Notes for the Physics GRE - Harvard University Department of Physics

9 Special Relativity (6%)∆s 2 = −c 2 ∆t 2 + ∆x 2 + ∆y 2 + ∆z 2 = ∆s ′ 2(311)is a Lorentz-invariant distance. We may define γ = 11−(v/c) 2√moving at speed v. Time dilation:for a particleτ = γt. (312)Here τ is the proper time, which is the time measured by a clock in the particle’srest frame. Remember that “moving clocks appear to tick slowly.”Length contraction:L ′ = L γ . (313)Here, L is the proper length of the object, as measured in its rest frame. Rememberthat “moving rods appear shorter.” Note that length contraction occursonly along the direction of motion: perpendicular distances are unaltered.Lorentz transformation for motion at velocity v:t ′ = γ(t ± vx ), (314)c2 x ′ = γ(x ± vt). (315)Note that the ± is the same for both time and position. To figure out whichsign should be used, consider the Newtonian limit for position. This tells youthe sign for time.Einstein’s famous formula generalizes toE = γmc 2 = γE rest = √ p 2 c 2 + m 2 c 4 . (316)To lowest order in v, this gives the Newtonian limit,The four-vector for momentum:Here, p i = γmv i . The fource four-vector:E = mc 2 + 1 2 mv2 . (317)p ν = (E/c, p x , p y , p z ). (318)F ν = dp νdτ . (319)Simultaneity is relative to the observer. A and B are simultaneous events withrespect to observer O if light rays emitted from A and B will reach O at thesame time. If ∆x ′ is the distance between events in the simultaneous frame,then∆t = γ(v/c 2 )∆x ′ (320)39