Lorentz transformations, time dilation, length contraction, spacetime intervals, energy-momentum, equivalence principle and curved spacetime.
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- What is a Lorentz transformation?
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A coordinate transformation that relates space and time measurements between observers in different inertial reference frames moving at constant relative velocity.- Write the Lorentz transformation formula for time t'.
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t' = gamma(t - vx/c^2), where gamma = 1/sqrt(1 - v^2/c^2)- Write the Lorentz transformation formula for position x'.
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x' = gamma(x - vt), where gamma = 1/sqrt(1 - v^2/c^2)- Define the Lorentz factor.
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gamma = 1/sqrt(1 - v^2/c^2), the dimensionless factor that appears in all relativistic formulas and Lorentz transformations.- State the relativistic velocity addition formula.
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u' = (u - v) / (1 - uv/c^2), where u is velocity in the original frame and v is the relative frame velocity.- How do the Lorentz transformations of transverse coordinates differ from parallel ones?
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Transverse coordinates y and z are unchanged (y' = y, z' = z), only coordinates parallel to motion are transformed.- Write the inverse Lorentz transformation for time t.
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t = gamma(t' + vx'/c^2)- What is time dilation?
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Moving clocks run slower relative to stationary clocks; a time interval measured in a moving frame appears longer when observed from the stationary frame.- Write the time dilation formula.
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Delta t = gamma * Delta t0, where Delta t0 is the proper time in the moving frame.- Define proper time.
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The time interval measured by a clock at rest relative to an event, the time in an object's own reference frame.- Explain the muon decay experiment and time dilation.
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Cosmic muons decay too quickly to reach Earth in lab time, yet they do because time dilation stretches their lifetime in the lab frame due to their relativistic speed.- What happens to time dilation as velocity approaches the speed of light?
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The time dilation factor gamma approaches infinity, so moving clocks run arbitrarily slowly in the stationary frame.- What is length contraction?
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Objects in motion are contracted in their direction of motion relative to a stationary observer; the moving length is shorter than the rest length.- Write the length contraction formula.
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L = L0 * sqrt(1 - v^2/c^2) = L0 / gamma, where L0 is the proper length.- Define proper length.
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The length of an object measured in its own rest frame, the maximum length the object can have.
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