LEARNER / PAGE 1 OF 5
Two Clocks. One Reunion.
Can a traveler return home having aged less than the people who stayed? Make a prediction, compare a round-trip model, and explain why arriving messages answer a different question from the clocks.
Core: grades 9-12, about 50-60 minutes. Use a calculator and paper or the response text file. The printed cases work without internet; the online lab is optional. Introductory-college/algebra extensions are on page 5. Planning times are estimates, not classroom-trial results.
Original lesson, tables and diagrams: The Last Transmission. CC BY-NC 4.0: share and adapt for noncommercial purposes with credit, a license link and an indication of changes. Full terms and exclusions: page 11.
Read or download the complete lessonhttps://darksciencedocs.com/exploration-guides/time-dilation/
Task 1 / Predict before calculating
Earth and the ship start together with clocks set to zero. The ship travels at 80% of light speed to a marker 6 light-years away in Earth’s frame, turns around, and returns at the same speed. Predict which clock has accumulated more time at reunion. Explain what event makes the comparison unambiguous.
Diagram guide: vertical position is Earth-frame time; horizontal position is distance from Earth. Earth is the solid vertical path. The ship follows the dashed path. The drawing is a model, with the cruise and instantaneous-turnaround assumptions stated on page 2.
Respond on paper or in the learner response text file.
LEARNER / PAGE 2 OF 5
Calculate the reunion
Task 2 / Change one input; compare both clocks
Use L = 6 light-years for the one-way Earth-frame distance. Write beta for v/c, t_E for Earth elapsed time, and tau for crew elapsed time. In these units, c = 1 light-year per year. A light-year is a distance. A year here is a Julian year of 365.25 days.
Earth elapsed time in years equals twice the one-way distance in light-years divided by the fraction of light speed, when c is one light-year per year.
Crew elapsed time equals Earth elapsed time multiplied by the square root of one minus beta squared.
For the 80% case: t_E = 2 × 6 / 0.8. Calculate the square-root factor, then tau and the difference t_E - tau. Keep years in every time answer.
| Speed beta | Earth years t_E | Crew years tau | Difference, years |
|---|---|---|---|
| 0.50 | 24.0000 | Calculate | Calculate |
| 0.80 | 15.0000 | Calculate | Calculate |
| 0.99 | 12.1212 | Calculate | Calculate |
a. Rank the missions by crew aging. b. If distance stays fixed, how do Earth time and crew time change as speed increases? c. Would a longer-distance voyage at a lower speed necessarily age the crew less? State which inputs must be controlled.
Respond on paper or in the learner response text file.
Optional online check
Use the lab’s Classroom settings button near Worksheet & educator notes, or set Speed & spaceflight to 80% and 6 light-years. Select Compare at reunion. For messages, set the same values in Plan your flight, then Launch & watch Earth messages. The workbench and message mission are separate experiments; check both settings. The 3D flight deck is optional.
The model boundary
Both cruise legs have constant speed in an inertial Earth frame. The turnaround takes zero time only as an idealized limit. Gravity, propulsion, finite acceleration, shielding and energy budgets are omitted. Both travelers experience their own local clocks normally; the crew changes inertial frames at turnaround.
LEARNER / PAGE 3 OF 5
Messages are not clock readings
Task 3 / Distinguish transit from the wait between receipts
Use the 80% mission from page 2. Earth sends one numbered message every 0.5 Earth year (6 months), starting with #1 at 0.5 year. The message carries its send timestamp. Each signal travels at c in vacuum. Other missions may use a different send interval; read the cadence label before comparing.
| Message | Earth send s | Earth-frame reception t_r | Crew clock at reception |
|---|---|---|---|
| #1 | 0.5000 | 2.5000 | 1.5000 |
| #2 | 1.0000 | 5.0000 | 3.0000 |
| #3 | 1.5000 | 7.5000 | 4.5000 |
| #4 | 2.0000 | 7.7778 | 4.6667 |
| #29 | 14.5000 | 14.7222 | 8.8333 |
| #30 | 15.0000 | 15.0000 | 9.0000 |
Earth-frame reception time is the coordinate time of the receipt event away from Earth, assigned in Earth’s frame. It is not the older Earth timestamp carried in the arriving message. #3 arrives exactly at turnaround; #30 is sent and received at the reunion point.
a. Find the Earth-frame transit duration t_r - s for #1, #2, #3, #4, #29 and #30. Does it decrease for successive messages during the return leg?
b. Subtract consecutive crew receipt readings for #1→#2, #2→#3, #3→#4 and #29→#30. Convert the waits to months using 12 months per year. Why can the return-leg wait stay constant while individual transit durations decrease?
c. May you subtract an Earth send-clock reading from a crew receive-clock reading and call it transit time? Explain which time convention each of your two duration calculations uses. Rows #5-#28 are omitted here; do not treat #4 and #29 as consecutive messages.
Respond on paper or in the learner response text file.
LEARNER / PAGE 4 OF 5
Compare gravity; defend a claim
Task 4 / Two contributions to a GPS clock
In Gravity & clocks, choose GPS satellite. Clock A is on the ground; Clock B orbits Earth. Compare the signed numerical contributions. The actual hand drift is tiny; any magnified animation is explicitly illustrative. Do not infer microseconds by counting animation frames.
| Contribution to satellite B relative to ground A | Approximate offset per day |
|---|---|
| Weaker gravity at the satellite | +45 microseconds |
| Satellite’s orbital motion | -7 microseconds |
| Combined offset | Calculate |
a. Add the two contributions. Does B gain or lose time relative to A without compensation? b. Explain why “every higher clock runs slower” is not a valid rule. c. Name one additional effect omitted from this rounded classroom comparison.
Respond on paper or in the learner response text file.
Task 5 / Claim, evidence, reasoning and boundary
A classmate says: “The ship gets messages faster coming home, so the crew must have aged more than Earth.” Evaluate this claim using the reunion clocks and two receipt events. Include the relevant time units, one idealization and a measurement that would be needed to test a real journey.
Respond on paper or in the learner response text file.
Exit check
1. Does a photon carry a working clock or have a rest frame? 2. Does zero proper time along a lightlike path mean a signal reaches the ship instantly in Earth’s frame? 3. At reunion, which readings give each traveler’s accumulated aging?
Respond on paper or in the learner response text file.
Optional: compare rest and motion in the uncalibrated flight-deck sky.
OPTIONAL LEARNER EXTENSIONS / PAGE 5 OF 5
Design a mission; derive a receipt
Task 6 / Algebra and introductory-college extension
Design an ideal cruise mission that returns after 100 Earth years while accumulating 10 crew years. First rearrange tau = t_E sqrt(1 - beta^2) to make beta the subject. Show the algebra, substitute the target ratio, then use L = beta t_E / 2 in light-years.
Respond on paper or in the learner response text file.
The online mission controls have finite slider precision. Select the nearest supported speed and distance, recalculate the actual clocks, and judge the stated challenge tolerances. Mathematical target values do not make the acceleration or energy requirements achievable.
Optional derivation / Where does a message meet the ship?
For the 80% mission, let t be Earth-frame years and x be light-years from Earth. Use x_ship = 0.8t outbound, and x_ship = 12 - 0.8t on return. A signal sent at Earth time s follows x_light = t - s. Set the two positions equal to derive each reception formula. State the send-time boundary that selects the correct cruise leg.
Check your formulas with messages #1, #3 and #4. Then derive the constant crew wait for consecutive messages on each cruise leg. A local inertial-frame comparison is not one global rest frame for the entire returning ship.
Respond on paper or in the learner response text file.
Optional gravity calculation / Two static hovering clocks
Near clock B is held at r = 4 R_s; far clock A is held at r = 20 R_s around a nonrotating black hole. Both are static, not orbiting. R_s is the horizon radius. The comparison is normalized to A, not to a clock at infinity.
The rate of the near static clock relative to the far static clock equals the square root of one minus one quarter, divided by one minus one twentieth.
How many minutes elapse on B while A advances one hour? Compare the percentage effect with the GPS microsecond-scale effect. The lab’s black-hole Comparison interval is in days on the farther clock; normalize to one hour for this calculation. Strong-field hovering and circular GPS orbits use different models.
EDUCATOR / PLANNING AND REUNION ANSWERS
Facilitate the investigation
Levels and routes
Grades 9-12 / ages about 14-18: pages 1-4 form the core. Prerequisites are ratios, signed arithmetic, square roots and reading a table. Introductory college or advanced high school: add page 5’s formula rearrangement and piecewise intersection derivation. No calculus is required.
Grades 6-8 / ages about 11-14: teacher-led conceptual enrichment. Supply the calculated crew values, compare and rank the cases, and discuss receipt cadence without deriving the Lorentz factor. Do not report this as completion of a separate middle-school performance expectation. Homeschool and independent learners can use the same level choices.
Suggested 50-60 minute core
Prediction and definitions: 8 minutes. Reunion calculations: 12. Message events: 15. GPS comparison: 8. Claim, discussion and exit check: 12. Optional extensions add about 20-35 minutes. These are planning estimates; the lesson has not been classroom-timed.
Print pages 1-4 for the core and page 5 only when needed; retain pages 6-11 for teaching and reference. Use a calculator, paper or the plain-text response file. Shared-device, teacher-projected and fully offline routes are supported. Student answers are not collected by this resource.
Answers / Tasks 1 and 2
Reunion is one shared event, so the co-located clocks can be compared directly. The crew has accumulated less proper time. For beta = 0.8, the square-root factor is 0.6, gamma is 5/3, Earth time is 15 years and crew time is 9 years. The difference is 6 years. At turnaround the crew clock reads 4.5 years and Earth-frame time is 7.5 years.
| beta | Earth years | Crew years | Difference, years |
|---|---|---|---|
| 0.50 | 24.0000 | 20.7846 | 3.2154 |
| 0.80 | 15.0000 | 9.0000 | 6.0000 |
| 0.99 | 12.1212 | 1.7099 | 10.4113 |
At fixed distance both times decrease as speed increases, while the crew-to-Earth ratio also decreases. Speed alone cannot rank aging for journeys with different distances. The round trip is asymmetric because the ship changes inertial frames; acceleration is not a magical slowing substance. Access and response options are on page 11.
EDUCATOR / ANSWERS AND ASSESSMENT
Separate the two durations
Answers / Tasks 3-5
Earth-frame transit durations for #1, #2, #3, #4, #29 and #30 are 2.0000, 4.0000, 6.0000, 5.7778, 0.2222 and 0 years. The late zero is the reunion endpoint at zero separation. Do not generalize it to distant messages.
The crew waits #1→#2 and #2→#3 are 1.5 years = 18 months each. #3→#4 and #29→#30 each take about 1/6 year = 2 months. For uniform cruise speed and a regular emitter, receipt spacing is constant on a cruise leg. Individual return-leg photon paths still shorten, so transit durations decrease. All photons travel at c.
A crew wait subtracts two events on the same crew clock. Earth-frame transit subtracts Earth-coordinate send and receipt times. Subtracting an Earth send reading from a crew receive reading mixes clock conventions and is not either of these durations.
The GPS reference sum is +38 microseconds per day: the satellite gains time without compensation. Higher static clocks tick faster in this comparison; orbital motion contributes in the opposite direction. Exact orbit, Earth rotation, nonspherical gravity and hardware corrections need a more detailed model. NIST’s rounded example is independent of the generated spacecraft cases.
A strong Task 5 response rejects the classmate’s inference, uses the 15-year/9-year reunion and a receipt comparison, states an idealization and proposes a clock or signal observation. No photon rest frame exists. A null path’s zero proper time does not eliminate Earth-frame propagation time.
Suggested 12-point rubric
| Criterion | Evidence for 2 points |
|---|---|
| Model and reunion | Identifies shared reunion, cruise legs and a model limit. |
| Clock calculation | Finds Earth 15 y, crew 9 y; keeps units consistent. |
| Transit duration | Uses Earth-frame t_r - s; return values decrease. |
| Receipt cadence | Finds 18 then 2 crew months; does not change c. |
| GPS contributions | Combines signs to +38 microseconds/day. |
| Explanation and limits | Uses numerical evidence; separates clock aging from received light. |
This rubric records evidence within one activity, not standards mastery or external certification. Assess the optional algebra separately; do not award or claim that alignment when learners only substitute into a supplied formula.
EDUCATOR / STANDARDS MAP
Standards / scientific practice
Publisher-authored mapping for the grades 9-12 core and optional algebra extension. The lesson supports selected NGSS science practices and crosscutting concepts. It does not claim an entire three-dimensional NGSS performance expectation, external approval or college accreditation.
| Target | Task and evidence |
|---|---|
| NGSS SEP 2 / Developing and using models | Tasks 1, 2, 5: worldline prediction, parameter comparison and an explicit model boundary. |
| NGSS SEP 4 / Analyzing and interpreting data | Task 3: interprets generated send/receipt events and compares both durations. |
| NGSS SEP 5 / Mathematics and computational thinking | Tasks 2-4: clock factors, time differences, months and signed GPS contributions. |
| NGSS SEP 6 / Constructing explanations | Task 5: a numerical claim connected to a principle, assumption and needed observation. |
| NGSS CCC 3 / Scale, proportion and quantity | Tasks 2-4: light-years versus years, clock ratios and microseconds versus years. |
| NGSS CCC 4 / Systems and system models | Tasks 1, 5: defines Earth, ship, signals, inputs and omitted physics. |
Assess these components through the scientific work, not through memorizing the names of practices. A generated model result should be distinguished from an empirical measurement. The separate NIST clock reference provides a real-world comparison.
Official NGSS practices and crosscutting-concept appendiceshttps://www.nextgenscience.org/resources/ngss-appendices
EDUCATOR / MATHEMATICS AND LEVELS
Standards / mathematics and levels
These are selected mathematics connections tied to completed tasks. A numerical answer without units or reasoning does not, by itself, establish mastery of a standard.
| Target | Student work |
|---|---|
| CCSS HSN.Q.A.1 / Quantities | Tasks 2-4: consistent units and time origins in multistep calculations. |
| CCSS HSA.CED.A.4 / Rearranging formulas | Task 6 only: isolates beta from tau = t_E sqrt(1 - beta^2), then checks the target. |
Choose a route for your learners
| Level | Suggested route | Mathematical support |
|---|---|---|
| Grades 6-8 enrichment | Teacher-led predictions, supplied clock values and message comparisons; about 30-40 minutes. | Ratios and reading tables. No separate middle-school PE claim. |
| Grades 9-12 core | Learner pages 1-4; about 50-60 minutes. | Ratios, signed arithmetic, square roots and consistent units. |
| Advanced high school / introductory college | Core plus page 5; add about 20-35 minutes. | Rearrange a formula; derive piecewise signal intersections. No calculus needed. |
Homeschool and independent learners may use any appropriate route. The introductory-college label describes intended use, not accreditation. Claim the formula-rearrangement connection only if Task 6 is completed; substituting into a supplied expression is not that task.
Official Common Core quantities targetshttps://www.thecorestandards.org/Math/Content/HSN/Q/
Official Common Core creating-equations targetshttps://www.thecorestandards.org/Math/Content/HSA/CED/
EDUCATOR / EXTENSION ANSWERS AND SOURCES
Check the model independently
Extension answers / Task 6 and static gravity
Rearranging gives beta = sqrt[1 - (tau/t_E)^2]. For 100 Earth years and 10 crew years, beta = sqrt(0.99) = 0.9949874371 and L = 49.74937186 light-years. Nearest slider values can give slightly different clocks; report the tested settings and values. No real mission feasibility is implied.
For message send time s, outbound reception is t_r = s/(1 - 0.8) = 5s, valid through s = 1.5 years. On return, t_r = (12 + s)/1.8. The boundary is the send time that produces reception at turnaround, not the time the ship turns. Crew receipt is 0.6 t_r. The static-clock ratio sqrt(15/19) is 0.8885233, so near clock B advances 53.3114 minutes per hour on far clock A.
Offline verification
Extract the teaching kit. Run python3 verify_analysis.py (Windows: py -3 verify_analysis.py). Python 3.9+ and its standard library are sufficient. The checker compares CSV values with analytic worldline intersections, the null signal path, proper-time factors and rounding tolerances. It makes no network request and does not modify the supplied cases.
These are generated ideal-model teaching data, not telemetry. The separate NIST clock/GPS comparison supplies a connection to measured technology. Full source references and stable links are included in the HTML lesson and source-links.txt.
Scientific sources
Einstein Online: the travelling twinshttps://www.einstein-online.info/en/spotlight/twins/
Einstein Online: light clocks and time dilationhttps://www.einstein-online.info/en/spotlight/light-clocks-time-dilation/
NIST: atomic-clock tests and the GPS comparisonhttps://www.nist.gov/atomic-clocks/a-powerful-tool-for-science/putting-einstein-test
MIT 8.962: Schwarzschild clocks, section 6.2.2https://web.mit.edu/8.962/www/lecnotes/8_962TA-lec-all.pdf
Standards sources
NGSS high-school practice elementshttps://www.nextgenscience.org/sites/default/files/HS%20Practices%20Combined.pdf
NGSS appendices and crosscutting conceptshttps://www.nextgenscience.org/resources/ngss-appendices
Common Core HSN.Q.A.1 and HSA.CED.A.4 are linked on page 9 and in the text edition. Standards descriptions are paraphrased; the issuing organizations own their standards. Sources and alignment reviewed November 7, 2026.
REFERENCE / ACCESS, LICENSE AND ATTRIBUTION
Use, adapt and share the lesson
Accessible formats and response choices
Use the semantic HTML lesson for reflow, selectable text, headings, table headers and diagram descriptions. The standalone HTML in the kit works offline. Use learner-responses.txt for local typed responses, or write on paper or dictate. No account, purchase or online answer submission is required.
The PDF uses tagged headings, tables with scoped headers, a logical reading sequence, descriptive links, heading bookmarks, embedded fonts, document language and text alternatives for figures and formulas. Running furniture and writing rules are marked as artifacts. Meaning is carried by labels and line styles as well as color.
Print on US Letter, 8.5 × 11 inches. For A4, choose Fit. Text and diagrams remain usable in grayscale. For larger type or high zoom, use the HTML edition or reflow where supported. The full online lab uses JavaScript; the core printed lesson does not need JavaScript, WebGL, animation or sound.
Review status
Reviewed November 7, 2026. PAC 2026 PDF/UA, WCAG and Quality checks report no errors or warnings. NVDA sound/reading check completed.
Creative Commons educational license
© 2026 DarkScienceDocs.com. Original lesson text, generated tables, response template and instructional diagrams by The Last Transmission are licensed under Creative Commons Attribution-NonCommercial 4.0 International. You may copy, distribute and adapt them for noncommercial purposes with appropriate credit, a license link and a statement of changes.
Read the CC BY-NC 4.0 licensehttps://creativecommons.org/licenses/by-nc/4.0/
This grant does not license the website software, branding, documentary footage, third-party standards or linked source material. Their separate terms continue to apply. Reused editorial artwork carries its own credit and is not part of the PDF’s original mathematical diagrams.
Suggested attribution
The Last Transmission, “Two Clocks. One Reunion.” Time Dilation teaching lesson, DarkScienceDocs.com, CC BY-NC 4.0. Add the resource link and identify adaptations.