# How can gravitational forces influence time?

How does it work that gravitational forces can affect time and what usable applications could arise from this?

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+1 good question, can we get Einstein in here? – Mark Schultheiss Apr 20 '11 at 14:48
You may be interested in the classic experiment that verified the existence of the effect: en.wikipedia.org/wiki/Pound-Rebka_experiment It was also one of the effects involved in the memorable Hafele-Keating experiment: en.wikipedia.org/wiki/Hafele-Keating_experiment – Ben Crowell Apr 6 at 19:08

## 4 Answers

Gravity is currently understood to be curvature of spacetime, so mathematically it's unambiguous why gravity should affect time. But to leave it at that would be cheating, no? So I'll try to give an explanation of why you'd intuitively expect gravity to affect time. This is called the Einstein Tower thought experiment. If you've seen the phenomenon of time dilation in special relativity before, you'll notice that the way time behaves depends a lot on how light behaves. It's not crucial you know this but the spirit of the following explanation is the same.

Let's assume for a moment that gravity doesn't affect light in any way. Suppose you create a photon with energy $E$ on the ground and fire it up to the top of a tower where this photon is converted into an equivalent mass $m=E/c^2$. This mass will fall back to the ground and once it reaches the ground it will have gained extra energy $mgh$ because it fell through the gravitational field. If you started with energy $E$, you end up with energy $$E'=E+mgh=E(1+gh/c^2)$$ So every time you do this, you create energy! Surely something is wrong -- it must've been our assumption that gravity doesn't affect light. The most obvious fix then is to assume that the photon lost energy as it climbed up the gravitational field and that its energy at the top is $$E_{top}=E(1+gh/c^2)^{-1}$$ This is (a first approximation to) the phenomenon of gravitational red shift where photons lose energy as they climb out of a gravitational field.

Now, if you remember that the photon frequency is related to its energy by $E=h\nu$, you'll see that the frequency (~$1/T$ which is the "internal clock" of a photon) obeys the same relation as above. This is indicative of gravity affecting time -- in fact it's immediately obvious that any clock based on the frequency of light will run at a slower rate higher up in the gravitational field.

In reality, the result is far more general, though you'll need the apparatus of general relativity to see this. See http://en.wikipedia.org/wiki/Gravitational_time_dilation for more.

There's probably no practical application you can milk out of gravitational time dilation (yet) since the effect is weak. But you definitely need to account for these effects in engineering projects such as GPS satellites.

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+1! That's a very nice argument you got there. – Olaf Apr 21 '11 at 12:44
Since the OP phrased the question in terms of "gravitational forces," it may be worth pointing out explicitly that the effect has absolutely no dependence on gravitational force. It depends only on a difference in gravitational potential. – Ben Crowell Apr 6 at 19:10

Since your question is from the How Things Work site I guess you're looking for some basic understanding rather than lots of equations. What follows is in that spirit ...

Your question doesn't really have an answer, because it isn't meaningful, or at the least it's misleading, to say that gravitational forces affect time. Mainly that's because, assuming you believe in General Relativity, gravity isn't really a force at all. At least it's not a force in the same way that there's an electric force between two charges or a strong force between two subatomic particles.

General Relativity is based on the idea that mass (and energy and even pressure) cause spacetime to curve. The path that objects moving through spacetime follow is affected by this curvature so objects moving through curved space don't move in straight lines. If you've watched any popular science programmes you've probably seen the analogy of balls rolling on rubber sheets. The key point is that the freely moving object thinks it's moving in a straight line and it doesn't feel any gravitational force.

For example, suppose you leap off a high cliff. As long as nothing is in your way you'll move in a locally straight line and you won't feel any force at all (ignoring wind resistance). That is, you won't experience any gravitational force. The trouble is that the presence of the Earth curves spacetime in your vicinity, and as a result your locally straight line will hit the surface of the earth. Assuming you land with no injury, you'll now find you feel a force holding you to the ground that you (and Newton!) would describe as a gravitational force. But the apparent force is just because you and the Earth are trying to move along different locally straight lines. There isn't really any force there.

So my point is that it isn't meaningful to ask how a gravitational force affects time, because there isn't any such thing as a gravitational force. What does affect time, or more precisely spacetime, is the presence of matter, and that produces the appearance of a gravitation force. Your question should really be "How can (the curvature of) time affect gravity?" i.e. exactly the reverse of what you asked!

So now you're going to ask "How does matter affect time", and that's a good question that doesn't have an easy answer. The Einstein equation tells you how matter affects spacetime, and you can feed in numbers and get the right answers. What it doesn't tell you is why matter affects spacetime. We tend to leave the "why" questions to the philosophers :-)

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The Schwarzschild metric is $$ds^2~=~(1~-~2GM/rc^2)(cdt)^2~-~(1~-~2GM/rc^2)^{-1}dr^2~-~r^2d\Omega^2$$ This has all the constants and the like here to illustrate how the time part of the metric has $O(c^2)$ larger dependency than the spatial part, so $|g_{00}~-~1|~>>$ $|g_{rr}~-~1|$. So we may approximate this with $$ds^2~\simeq~(1~-~2GM/rc^2)(cdt)^2~-~dr^2~-~r^2d\Omega^2$$ So we may then consider a stationary situation with $dr~=~0$ so the propertime is $$d\tau~=~\sqrt{1~-~2GM/rc^2}dt~\simeq~(1~-~GM/rc^2)dt,$$ which for near Earth surface gravity the $1~-~GM/r~\simeq~1~+~gh/c^2$, $r~=~R~+~h$. This recovers the dbrane result. For a moving body $dr~=~vdt$ we may the compute a gamma factor $$ds^2~\simeq~(1~-~2GM/rc^2~-~(v/c)^2)dt^2$$ which can be used to derive modified Lorentz transformations of spacetime. These then may be used to compute time dilation results for satellites in Earth orbit.

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I changed the last equation.  In looking at it I see I made a small mistake. – Lawrence B. Crowell Apr 21 '11 at 22:46

The easy answer is that in day to day life, it can't. And so not any really usable applications.

The hard answer is that in special circumstances it can (see Special relativity), or more accurately the speed you travel at can affect how you experience time. To get a really good understanding of why would require a university degree in both maths and theoretical physics, which I don't have and neither would I have chance to explain in a couple of paragraphs.

Put simply, the closer to the speed of light you travel, the slower you experience time, so that if you could theoretically reach the speed of light (which you can't without an infinite amount of energy, unless you have zero mass, which is a mind bender in its self) time would appear to stop (from your point of view at least).

Consider this (sorry its quoted directly from wikipedia but its illustrates it quite well): Since one can not travel faster than light, one might conclude that a human can never travel further from Earth than 40 light years, if the traveler is active between the age of 20 and 60. One would easily think that a traveller would never be able to reach more than the very few solar systems which exist within the limit of 20-40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, he can travel thousands of light years during his 40 active years. If the spaceship accelerates at a constant 1G, he will after a little less than a year (mathematically) reach almost the speed of light, but time dilation will increase his life span to thousands of years, seen from the reference system of the Solar System, but his subjective lifespan will not thereby change. If he returns to Earth he will land thousands of years into its future. Even if he should accelerate for a longer period, his speed will not be seen as higher than the speed of light by observers on Earth, and he will not measure his speed as being higher than the speed of light. This is because he will see a length contraction of the universe in his direction of travel. And during the journey, people on Earth will experience much more time than he does. So, although his (ordinary) speed cannot exceed c, his four-velocity (distance as seen by Earth divided by his proper (i.e. subjective) time) can be much greater than c. This is similar to the fact that a muon can travel much further than c times its half-life (when at rest), if it is traveling close to c.

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"the closer to the speed of light you travel, the slower you experience time" could be misleading. A person moving near the the speed of light will age slower relative to someone who is moving slower, but each person will experience time the same way. 40 years would still feel like 40 years to both people. As described in the Wikipedia article, the traveller returning to earth would find that much more time has passed on earth, even though the traveller himself has only experienced 40 years of life as a high-speed astronaut. – e.James Apr 20 '11 at 17:46
That's a mind bender. Not a scientific reference by any means, but the Stargate episode "A Matter of Time" illustrates this theory if you're interested in a less technical explanation. – Michael Apr 20 '11 at 20:34

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Note: special relativity specifically does not deal with gravity: the special means it applies in inertial reference frames. To answer this question, one must appeal to general relativity. – dmckee♦ Apr 20 '11 at 22:44

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