An Introduction to Modern Astrophysics · Problem 4.11

The Interaction of Light and Matter — Carroll & Ostlie

Physics — Astrophysics · spacetime-interval

Problem

The spacetime interval, Δs\Delta s, between two events with coordinates (x1,y1,z1,t1)and(x2,y2,z2,t2)(x_1, y_1, z_1, t_1) \quad \text{and} \quad (x_2, y_2, z_2, t_2) is defined by (Δs)2≡(cΔt)2−(Δx)2−(Δy)2−(Δz)2.(\Delta s)^2 \equiv (c\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2. (a) Use the Lorentz transformation equations (4.16–4.19) to show that Δs\Delta s has the same value in all reference frames. The spacetime interval is said to be invariant under a Lorentz transformation. (b) If (Δs)2>0(\Delta s)^2 > 0, then the interval is timelike. Show that in this case, Δτ≡Δsc\Delta \tau \equiv \frac{\Delta s}{c} is the proper time between the two events. Assuming that t1<t2t_1 < t_2, could the first event possibly have caused the second event? (c) If (Δs)2=0(\Delta s)^2 = 0, then the interval is lightlike or null. Show that only light could have traveled between the two events. Could the first event possibly have caused the second event? (d) If (Δs)2<0(\Delta s)^2 < 0, then the interval is spacelike. What is the physical significance of −(Δs)2\sqrt{-(\Delta s)^2}? Could the first event possibly have caused the second event? The concept of a spacetime interval will play a key role in the discussion of general relativity in Chapter 17.

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