Theory of Relativity
Answer all 60 questions to test your understanding. Your results will be shown at the end.
A tracking system records an event at t = 10 s and x = 8 light-seconds in frame S. Frame S′ moves at +0.60c relative to S, and their origins coincide at t = t′ = 0.
Which coordinates should the system assign to the event in S′?
Two synchronized laboratory sensors fire at t = 0, one at x = 0 and one at x = 10 light-seconds. Frame S′ moves at +0.60c relative to the laboratory; define every difference as the second sensor's event minus the first.
Which transformed differences and interpretations are correct? Select all that apply.
- In S′, Δt′ = −7.5 s, so the second sensor fires earlier.
- In S′, Δx′ = 8 light-seconds because the separation contracts.
- In S′, Δx′ = 12.5 light-seconds for these two firing events.
- In S′, Δt′ = 0 because the laboratory clocks were synchronized.
Four event pairs have laboratory separations (Δt, Δx) of (0 s, 4 light-seconds), (5 s, 3 light-seconds), (4 s, 0 light-seconds), and (3 s, 5 light-seconds). Frame S′ moves at +0.60c relative to the laboratory.
Match each laboratory separation to its Lorentz-transformed separation (Δt′, Δx′).
Two events are separated by Δt = 2 s and Δx = 5 light-seconds. An analyst proposes a frame moving at +0.40c in which the events have equal time coordinates.
The proposed simultaneity is physically possible because the event separation is ___.
A fixed laboratory detector records two flashes at its own location, separated by 5 s. Engineers ask whether a moving observer can assign equal time coordinates to those same events.
Assertion: An appropriate inertial frame can make the two detector events simultaneous. Reason: The events have a positive invariant interval c²Δt² − Δx².
A particle travels at 0.80c between two laboratory detectors whose synchronized clocks record a flight time of 10 μs. A clock moving with the particle records its own elapsed time.
What elapsed time does the moving clock record?
A spacecraft coasts at 0.60c between laboratory beacons. The synchronized beacon clocks report a 12 μs transit; use c = 300 m/μs.
Which statements correctly describe the transit? Select all that apply.
- The spacecraft clock advances 15 μs because moving clocks are time-dilated.
- The spacecraft clock advances 9.6 μs between the beacon encounters.
- The 12 μs reading is proper time because the beacon clocks are synchronized.
- The beacon separation in the laboratory is 2160 m.
Four clocks undergo inertial segments described in a laboratory frame: 10 s at 0.60c, 10 s at 0.80c, 5 s at rest, and 15 s at 0.80c. Each listed duration is laboratory coordinate time.
Match each segment to the time recorded by the clock following it.
An engineer subtracts the readings of two synchronized laboratory clocks at a spacecraft's departure and arrival. The spacecraft moves at 0.60c, so the two events occur at different laboratory positions.
Relative to the spacecraft's journey, the laboratory subtraction supplies ___, not the elapsed time on the spacecraft's single clock.
Four ideal clocks each travel inertially for exactly 10 s of laboratory coordinate time, at speeds 0.60c, 0, 0.96c, and 0.80c. All clocks start at zero.
Order the clocks from least to greatest accumulated proper time.
A spacecraft has a nose-to-tail length of 100 m in its own rest frame and passes a laboratory at 0.80c. The laboratory records both endpoint positions at the same laboratory time.
What length does this procedure measure?
A rod moving parallel to its axis at 0.60c measures 80 m long in the laboratory using simultaneous endpoint records. Its transverse width is 2 m in its rest frame.
Which conclusions follow? Select all that apply.
- The rod's proper length is 100 m.
- The laboratory endpoint-recording events are simultaneous in the rod frame.
- The laboratory measures the transverse width as 2 m.
- A rest-frame ruler attached to the rod measures a length of 100 m.
Four identical rods each have proper length 10 m and move along their lengths at speeds 0, 0.60c, 0.80c, and 0.96c. Laboratory instruments sample each rod's endpoints simultaneously.
Match each speed to the measured laboratory length.
A moving rod measures 12 m in the laboratory at 0.60c. A ruler attached to the rod measures 15 m, and an engineer argues that this larger reading must be a relativistic measurement error.
The attached ruler instead measures the rod's ___, which remains 15 m.
A 100 m rest-length spacecraft passes a laboratory at 0.80c. One system records simultaneous laboratory endpoint positions, while another takes an ordinary photograph.
Assertion: The simultaneous endpoint-position system measures a length of 60 m. Reason: Simultaneous arrival of light from both endpoints guarantees that the light was emitted simultaneously in the laboratory.
A spacecraft moves at +0.60c relative to Earth and launches a probe at +0.60c relative to the spacecraft. Both velocities lie along the same axis.
What probe velocity does Earth measure?
A spacecraft moves at +0.80c relative to Earth. It can launch one probe at +0.50c or another at −0.50c relative to itself.
Which Earth-frame launch velocities are correct? Select all that apply.
- The forward probe travels at +(13/14)c.
- The forward probe travels at +1.30c.
- The backward probe travels at +0.50c.
- The backward probe travels at +0.30c.
A spacecraft moves at +0.80c relative to Earth. It separately emits a probe backward at −0.80c, a probe forward at +0.80c, and a backward-directed light pulse, all specified in the spacecraft frame.
Which Earth-frame results are consistent with relativistic velocity addition? Select all that apply.
- The backward light pulse travels at −0.20c.
- The backward probe is stationary relative to Earth.
- The forward probe travels at +(40/41)c.
- The backward light pulse travels at −c.
A spacecraft moves at +0.60c relative to Earth and emits four objects or signals with spacecraft-frame velocities −c, −0.60c, 0, and +c. All motion is collinear.
Match each spacecraft-frame velocity to its Earth-frame velocity.
A simulation composes two collinear boosts of 0.60c and incorrectly obtains 1.20c by adding their velocities. A reviewer instead adds the parameters 0.6931 and 0.6931, then takes the hyperbolic tangent to obtain approximately 0.882c.
The reviewer is adding ___ rather than velocities.
On a spacetime diagram with x horizontal and ct vertical, event B lies 5 light-seconds to the right and 3 seconds later than event A. Signals are restricted to speeds at or below c.
Which interpretation is correct?
Event A is at the origin and event B is at t = 5 s, x = 3 light-seconds in an inertial frame. Consider all inertial frames moving along x.
Which statements about this pair are correct? Select all that apply.
- A massive object traveling at 0.60c can connect A to B.
- A frame moving at +0.60c places A and B at the same position.
- A suitable inertial frame makes A and B simultaneous.
- Every inertial frame places B later than A.
An engineer plots x horizontally and ct vertically using equal physical scales. A particle leaves the origin with constant velocity +0.60c.
Assertion: The particle's worldline has slope 5/3 and lies inside the future light cone. Reason: On these axes, the slope of a constant-velocity worldline is ct/x = c/v.
Four candidate events B are specified relative to event A by (Δt, Δx): (5 s, 3 light-seconds), (3 s, 5 light-seconds), (4 s, 4 light-seconds), and (−5 s, 3 light-seconds).
Match each separation to its causal classification relative to A.
Four signals leave x = 0 at t = 0 and travel toward a stationary detector at x = 3 light-seconds. Their straight worldlines correspond to speeds 0.60c, c, 0.50c, and 0.75c.
Order the detector-reception events from earliest to latest on the spacetime diagram.
Two particles, each with rest mass 1 GeV/c², approach each other at speeds 0.60c in their center-of-momentum frame. They merge into one composite object without emitting radiation.
What is the composite object's rest mass?
Use units c = 1, so mass, momentum, and energy have compatible numerical units. A particle of rest mass 1 moving at +0.60 collides with a stationary particle of rest mass 1, and they stick without radiation.
Which results for the composite are correct? Select all that apply.
- Its rest mass is exactly 2 because the particles stick.
- Its rest mass is √4.5, approximately 2.121.
- Its velocity is 0.375.
- Its velocity is 1/3.
Use units c = 1. A parent particle of rest mass 10 decays at rest into two identical particles, each of rest mass 3, with no other products.
What energy and momentum magnitude does each daughter have?
Use units c = 1 and positive momentum to the right. Consider four isolated initial systems: opposite photons of energies 2 and 2; a right-moving photon of energy 3 plus a left-moving photon of energy 1; a mass-3 particle moving right at 0.80; and a mass-5 particle at rest.
Match each system to its total energy and signed total momentum (E, p).
Use units c = 1. Two oppositely directed photons each have energy 3, giving total energy 6 and total momentum zero; an analyst claims that a system made only of massless particles must itself have zero mass.
The analyst has overlooked the system's ___, which equals 6.
A source emits a spectral line at 100 MHz in its rest frame while receding directly from a stationary receiver at 0.60c. Ignore gravity and propagation through matter.
What frequency does the receiver measure?
Two spacecraft emit the same known rest-frame spectral line along their line of motion toward an observer. One is observed at twice the rest frequency, and the other at half the rest frequency; gravitational and cosmological shifts are absent.
Which velocity inferences are correct? Select all that apply.
- The doubled frequency implies approach at 0.60c.
- The doubled frequency implies approach at 0.50c.
- The halved frequency implies recession at 0.50c.
- The halved frequency implies recession at 0.60c.
Four sources move directly toward or away from a receiver, with no gravitational shifts: recession at 0.60c, approach at 0.60c, recession at 0.80c, and approach at 0.80c. All quoted emission frequencies are measured in each source's rest frame.
Match each motion to the ratio of received frequency to rest-frame emission frequency.
A source moves at 0.60c in a laboratory, and the photons reaching a detector propagate perpendicular to the source velocity as measured in that laboratory. The detector measures 0.80 times the source's rest-frame emission frequency, despite the laboratory transverse geometry.
The frequency reduction that survives in this geometry is associated with ___.
Two separate laboratory measurements use sources moving at 0.60c: one source recedes directly, while the other is observed with photon propagation perpendicular to its velocity in the laboratory frame. Both sources emit the same rest-frame frequency f₀.
Assertion: The directly receding source is observed at 0.50f₀. Reason: The source observed in the laboratory-transverse geometry is observed at 0.80f₀.
A small rigid elevator accelerates upward through otherwise empty flat spacetime, with floor proper acceleration g and floor-to-ceiling proper height h. A floor source sends light upward; assume gh/c² ≪ 1 and compare frequencies measured by local floor and ceiling clocks.
What frequency does the ceiling receiver measure to first order?
An elevator is released into free fall near a planet, and a passenger releases a small ball initially at rest beside them. The cabin is small enough that tidal effects are negligible for a brief central experiment, but larger-scale measurements remain possible.
Which predictions and limitations are justified? Select all that apply.
- The ball initially remains beside the passenger rather than falling toward the floor.
- A sufficiently local light experiment agrees with an inertial laboratory to leading order.
- The passenger can conclude that spacetime curvature vanishes throughout the cabin.
- Measurements across a larger region may reveal relative accelerations caused by tides.
A rigid elevator accelerates upward in flat spacetime and has a floor clock, a ceiling clock, and a light beam launched horizontally from one wall. Use the steady accelerating-frame description with floor acceleration g and height h.
Which statements are consistent with this description? Select all that apply.
- Every fixed cabin point must have exactly the same proper acceleration to preserve rigidity.
- The ceiling clock advances faster than the floor clock by a factor approximately 1 + gh/c².
- The horizontal light beam bends toward the floor in cabin coordinates.
- The cabin's apparent gravity requires nonzero spacetime curvature.
A supported elevator near a spherical planet is compared with an accelerating elevator in flat spacetime. The cabins have matching accelerometer readings at their centers, but passengers can track freely floating objects throughout each finite cabin.
Assertion: Matching the central accelerometer readings guarantees that every experiment throughout the cabins gives identical results. Reason: Tidal relative accelerations can reveal curvature that is absent from the flat-spacetime accelerating cabin.
Passengers want to distinguish a finite free-falling cabin near a planet from a local inertial cabin in flat spacetime. They release a horizontal pair of freely floating beads and use reconstructed relative acceleration, rather than wall-relative acceleration, as their tidal diagnostic.
Arrange the analysis in dependency order, from gathering evidence to stating the limit of equivalence.
Use c = 1 and the metric dτ² = 4dt² − dx². During 0 ≤ t ≤ 5, clock A stays at x = 0, while clock B follows x = 1.2t; both clocks start at zero.
What times do A and B record over the specified coordinate-time interval?
Use c = 1 and the metric dτ² = (1 + x/10)²dt² − dx² in a region with x > −10. Clock A remains at x = 0 and clock B at x = 2, each from t = 0 to t = 10.
Which calculations or interpretations are correct? Select all that apply.
- Clock A records 10 units of proper time.
- Both clocks record 10 because neither changes spatial coordinate.
- Clock B records 12 units of proper time.
- Equal coordinate-time intervals need not produce equal local clock readings.
Use c = 1 and dτ² = dt² − dx². Between shared events (t, x) = (0, 0) and (10, 0), clock A stays at x = 0, while clock B moves at +0.60 for 5 coordinate-time units and −0.60 for 5; idealize the turnaround as instantaneous.
Which comparison follows from integrating along the two worldlines?
Use c = 1 and dτ² = 4dt² − dx². Four clocks move at constant coordinate velocities dx/dt = 0, 1.2, 1.6, and √3, each over a coordinate-time interval of 5.
Match each coordinate velocity to its accumulated proper time.
Use c = 1 and dτ² = 4dt² − dx². A proposed clock trajectory has dx/dt = 3, and a developer tries to repair the resulting negative dτ² by taking its absolute value.
The repair is invalid because the proposed trajectory is ___ and cannot be followed by an ordinary physical clock.
Outside a nonrotating spherical mass, a stationary clock obeys dτ = √(1 − r_s/r)dt, where t matches proper time for a stationary observer at infinity. At the clock's radius, r_s/r = 0.36, and 10 s of coordinate time elapse.
How much time does the stationary clock record?
In a static spacetime, stationary lower and upper clocks have lapse factors N_lower = 0.80 and N_upper = 1.00, with dτ = Ndt. The lower station emits an upward light signal of locally measured frequency 100 MHz.
Which statements are correct, assuming no relative station motion and no intervening absorption? Select all that apply.
- The upper station receives 125 MHz because its clock runs faster.
- The upper station receives 80 MHz.
- During 10 coordinate seconds, the lower clock records 8 s.
- Both stations measure the same frequency because photon energy is conserved locally.
Two stationary clocks are separated vertically by a small height h in a weak, approximately uniform gravitational field. The supplied parameter is gh/c² = 10⁻¹²; neglect rotation and terms quadratic in this parameter.
Which first-order results and assumptions are appropriate? Select all that apply.
- The upper-to-lower clock-rate ratio is approximately 1 + 10⁻¹².
- Light sent upward has received-to-emitted frequency ratio approximately 1 − 10⁻¹².
- The upper-to-lower clock-rate ratio is approximately 1 + 2 × 10⁻¹².
- Treating g as constant across h is part of this approximation.
A lower stationary transmitter and upper stationary receiver have lapse factors 0.90 and 1.00 in a static spacetime. A 100 MHz locally emitted signal is received at 90 MHz even though the stations have no relative motion.
The measured 10% frequency reduction is a ___, rather than a kinematic Doppler shift.
A clock is held stationary at r = 4r_s outside an ideal nonrotating spherical mass. Schwarzschild coordinate time t is normalized to the time kept at infinity.
Assertion: The clock records approximately 0.866 s per coordinate second. Reason: Along this stationary worldline, the metric gives dτ² = (1 − r_s/r)dt².
An ideal nonrotating, uncharged black hole has mass 10 solar masses. Use r_s = 2GM/c² = 2.95 km × (M/M_sun), and interpret r as Schwarzschild areal radius.
How should a location at r = 40 km be classified?
Three ideal Schwarzschild black holes have masses 2, 10, and 20 solar masses. Use r_s = 2.95 km × (M/M_sun), with every listed location expressed as areal radius.
Which radius calculations and location classifications are correct? Select all that apply.
- For 2 solar masses, r_s = 5.90 km and r = 4 km is inside.
- For 10 solar masses, r_s = 59.0 km and r = 40 km is inside.
- For 20 solar masses, r_s = 29.5 km and r = 40 km is outside.
- For 20 solar masses, r_s = 59.0 km and r = 40 km is inside.
Consider four ideal nonrotating, uncharged black-hole cases: 1 solar mass at r = 4 km, 2 solar masses at r = 4 km, 10 solar masses at r = 29.5 km, and 20 solar masses at r = 100 km. Use r_s = 2.95 km per solar mass and treat the supplied numbers as exact for classification.
Match each case to its horizon radius and location classification.
An ideal spherical body of fixed mass 2 solar masses contracts from physical radius 100 km to 4 km without losing mass. Using r_s = 2.95 km per solar mass, an engineer claims that the mass-based threshold radius must shrink along with the body.
The engineer is incorrect: the body's ___ remains 5.90 km throughout this fixed-mass contraction.
For an ideal one-solar-mass Schwarzschild black hole, take r_s = 2.95 km. A mission planner considers locations at areal radii 3.00 km and 2.00 km and assumes the horizon is a material surface.
Which corrections to the plan are justified? Select all that apply.
- The 3.00 km location is outside the event horizon.
- The 2.00 km location is inside, and no timelike observer can remain at constant r there.
- Crossing r = 2.95 km necessarily means striking a physical shell of matter.
- From r = 2.00 km, no future-directed light signal can escape to distant observers.
An experiment measures a moving unstable particle's mean laboratory lifetime at speed 0.60c as 12.4 ± 0.2 μs, where 0.2 μs is one standard uncertainty of the estimated mean. Its independently known rest lifetime is 10.0 μs, so the relativistic and nonrelativistic predictions are 12.5 μs and 10.0 μs; neglect prediction uncertainties.
What are the absolute standardized discrepancies from the relativistic and nonrelativistic predictions, respectively?
A particle-lifetime test reports 12.4 ± 0.2 μs at 0.60c, with one-standard uncertainty stated and negligible prediction uncertainties. The relativistic prediction is 12.5 μs and the nonrelativistic prediction is 10.0 μs; percentage discrepancies are to use each prediction as its denominator.
Which statements belong in an evidence-based conclusion? Select all that apply.
- The 0.1 μs mismatch indicates systematic bias because statistical uncertainty describes individual decays, not the estimated mean.
- The absolute percentage discrepancy from relativity is 0.8%.
- The absolute percentage discrepancy from the nonrelativistic prediction is 24%.
- Under the stated uncertainty model, the result strongly favors the relativistic prediction over this alternative.
A repeat of the same particle-lifetime test gives 12.4 ± 2.0 μs rather than 12.4 ± 0.2 μs. Treat 2.0 μs as one Gaussian standard uncertainty of the estimated mean, keep predictions at 12.5 μs and 10.0 μs, and neglect their uncertainties.
How should the interpretation change? Select all that apply.
- The nonrelativistic discrepancy is now only 1.2 standard uncertainties.
- The unchanged central value provides equally strong discrimination between the models.
- The result is compatible with both predictions at the approximate two-standard-uncertainty level.
- The nonrelativistic prediction remains excluded because the 24% discrepancy is unchanged.
The reported particle lifetime is 12.4 ± 0.2 μs, versus predictions of 12.5 μs and 10.0 μs. A reviewer notes that the uncertainty includes only counting statistics and that an untested electronics delay could add a constant offset to every inferred lifetime.
Which follow-up actions or cautions are justified? Select all that apply.
- An unaccounted +2.4 μs offset could reproduce the difference from the nonrelativistic prediction.
- Independent timing calibration and measurements across multiple speeds would test the offset explanation.
- The quoted uncertainty suffices for model comparison because a constant delay would shift both predictions equally.
- The quoted statistical discrepancy alone does not establish that all relevant systematic errors are controlled.
A colleague is drafting a 300-word conclusion for a lifetime experiment reporting 12.4 ± 0.2 μs at 0.60c, compared with relativistic and nonrelativistic predictions of 12.5 μs and 10.0 μs. Independent calibration supports the stated Gaussian uncertainty, prediction uncertainties are negligible, and the conclusion must distinguish strong evidence from universal proof.
Which central claim should anchor the conclusion?