# B Gravitational Time Dilation

#### PhDnotForMe

So I know gravity correlates with time dilation. If you have two individual equal size black holes close to each other, then at a point between them, gravity is equal to zero. Would the time dilation at that point be a sum of each individual black holes gravity or would the two time dilation effects cancel out making the speed of time at that point equal to the speed of time at a point in the middle of space somewhere? I know my question probably has a few misunderstandings in it, but I believe what I am trying to ask is evident. Please help!

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#### Nugatory

Mentor
So I know gravity correlates with time dilation.
It does not.
Gravitational potential does correlate with time dilation, but that's something very different - and note that gravitational potential is only a meaningful concept for some spacetimes and one containing two black holes orbiting one another is not one of them.

In any case the Einstein field equations are non-linear, meaning that you cannot get the solution to a two-body problem by adding the the solutions for each individual body. In the case that you're asking about, the solution for the two bodies orbiting one another (it's irrelevant whether they're black holes or not) looks nothing like what you'd get by adding (it's not even clear what what you've be adding) the solutions for each one in isolation.

#### Janus

Staff Emeritus
Gold Member
Gravitational time dilation depends on the difference in gravitational potential, or the amount of energy per unit mass it would take to move from one point to another in the gravity field.
So imagine you started at that point exactly halfway between those black holes, and moved away from it along a line so that you keep an equal distance from each black hole. As you move away from that point, the forces no longer cancel out in all directions, and you will start to feel a force pulling on you opposite to the direction you are trying to move. It will take energy to increase your distance from the black holes, thus you will be increasing your gravitational potential. Clock at higher gravitational potential run faster than those at lower potential. Since you have to increase your potential to travel from this midpoint to some far removed point in space, a clock at that far removed point will run faster than one at the midpoint between the black holes.
The same would hold for an clock sitting at the center of the Earth, it would be at a point where there is zero g-force, But it is also at a lower gravitational potential than a clock on the surface, so it would run slower than a clock on the surface.

#### PhDnotForMe

It does not.
Gravitational potential does correlate with time dilation, but that's something very different - and note that gravitational potential is only a meaningful concept for some spacetimes and one containing two black holes orbiting one another is not one of them.

In any case the Einstein field equations are non-linear, meaning that you cannot get the solution to a two-body problem by adding the the solutions for each individual body. In the case that you're asking about, the solution for the two bodies orbiting one another (it's irrelevant whether they're black holes or not) looks nothing like what you'd get by adding (it's not even clear what what you've be adding) the solutions for each one in isolation.
Ok, so in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero. I would say this would mean that any time dilation effects caused by one body (black hole A) would be cancelled out by the other body (black hole B) due to gravitational potential being equal to zero at that point. Thanks.

Staff Emeritus
Ok, so in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero
There is not. We can say there is, but that won't make it so. (I think that was Abraham Lincoln)

#### Janus

Staff Emeritus
Gold Member
Ok, so in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero. I would say this would mean that any time dilation effects caused by one body (black hole A) would be cancelled out by the other body (black hole B) due to gravitational potential being equal to zero at that point. Thanks.
Gravitational potential is not an absolute thing. You could assign a value of zero to such a spot, but that would mean that a spot far removed would have a positive gravitational potential relative to that spot. Or, we could assign zero potential to being infinitely far from the black holes, but that would make the gravitational potential at the spot as having a negative value, and would not change the fact that the spot was at a lower gravitational potential. The actual choice of zero gravitational potential is arbitrary and it generally chosen for its convenience.
It is the difference in potential between two clocks that determines which one runs fast and by how much.
It's like standing on the side of a hill. You have an altimeter with you. It doesn't matter what elevation you set to be zero for the altimeter, walking uphill increases your altitude and walking down hill decreases it. And the setting you choose for your altimeter doesn't have any effect on whether someone else is lower or higher than you are. So, with gravitational time dilation all that would matter is whether the clock is uphill or downhill of you and by how much, not where you are on the hill as compared to some arbitrary reference.

#### Ibix

say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero.
It isn't gravitational potential that's interesting for time dilation - it's the gravitational potential difference between the two clocks whose tick rates you are comparing. And gravitational potential is not the same as gravitational "force" (scare quotes because it's not a force in general relativity). Force is the gradient of potential, so it can be zero where the potential is not zero. Furthermore, gravitational potential is not defined in non-stationary spacetimes like orbiting black holes, so you can't use it as a short cut for evaluating clock tick rates in this case anyway, and you have to do parallel transport.

As an example of a situation where potential is defined, take the Earth. Dig a tunnel to the centre. Outside the Earth your weight increases as you move closer to the centre; inside it your weight decreases to zero as you move closer to the centre [1]. But your gravitational potential always decreases as you move down, whether you are inside the Earth or out. So clocks above you always tick faster and clocks below you always tick slower.

[1] For an idealised version of the Earth with uniform density. The real Earth varies in density, so weight increases until you are near the core before dropping off to nothing at the centre.

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#### PeterDonis

Mentor
in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero
Gravitational potential doesn't work that way. Think of the gravitational potential of each black hole (this is a highly heuristic analogy but it works for this particular aspect of gravitating bodies) as like a pit. If two pits overlap, they don't cancel each other out.

#### Ibix

Think of the gravitational potential of each black hole (this is a highly heuristic analogy but it works for this particular aspect of gravitating bodies) as like a pit.
@PhDnotForMe - in this analogy, gravitational potential is how deep the pit is at your location. Gravitational acceleration is how steep the slope is at your location. The former matters for time dilation, the latter for how you fall towards the bottom.

#### David Lewis

It is the difference in potential between two clocks that determines which one runs fast...
Would a person standing next to one of the clocks find that his clock is running at the correct rate (neither fast nor slow)?

#### Ibix

Would a person standing next to one of the clocks find that his clock is running at the correct rate (neither fast nor slow)?
If two clocks are at the same gravitational potential they will tick at the same rate. So yes. Note that I seem to recall reading that modern atomic clocks are precise enough to detect height differences of less than a meter on Earth, so the tolerance for "the same gravitational potential" is quite tight.

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#### David Lewis

Thanks. Then to say one clock runs fast, it only means faster than another clock at a lower potential -- not running fast in the ordinary sense (as an inaccurate clock might do).

#### Ibix

Thanks. Then to say one clock runs fast, it only means faster than another clock at a lower potential -- not running fast in the ordinary sense (as an inaccurate clock might do).
A clock running fast or slow always means with respect to another one - even when you only have one clock you can time it against a day, or your own personal counting.

But yes, all clocks in relativity thought experiments are assumed to be accurate clocks, correctly reporting their own proper time, unless otherwise noted. Relativistic effects never do anything to break clocks - it's just that there isn't a single shared definition of time, so what one clock is measuring may not correspond to what another one is measuring. But if they are in the same place under the same conditions then they will be measuring the same thing, so ticking at the same rate.

#### David Lewis

The low potential man will claim that his wristwatch is not running fast, and the high potential man will say that his watch is not running fast either. They will both be correct.

#### Ibix

Yes. The low man will see the high man's clock running fast but his own running normally, and the high man will see the low man's clock running slow but his own running normally. If they then meet up they will find that their clocks do not show the same elapsed time since whenever they were synchronised (perhaps they met up earlier, like in the twin paradox), but they are now ticking at the same rate.

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#### stevendaryl

Staff Emeritus
A clock running fast or slow always means with respect to another one - even when you only have one clock you can time it against a day, or your own personal counting.
I would say that a little differently. You can't actually directly compare two clocks that are not side-by-side in an observer-independent way. So I would say that when we say that a clock is running fast or slow, it's with respect to a time coordinate.

Or more generally, as one clock ticks away, there is an associated sequence of events: $tick_1, tick_2, ...$. A different clock produces a different sequence of events: $tick_1', tick_2', ...$. To say that one clock is running faster than another, you need a procedure for associating events from the first sequence to events from the second sequence.

In the case of static gravitational fields or constant proper acceleration, you can set up a correspondence this way: Send a light signal from one clock to the other and back. Then let $tick_{send}$ be the sending event, $tick_{receive}'$ be the receiving event at the second clock, and $tick_{reply}$ be the receiving of the reply at the first clock. Then we can conventionally say that $tick_{receive}'$ took place half-way between $tick_{send}$ and $tick_{reply}$. Armed with this correspondence between events on one clock and events on the other clock, we can say that, relative to the correspondence, one clock is ticking faster or slower than the other.

#### sweet springs

Ok, so in the example above say there is a spot between the two black holes where GRAVITATIONAL POTENTIAL is equal zero.
Gravitaitonal potential is below zero everywhere except zero point infinite far away from the source.

#### Ibix

Gravitaitonal potential is below zero everywhere except zero point infinite far away from the source.
That depends where you choose to set the zero. Conventionally it's set to zero either at infinity or whatever you are calling ground level, but you can pick any value.

#### sweet springs

I know it and fully agree with you. In OP's expectation gravitational potential shoud be an absolute quantity, otherwise his "cancel out" would become intentional and not trustable one. If he would rely on gravitational potential to explain time dilation, it should not contaion indefinite integral constant.

#### David Lewis

The low man will see the high man's clock running fast but his own running normally, and the high man will see the low man's clock running slow but his own running normally.
Is the high man farther in the future than the low man?

#### Orodruin

Staff Emeritus
Homework Helper
Gold Member
2018 Award
Is the high man farther in the future than the low man?
Define what you mean by "future".

#### David Lewis

Would it be possible for the high man to read a newspaper that, from the low man's standpoint, hasn't been printed yet?

#### PAllen

Would it be possible for the high man to read a newspaper that, from the low man's standpoint, hasn't been printed yet?
Let's count the problems in this question:

- who prints the newspaper (high man, low man, or somewhere else)?
- does "from the low man's standpoint" imply distant simultaneity? [then, the answer is fundamentally inteterminate, since simultaneity at distance is a matter of convention, even more in GR than in SR]

If you refer to realizable observations, then, if low man prints newspaper, high man's reading is in the causal future of low man's reading it. If high man prints it, then low man's reading is in the causal future of high man's reading it. (Assuming, for simplicity, the ability to instantly print and read locally).

#### Ibix

Is the high man farther in the future than the low man?
I don't think this question makes sense as written. I'm farther in the future now than when I started typing this. To ask if I'm farther in the future than someone else, though, you'd have to mean "am I now further in the future than someone else is now", which would seem to answrker itself.
Would it be possible for the high man to read a newspaper that, from the low man's standpoint, hasn't been printed yet?
This question has essentially the same problem. For the low man to say when the high man starts reading a newspaper, he needs to come up with a definition of "now". And when he does that, the printing of the newspaper will be in the past by any definition that regards the reading as "now".

@PAllen approached your question in a completely different way - the difference lies in how we chose to interpret your question, not in the physics. One addition I'd make is that the low man will eventually be able to see the high man's watch showing the same time as his, and eventually times ahead of his own. The only conclusion you should draw from this is that you can't construct a sensible global definition of "now" just from people's wristwatch readings.

#### FactChecker

Gold Member
2018 Award
Would it be possible for the high man to read a newspaper that, from the low man's standpoint, hasn't been printed yet?
No. Their clocks just run at different rates. There is never a true "cause-effect" paradox. Suppose you had a friend whose watch displayed the wrong date and he told you that he read the Wednesday newspaper on Tuesday. You would know that his date was just wrong. You would not think that he had time-traveled.

"Gravitational Time Dilation"

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