Astronomy 2e · Black Holes and Curved Spacetime
Time in General Relativity
On this page 9 sections
In 30 seconds
Before Einstein, physicists treated time as a universal clock ticking identically everywhere. General relativity (GR) destroys that assumption: gravity changes the rate at which time flows. A clock sitting deep in a gravity well — closer to a massive object — runs slower than an identical clock higher up or farther away. This effect, called Gravitational time dilation The slowing of clocks in stronger gravity relative to clocks in weaker gravity Full entry →, is tiny on Earth but real enough that GPS satellites must be corrected for it every single day, and it becomes extreme near a black hole, where a falling clock appears to freeze at the Event horizon The boundary around a black hole from which nothing, not even light, can escape Full entry →.
The physics behind this is the Equivalence principle The claim that gravity and acceleration are locally indistinguishable Full entry →: a person in a sealed elevator cannot tell whether the elevator is accelerating upward or sitting in a gravitational field. Because accelerating frames naturally have time running faster at the "front" and slower at the "rear," gravity — which mimics acceleration — must do the same to time. Time dilation is not a quirk of particular clocks; it is a property of spacetime itself. Every clock and every process runs slow in stronger gravity.
Why this matters
Gravitational time dilation is the most used prediction of general relativity in everyday life: GPS receivers work only because the system corrects for the fact that satellite clocks run fast relative to clocks on the ground. The same physics explains why light from a collapsing star fades and reddens forever from our viewpoint and why matter falling toward a black hole appears to freeze at the event horizon. Understanding time dilation is also the key to not being fooled by one of astronomy's most common misconceptions: from the outside, nothing ever appears to cross the horizon, even though from the infalling object's own perspective it crosses quickly. On exams, questions about "what does an observer see near a black hole" are almost always time-dilation questions in disguise.
The college version
Core Concepts
Gravity slows clocks
In GR, the rate at which time passes depends on where you are in a gravitational field. An observer far from any mass measures that a clock at distance r from a mass M ticks at a rate given (for a non-rotating mass) by the commonly taught relation
Δtfar = Δtclock1 - 2GMrc2
where G is the gravitational constant and c is the speed of light. The term Δt~clock~ is the time that passes on the clock itself (its Proper time The time measured by a clock riding along with an object Full entry →); Δt~far~ is what a distant observer measures. When r is large, the correction is tiny; as r shrinks toward 2GM/c² (the Schwarzschild radius The radius of the event horizon of a non-rotating black hole, 2GM/c² Full entry →), the denominator approaches zero, and the distant observer sees the clock's time stretch out — appearing to stop entirely at that radius. On Earth's surface the effect is only about a nanosecond per day's difference per kilometer of height, but it is measurable and matters for precision timing.
The equivalence principle: gravity is acceleration
Einstein's key insight was that gravity and acceleration are locally indistinguishable. In a rocket accelerating through empty space, a clock at the nose runs faster than a clock at the tail — the rear of the rocket is always "falling away" from signals sent forward, stretching them. Because you cannot distinguish the rocket's acceleration from standing on Earth's surface (the classic elevator thought experiment), gravity must do the same: clocks lower in the well run slower. This one idea links Gravitational redshift Light stretching to longer wavelengths as it climbs out of a gravity well Full entry → (light climbing out loses energy) to time dilation (clocks down there run slow) — they are the same phenomenon seen two ways.
GPS: relativity in your pocket
GPS satellites orbit about 20,200 km above Earth (a commonly cited reference altitude). Two relativistic effects compete:
- Gravity (GR): clocks higher up experience weaker gravity, so they run fast — about +45 microseconds per day relative to Earth's surface.
- Motion (special relativity): the satellites' orbital speed (about 3.9 km/s, a commonly cited value) slows their clocks — about −7 microseconds per day.
The net result is that satellite clocks run fast by about 38 microseconds per day (a commonly cited reference value). In 38 microseconds, light travels roughly 11 kilometers — so without correction, GPS positions would drift kilometers per day and become useless within minutes. The system corrects for these effects, making GPS a working, everyday demonstration that general relativity is real.
Direct clock experiments
Physicists have measured gravitational time dilation in the lab and in airplanes. The Pound–Rebka experiment (1960) detected the gravitational redshift over a 22-meter tower. In 1971, the Hafele–Keating experiment flew cesium clocks around the world on commercial jets; the flying clocks differed from ground clocks by amounts consistent with the predicted combination of speed and altitude effects (commonly cited results). Modern optical lattice clocks are so precise they can detect the time dilation caused by a height difference of only a few centimeters — the effect is no longer exotic, just small.
Time near a black hole
As a clock falls toward a black hole, the gravitational time dilation grows without limit. To a distant observer:
- The clock ticks slower and slower, and its emitted light is redshifted toward invisibility.
- The falling object appears to take infinite time to reach the event horizon — it seems to freeze just outside.
To the falling clock itself (and to anything riding with it), none of this happens: its own time (proper time) flows normally, and it crosses the horizon in finite time. Both descriptions are correct — they are measurements made in different frames, and this observer-dependent view of time is the heart of the topic.
Time travel? Keep the physics honest
Some solutions of Einstein's equations (rotating black holes, wormholes, Gödel's rotating universe) allow "closed timelike curves" — paths that loop back in time. These are mathematically interesting but speculative: no known physical object can traverse them, and they likely violate energy conditions. On an exam or in popular science, treat time machines as science fiction unless the question is specifically about the exotic mathematics of GR.
Common Confusions
| Do Not Confuse | With | Difference |
|---|---|---|
| Gravitational time dilation | Time dilation from motion (special relativity) | Gravity slows clocks in stronger gravity (GR); motion slows clocks relative to moving observers (SR). GPS needs both corrections |
| "Satellite clocks run fast" | "Satellites' clocks are broken" | They run fast because of weaker gravity, exactly as GR predicts; the system corrects for it |
| "Time stops at the event horizon" | "Time stops for the infalling astronaut" | It stops as seen from far away; the astronaut's own clock ticks normally |
| The "frozen star" appearance | The real behavior of infalling matter | Freezing/reddening is what distant observers see; the object itself crosses the horizon quickly |
| Gravitational redshift | Doppler shift from motion | One is caused by gravity, the other by relative motion; both stretch light |
| "Time is an illusion" pop claims | Measured, quantitative time dilation | The effect is real, tiny, and precisely measured — not a philosophical slogan |
Eli explains
The same idea, in plain words
Explain it like I’m 10
A strong magnet slows a toy car's clock if the clock were made of magnets — in the same way, strong gravity slows real clocks. A clock on the floor runs a tiny bit slower than a clock on a shelf, and a clock on a satellite runs faster than one on the ground. That's why GPS satellites need to be fixed every day. Near a black hole, gravity is so strong that someone far away would see your clock almost stop — but you, falling in, wouldn't notice anything weird about your own watch.
Worked example
Suppose you are the engineer who must make GPS accurate to a few meters. Your satellites carry atomic clocks, and you notice that after a day their time disagrees with ground stations. You separate the two relativistic contributions. First, the satellites are 20,200 km up, where gravity is weaker — GR says their clocks run fast, gaining about 45 microseconds a day. Second, they move at 3.9 km/s — special relativity says fast-moving clocks run slow, losing about 7 microseconds a day. Net: +38 microseconds per day. You then realize what 38 microseconds means: light travels 300,000 km/s, so in 38 microseconds a signal travels about 11 km. A distance error of 11 km a day would make the navigation system useless — your position could be off by kilometers by afternoon. The fix is built into the system: satellite clock rates are tuned (or mathematically corrected) so that ground receivers report positions consistent with Earth's surface. You have just used general relativity to make a consumer product work — and every phone with GPS proves it daily.
Key takeaways
- Gravitational time dilation: clocks deeper in a gravity well run slower relative to clocks farther out; stronger gravity = slower time.
- The equivalence principle (gravity ≈ acceleration) is the reason gravity must affect time.
- GPS correction: satellite clocks run fast by about 38 microseconds/day (commonly cited net: +45 µs gravity minus −7 µs motion); without correction, positions drift kilometers per day.
- Pound–Rebka (1960) measured gravitational redshift over a 22-meter tower; modern optical clocks detect time dilation over centimeters.
- Near a black hole, a distant observer sees infalling matter freeze and redden at the event horizon, but the infalling object's own clock runs normally — both views are correct in their own frame.
- Time dilation is not clock malfunction; it is a property of spacetime.
- Exotic "time travel" solutions of GR are speculative, not demonstrated physics.
Check yourself
5 review questions from the chapter. Try each one, then open the answer.
Does a clock on the ground or a clock on a mountaintop run faster, and why?
Show answer
The clock on the mountaintop runs faster: it is higher in the gravity well, where gravity is weaker, so time flows slightly more quickly there.
Why does the equivalence principle force gravity to affect time?
Show answer
Gravity and acceleration are locally indistinguishable (equivalence principle). Because time runs at different rates at the front and rear of an accelerating rocket, gravity must produce the same effect — otherwise you could tell the two apart.
What are the two relativistic corrections in GPS, and what is their net effect on satellite clocks?
Show answer
Gravity makes satellite clocks run fast (~+45 µs/day, weaker gravity at altitude); orbital motion makes them run slow (~−7 µs/day). Net: satellite clocks run fast by about 38 microseconds per day, and the system corrects for it.
From the perspective of a distant observer, what happens to a clock falling toward a black hole's event horizon — and what does the clock itself experience?
Show answer
The distant observer sees the clock slow down, its time stretch toward infinity, and its light redshift toward invisibility as it approaches the horizon. The clock itself (proper time) ticks normally and crosses the horizon in finite time.
How did the Pound–Rebka experiment and modern optical clocks demonstrate gravitational time dilation?
Show answer
Pound–Rebka detected the gravitational redshift of gamma rays over a 22-meter tower; modern optical lattice clocks measure time dilation across height differences of centimeters because gravity measurably alters ticking rates.
Study tools & related lessonsKey vocabulary · Related
Key vocabulary
- Gravitational time dilation
- The slowing of clocks in stronger gravity relative to clocks in weaker gravity
- Proper time
- The time measured by a clock riding along with an object
- Equivalence principle
- The claim that gravity and acceleration are locally indistinguishable
- Gravitational redshift
- Light stretching to longer wavelengths as it climbs out of a gravity well
- Event horizon
- The boundary around a black hole from which nothing, not even light, can escape
- Schwarzschild radius
- The radius of the event horizon of a non-rotating black hole, 2GM/c²
Sources & references
This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.
Educational content only. It is not medical, legal or professional advice. Found an error? Tell us.