The mass of the earth creates a time gradient that drags us down. We call this "gravity."
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Our Mission
Technical and non-technical audiences alike can benefit from this unusual exploration of Albert Einstein's General Theory of Relativity. As we approach another wonderful solar eclipse we hearken back to the May 29, 1919 observation of gravitational lensing which confirmed his theory. Let's celebrate that scientific and public relations conquest by revisiting how Einstein explains gravity and its connection to time.
Curved Time Presentation
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Since the purpose of today’s narrative is to unveil the time warp in which you live, it seems fitting that I violate normal chronology by telling the punch line before the setup. I am counting on you noodling it in the background, so to speak, as we then backtrack to tell the story.
Since 1915 we have known (or should have known) that the reason things fall in gravitational fields (the only reason things fall in our low velocity day to day world), is that the closer you are to the ground the more slowly time proceeds. The higher up you are the faster time flows. Your feet are marching into the future more slowly than your head. That’s what keeps them on the ground.
Einstein first grappled with this in 1907 and later tagged it gravitational time dilation (enlargement). Like the dilated eyes you might experience when engaged in certain recreational activities. This is commonly understood to mean that as gravity intensifies closer to the source it slows clocks down – so a second gets bigger. The difference in time rates at different heights produces gravitational attraction.
So that is the punchline. Often discussions of general relativity – Einstein’s gravity theory - gallop breathlessly into the moments after the big bang or neutron stars or the incomprehensible interiors of black holes but tonight we are investigating the time warp that is our day to day lives.
To get our arms around this we will spend the rest of the evening covering the following
1.) How would time dilation cause things to fall?
2.) The evidence for time dilation.
3.) Einstein’s near miss with reputational disaster because he was confused about space.
The above is a spacetime diagram. Two dimensions – one space and one time. No room for any additional dimensions on this slide. This curve is the same as this apple dropping. Doesn’t look like a curve here in the real world but it does on the screen. How come? Because I’ve turned time into a fake space dimension. I could throw the ball and then it would track a second space dimension, length, at a steady rate with the flow of time, and so you would see the curve in real life. But we are going to stick with the simple drop to expose something hidden in the world.
On the chart, down is down and across is the passage of time. The bend in the line is the acceleration: 32 feet per second per second. In free fall you are locally weightless. It’s just like an orbit except in that case you keep missing the Earth. As Douglas Adams said,” “It's not the fall that kills you; it's the sudden stop at the end.”
Now there is something off with this picture because I have time flowing at a certain rate down here but, as I mentioned, it’s different at different heights.
So let’s illustrate a gradient and put a grid over it:
Each one of these purple lines represents a slightly different rate of time flow. They are elements of a continuous gradient illustrated by the deepening background color as you descend. The green line that diverges from the gold vertical shows the passage of one second at different heights. Why would that cause the curve? Before the stem snaps that apple is traveling parallel to those lines. When it’s freed up why the curve? Let’s discuss a few meat and potatoes physical analogies from everyday life.
If you are pushing a lawn mower and it hits the border between the grass and the concrete at an angle so that the left wheel goes to the concrete while the right is on the grass – what does the lawn mower do? It turns towards the grass. A differential in the speed of the left and right flanks causes a change in direction towards the retarded flank.
Tanks and bulldozers steer by adjusting the relative speeds of the right and left tracks accordingly.
Light curves in a lens or glass of water or prism because of the difference in light speed in water or glass as opposed to air.
Imagine a group of ice skaters skating forward in a row, arms locked in the fashion of a chorus line. They are all trained to put the identical effort so that the line stays straight and goes straight ahead. Now they come upon a stretch of ice which is a slush gradient. The further you are to the right (as they are facing) the more slushy the ice. Because they are all trained to make equal effort, what happens to the direction of travel? The line inexorably starts curving right. So goes the apple.
This is why things fall. This is what we experience as the force of gravity. This is the pressure on your seats.
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Now that we see how a time gradient creates gravitational attraction, what’s the evidence that clocks slow near the earth? The evidence is incontrovertible.
But first let’s do a little soul searching.
Little kids, when they first grapple with time, come early to the idea of sequence. At my fifth birthday party, I remember staring with some concern at the blue plastic 5 on the cake and asking my mother whether the rest of my birthdays were necessarily going to arrive in sequence. I wouldn’t mind being two again later. I was a little slow.
Generally, little kids get sequence early – it’s part of the core intuition of time. It takes them much longer to get the idea that time is the same everywhere. That the word “now” applies as well to Jupiter or Vega or the Andromeda galaxy as it does to us in this room. Adults prior to Einstein, on the other hand, assumed the physical world unfolds against a universal backdrop of completely homogeneous and rectilinear time and space.
This is an unwarranted projection from limited experience. Newton was mistaken, and the child’s openness to time being different in different locations turns out to be wise.
Now we all know that the time in Bern is not the time in Chicago. And when Einstein began contemplating these questions, he faced a train system that did not utilize hour unit time zones, but hundreds of local zones based on station location. This exotic scheme may have pushed him in the right direction.
In 1905, Einstein jettisoned the illusion that simultaneity was objective. Two observers, in motion relative to each other, could disagree as to whether two events happened at the same time. This is special relativity, which is not the main topic presently, but necessary to touch upon. Thereafter, space and time, separately considered, were relegated to fluid perspectival phenomena. Two observers would disagree about how fast the other’s clock was ticking and how reliable were their yardsticks. However, when the time and space data of these contrary observations were combined into a kind of spacetime (in other words, when you measured time as a kind of fake fourth spatial dimension) the observers could once again agree. Their combination, as in our chart here, was raised up to be the new objectivity.
Visualize time as a fake spatial dimension and you construct a picture of the world with which all observers agree. In other words, a mathematical model was proven to be more real than the world we experience.
Special relativity does not encompass gravity, and for Einstein this was an itch he had a hard time scratching but could not stop reaching for.
So in 1907 Einstein realized that gravitational attraction was the result of variations in the rate at which clocks tick at different distances from the gravitational source. But is this true? Are your feet really older than your ears? Not measurably by any pedestrian standard.
Fortunately, we have sensitive clocks, called atomic clocks, and decades ago when we synced two of them, and placed them in the top and bottom of a tower at Harvard University we were on our way to proof. How long do you imagine it took before they were noticeably out of sync?
As Kip Thorne explains in his book The Science of Interstellar:
The simplest, quantitative form of Einstein's law of time warps is this: Place two identical clocks near each other, and at rest with respect to each other, separated from each other along the direction of the gravitational pull that they feel. Denote by R the fractional difference in their ticking rates, by D the distance between them, and by g the acceleration of the gravity they feel (which points from the one that ages the fastest to the one that ages the slowest). Then Einstein's law says that g=Rc2 /D. For the Pound-Rebka experiment in the Harvard tower, R was 210 picoseconds in one day, which is 2.43 X 10-15, and the tower height D was 73 feet (22.3 meters). Inserting these into Einstein's law, we deduce g= 9.8 meters/second2, which indeed is the gravitational acceleration on Earth.
When you do the math, you can generate Newton’s gravitational constant from this time flow discrepancy.
And we re-prove it every moment when we use the GPS in our phones. Our phones use software to measure discrepancies from clock broadcasts from multiple satellites. It’s a version of triangulation based on the fact that the signals from satellites that are different distances from your phone arrive at slightly different times. But the calculation requires built-in fine tuning to accommodate two extraordinary results of the fluid nature of time:
1.) Special Relativity: because of the relative speed involved, clock calculations have to be adjusted against their slowness
2.) General Relativity: because the satellites are further away from the earth than you are, the satellite clocks have to be adjusted even more in the other direction because, relative to your phone, they are ticking too fast
Einstein’s math provides precisely the correct adjustments and without these adjustments your GPS would be worthless. I hear the engineers didn’t believe this would happen so they had these adjustments on a kind of toggle switch. Once up and running they quickly embraced the need to turn the Einstein switch on.
To review:
1.) We see how a time gradient can make something fall
2.) We see there is a time gradient.
3.) The time gradient is just perfect for making things fall the way they do.
So we are now back to the original punchline. It’s the variation in the rate of time according to height that causes what we experience as gravitational attraction – things falling.
But focusing on warped time, and ignoring warped space, for explaining gravitational attraction is only useful in normal situations where high speeds are not involved. When things are traveling close to the speed of light, or when light itself is under discussion, then a unified spacetime curvature must be invoked.
This unified spacetime is present right here but the distortion in space in this room is insignificant – only the time component keeps us planted here.
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As you will recall Einstein in 1905 weaved space and time together as spacetime. But in 1907 his theory of gravity only dealt with distortions in time. He could not get his arms around a spacetime model of gravity for another eight years – and he tried. For much of this time he was consumed with this quest.
As WWI broke out he still had only the time component nailed down but thought he was close enough to make a prediction regarding starlight curving around the sun. He imagined that during a solar eclipse the background stars would appear to shift a little because of the sun’s gravitational field bending the starlight. He was taking into account the effect of the time gradient but did not yet have a model of time and space distortion together. As he only later realized, when something is traveling at the speed of light, the amount of time gradient effect is matched, and therefore doubled, by the compression of space. So his prediction understated the light bending by 50%!
There was going to be a solar eclipse in the Crimea on August 21, 1914 and he saw this as an opportunity to prove the theory. He sent some colleagues led by Erwin Finlay-Freundlich to take some snapshots but the locals did not dig Germans running around with telescopes, and they ended up jailed in Russia as suspected spies. An American, William Wallace Campbell, got through but clouds got in the way. Thus, Einstein was saved from a public humiliation that might have hampered his career – and would in any case have emboldened his opponents (some jealous, some antisemitic, not a few simply perplexed).
At the end of 1915, the theory jelled. A unified spacetime distorted by mass and energy causes apparent attraction. This theory led to the correct prediction of the amount of displacement a background star image undergoes during a solar eclipse – double what he had earlier predicted. Einstein was vindicated to world acclaim by British astronomers Eddington and Dyson when they photographed stars that were behind the sun during a May 29, 1919 total eclipse. The New York Times headline was “Lights All Askew in the Heavens.”
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background details
Einstein’s pursuit of gravity’s secrets is a startling story of dogged detective work and uncanny creativity against the backdrop of European crisis and the first World War. The theory coalesced in stages over a decade. In 1914 he thought he had nailed it and predicted the extent to which starlight would curve around the sun. But the German astronomers who went to the Crimea to photograph a solar eclipse as proof were arrested as German spies. Dumb luck. Einstein’s prediction was wrong by half but he corrected and completed the theory before his next shot at vindication - the solar eclipse in Brazil in 1919. By unpacking the story of how he got it wrong and then right, we gain unique insight into gravity’s breadth from the expanding cosmos to the apple falling from the tree.
a breakthrough in teaching relativity
In normal circumstances such as those we experience on Earth, the cause of gravitational attraction (the reason things fall) is the disparity of the rates at which time flows at different distances from the Earth’s surface. Curved time. Although this fact is broadly appreciated by experts in the field, it is not universally understood even by physicists, and is seldom mentioned in popularizations. Some happy exceptions are mentioned below.
See video from Eugene Khutoryansky of time dilation causing gravitational attraction:
https://www.youtube.com/watch?v=gcvq1DAM-DE
An alternate visualization from Lewis Carroll Epstein and Rickard Jonsson:
https://www.youtube.com/watch?v=DdC0QN6f3G4
A PBS explanation:
https://www.youtube.com/watch?v=UKxQTvqcpSg&t=4s
A revolutionary approach to teaching relativity in secondary school from Magdalena Kersting, Richard Toellner, David Blair and Ron Burman:
https://iopscience.iop.org/article/10.1088/1361-6552/ab56d7
More typical explanations depending on the technically correct “curved spacetime” are often misunderstood as “curved space”. As we sit watching the apple tree, we see no visual evidence of curved space beyond the apple’s fall. As a result, explanations using the phrase “curved spacetime” often lead to confusion and even cynicism among the uninitiated. Thus the name of this site: “Curving Time.” Our presentation is designed to correct the common pedagogical error and make general relativity come alive for everyone.
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historical Background of curving time
1907
Einstein partially solved the time component of the puzzle of gravity several years before he published his general relativity which is a theory of gravity as spacetime curvature. In 1911 Einstein published On the Influence of Gravitation on the Propagation of Light documenting his 1907 breakthrough. He explains that light’s behavior in a gravitational field demonstrates that time flows slower near the gravitational source. The marker for this is an apparent variation in the speed of light. The show-stopper comes in the middle of page 107 as this is reprinted in the Dover paperback The Principle of Relativity.
“From the proposition which has just been proved, that the velocity of light in the gravitational field is a function of the place, we may easily infer, by means of Huyghens’s principle, that light rays propagated across a gravitational field undergo deflexion.”
This is followed by a little diagram showing the plane light-wave changing direction as a result of crossing a time gradient. This is equivalent to optical refraction where varying thickness of a retarding medium alters the path of light (which is why he has dragged in Huyghens.)
Think of the wave as a group of ice skaters in chorus line formation, shoulder to shoulder and arms locked, all skating with equal effort. As you move from right to left on the line (as they are facing) the ice is increasingly slushy (they are skating across a temperature gradient), the slush retards each skater to a different degree and the line inexorably turns left. Time moves more slowly as you get closer to the surface of the sun. So the equivalent of turning left is to bend toward the sun – otherwise known as falling.
If the 1907 theory were thrown out in 1915 then this discussion might be a sterile curiosity. But that did not happen. Time curvature is absorbed into general relativity as the Newton limit - applicable to those circumstances in which Einstein and Newton predictions are identical. It was not a complete description of bending light because light is covering as much space as it is time, so space curvature and time curvature play equivalent and reinforcing roles and Einstein had not yet gotten a grip on curved space. But re-contextualized, as a component of spacetime, time curvature remains an accurate explanation and predictor for the dropping apple.
1915 - (The reader may skip the math below without losing the gist.)
In The General Theory of Relativity section of The Meaning of Relativity – Fifth Edition MJF Books - Einstein presents the general relativity equation on page 79 for the “motion of a material particle, under the action only of inertia and gravitation” and then asks on page 80, “How are these [GR] equations connected with Newton’s equations of motion?” On page 81 he answers “Let the velocity of the material particles be very small compared to that of light…dx₁/ds , dx₂/ds, dx₃/ds [the curvature components of the three spatial dimensions] will vanish compared to dx₄/ds [time curvature]…This [resulting] equation is identical with Newton’s equation of motion for a material particle in a gravitational field…”
In The Foundation of the General Theory of Relativity in a section titled Newton’s Theory as a First Approximation Einstein goes through a similar exercise and states on page 158,9 of The Principle of Relativity, “If we restrict ourselves to the case which almost exclusively offers itself to our experience, of v being small as compared with the velocity of light…What is remarkable in this result is that the component g₄₄ [the time component of spacetime curvature (in modern notation: g₀₀)] of the fundamental tensor alone defines, to a first approximation, the motion of a material point.”
In page 127 of the Crown Publishers paperback, Relativity The Special and General Theory, which he wrote (in contrast with the above sources) for mere civilians, Einstein provides a different formulation. He describes the confirmed angle of deflection of starlight around the sun and then, “It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification (“curvature”) of space caused by the sun.”
This unfortunate formulation (apparently in service of keeping it simple) gives the impression that general relativity might be something pasted on to Newton instead of subsuming it. It also gives the impression that geometrical considerations (curvature) only apply to space as opposed to spacetime. It is only from the standpoint of the actual theory as indicated by the previous references that we can recapitulate Einstein by substituting “time curvature” for “Newtonian field of attraction.” This pedagogical misstep, and related ones by popularizers since, are at the heart of the current confusion on this point. See this Physics Forum thread for an example of the trouble caused by the Einstein quote.
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