would be 9.76 μs - 1.79 μs (or 7.97 x 10^-6 seconds). After finding Δs^2 and confirming that the interval is timelike, meaning it is possible for a single observer to be present at both events, we can extract the proper time by taking the square root of the positive value of Δs^2/c^2 (since proper time is a real number).