Two firecrackers explode simultaneously (same t) but at different locations x₁ and x₂ in frame S. What does the Lorentz transformation predict about these events in frame S′, moving at velocity v relative to S?
AThey are simultaneous in S′ as well, since simultaneity is a physical fact independent of reference frame
BThey are generally not simultaneous in S′, because t′ = γ(t − vx/c²) depends on x as well as t
CThey are not simultaneous in S′ only if v > 0.5c
DThey are not simultaneous in S′ only if the events are causally connected
The key non-Galilean feature is that t′ depends on both t and x. For two events with equal t but different x, the t′ values differ: t′₁ = γ(t − vx₁/c²) ≠ γ(t − vx₂/c²) = t′₂ unless x₁ = x₂. This is relativity of simultaneity — a direct physical consequence of the x-term in the time transformation. In the Galilean transformation, t′ = t for all events regardless of position, so simultaneity is absolute. The Lorentz transformation replaces this with frame-dependent simultaneity at all nonzero velocities.
Question 2 Multiple Choice
A student applies t′ = γt to compute the time interval between two events at locations x₁ = 0 and x₂ = 100 m in frame S. What error has the student made?
ANo error — t′ = γt is the correct Lorentz time transformation
BThe student dropped the vx/c² term: the full transformation is t′ = γ(t − vx/c²), and setting x = 0 is only valid when both events occur at the same location in S
CThe student should have used the inverse transformation t = γ(t′ + vx′/c²) instead
DThe error is using γ rather than 1/γ for time dilation
The simplified formula t′ = γt is valid only when x = 0 — that is, when both events occur at the same spatial location in frame S (like two ticks of a clock at rest in S). For events at different locations, the full form t′ = γ(t − vx/c²) must be used, and the x-dependent term is the relativity-of-simultaneity correction. Applying the simplified formula to spatially separated events conflates time dilation (a clock-rate effect) with the full Lorentz transformation and gives a wrong answer. This is one of the most common errors in special relativity calculations.
Question 3 True / False
The Lorentz transformation predicts that the spacetime interval s² = c²t² − x² takes the same numerical value in all inertial frames.
TTrue
FFalse
Answer: True
The invariant interval is the spacetime analog of Euclidean distance: just as r² = x² + y² is unchanged by spatial rotations, s² = c²t² − x² is unchanged by Lorentz transformations. This is verifiable by direct substitution of the transformation equations. The invariance means that while individual coordinates (t, x) are frame-dependent, their combination s² is an objective, frame-independent property of any pair of events. The minus sign (unlike the plus sign in Euclidean distance) is what gives spacetime its hyperbolic geometry and prevents motion faster than light.
Question 4 True / False
In the Lorentz transformation, the time coordinate in one frame depends primarily on the time coordinate in the other frame, not on spatial position.
TTrue
FFalse
Answer: False
This is precisely the common misconception to avoid. The Lorentz time transformation is t′ = γ(t − vx/c²), which explicitly includes the spatial position x. This mixing of space into the time coordinate — absent in the Galilean t′ = t — is the mathematical expression of relativity of simultaneity. Two events at the same time but different locations in S are NOT at the same time in S′. The vx/c² term is not a small correction; it is the central non-Newtonian content of special relativity.
Question 5 Short Answer
Why is the fact that t′ depends on x in the Lorentz transformation a genuine physical statement and not merely a mathematical coordinate convention?
Think about your answer, then reveal below.
Model answer: In Newtonian mechanics, t′ = t: all observers agree on when events happen, regardless of where they are. The Lorentz transformation replaces this with t′ = γ(t − vx/c²): the time of an event in frame S′ depends on both its time and its location in frame S. Two events at the same t but different x have different t′ — they are not simultaneous in S′. This is not a convention about how we label events; it is a physical disagreement between inertial observers about which events happened at the same time. Experiments confirm this: moving clocks lose synchronization in ways that depend on their spatial separation, not just their relative velocity. Relativity of simultaneity has measurable consequences in particle physics and GPS timing corrections.
The contrast with Galilean relativity is the key. The Galilean transformation also changes spatial coordinates between frames, but time is untouched — every observer agrees on the time order and simultaneity of events. The Lorentz transformation breaks this by coupling time to space, producing a fundamentally different causal structure. The invariance of the spacetime interval (rather than time alone or distance alone) is the new invariant that replaces Newtonian absolute time.