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20 Cards in this Set
- Front
- Back
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What is β?
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β = v/c
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What is γ?
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γ = 1/√(1 - β^2)
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What is the Lorentz Transformation Matrix?
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What are the Lorentz transformations of intervals?
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The same as the normal transformation just with intervals (of space and time) replacing the absolute values.
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Formula for velocity transformation
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where u is the object's velocity and v is that of the reference frame
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Derivation of velocity transformation
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Find dx' and dt' from the normal transformation and then divide and simplify to get dx'/dt' = u'
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Formula for Relativistic Doppler effect
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Deriving formula for Relativistic Doppler effect
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- Start with normal Doppler wavelength equation: λ' = (c - vs)T - Account for time dilation: T = γTs - Recall that f' = c/(λ') and 1/Ts = fs - From this obtain f' in terms of fs and β - Other quantities can be found from this |
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Fractional Wavelength Change, z,
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z = (λ' - λs)/λs = sqrt((1 + β)/(1 - β)) - 1
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Features of Space-Time 4-vector, R
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- R = (ct, r), ct is 'timelike', r is 'spacelike' - (ct)^2 - r^2 is the same in any frame (since Lorentz transformations are rotations) |
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Velocity 4-vector, U
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- U = dR/d𝛕 = γ(c, v) - - Since d𝛕 = dt/γ |
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What is 'proper time', 𝛕
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Time according to the particle's own frame
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Time dilation (formula and derivation)
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- t = γ𝛕, where 𝛕 is proper time - - To derive apply Lorentz to time interval and set space interval to zero |
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Length Contraction (formula and derivation)
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- L = Lo/γ - - To derive apply Lorentz to space interval and set time interval to zero |
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Momentum 4-vector, P
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P = (E/c, p)
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Time-like interval
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(R1 - R2)^2 > 0, i.e. the distance travelled by light in the time between two events is greater than the distance between them
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Space-like interval
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(R1 - R2)^2 < 0, i.e. the distance travelled by light in the time between two events is less than the distance between them
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The light cone
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- ct on y axis and x on the x axis - Event inside the light cone have a time-like interval with the origin and can be causally connected with it - Events outside the cone have space-like intervals wrt the origin and cannot be causally connected to it |
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Relativistic energy
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E = K + rest energy = γmc² |
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Relativistic momentum
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P = γmu
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