Use spacetime diagrams to prove the following:(a) Two events that are simultaneous in one inertial frame are not necessarily simultaneous in another. More specifically, if frame F̅ moves with velocity v(vector) = βe(vector)_x as seen in frame F, where β > 0, then of two events that are simultaneous in F̅ the one farther “back” (with the more negative value of x̅) will occur in F before the one farther “forward.”

(b) Two events that occur at the same spatial location in one inertial frame do not necessarily occur at the same spatial location in another.

(c) If P₁ andP₂ are two events with a time like separation, then there exists an inertial reference frame in which they occur at the same spatial location, and in that frame the time lapse between them is equal to the square root of the negative of their invariant interval,

(d) If P₁and P₂ are two events with a spacelike separation, then there exists an inertial reference frame in which they are simultaneous, and in that frame the spatial distance between them is equal to the square root of their invariant interval

(e) If the inertial frame F̅ moves with speed β relative to the frame F, then a clock at rest in F̅ ticks more slowly as viewed from F than as viewed from F̅ —more slowly by a factor γ⁻¹ = (1 − β²)^1/2. This is called relativistic time dilation. As one consequence, the lifetimes of unstable particles moving with speed β are increased by the Lorentz factor γ .

(f) If the inertial frame F̅ moves with velocity v(vector) = βe(vector)_x relative to the frame F, then an object at rest in F̅ as studied in F appears shortened by a factor γ⁻¹ = (1 − β²)^1/2 along the x direction, but its length along the y and z directions is unchanged. This is called Lorentz contraction. As one consequence, heavy ions moving at high speeds in a particle accelerator appear to act like pancakes, squashed along their directions of motion.

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Δt = Δt = V-(Δg).