# GR time dilation cares only about local curvature, right?

Regarding this remark on Worldbuilding SE and the discussion leading up to it:

Can someone properly knowledgeable on the subject please explain whether the time dilation due to being in a gravitation field is in fact due only to the curvature of spacetime in the region where the clock is found, or if a flat "plateau" somehow knows that it is just a shelf along a larger slope?

In particular, would using compensating masses to form a nearly flat spacetime in a small region (as described by Robert L. Forward) cause a clock to experience only "flat spacetime" or would the clock be slowed anyway because this prepared region is itself in a gravitation well?

E.g. a prepared flat region in the inner solar system vs. one out in interstellar space: is there any difference in time rate when the local curvature of spacetime surrounding the clock is the same?

Edit for those who found it unclear: look at previous paragraph for a specific physics question.

• A key point for understanding these questions is to ask yourself "slowed relative what?" and perhaps "How will I measure the difference?" – dmckee --- ex-moderator kitten Aug 6 '15 at 23:46

A simple way of illustrating gravitational time dilation is using the gravitational redshift. If I shine light of frequency $\nu_{me}$ from the Earth's surface up to you on the International Space Station then when you received it the light will have reduced in frequency to $\nu_{you}$. We explain this by saying that my time is running more slowly than yours, so I measure the time between peaks in the light waveform to be smaller than you do. The relative time dilation between me and you is then simply:

$$\frac{\tau_{me}}{\tau_{you}} = \frac{\nu_{you}}{\nu_{me}}$$

This should make it obvious that time dilation is not a local phenomenon because it depends on where I am, where you are and what is in between us.

In most cases we are likely to encounter gravity is relatively weak, and we can describe the spacetime geometry using the weak field approximation. In this case the relative time dilation for any two observers is given by:

$$\frac{\Delta t_1}{\Delta t_2} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}}$$

where $\Delta\Phi$ is the difference in the (Newtonian) gravitational potential difference between the two observers.

It is not sufficient to have a flat spacetime.

For instance if you had a region of flat spacetime then you could place a spherical shell of matter around a person sitting in the flat section. They will become time dilated because they will observe the rest of the universe speeding up as the sphere is placed around them. But spacetime will stay flat right near them. In fact a person sitting nearby in the formerly flat region will now be in a curved region due to the shell placed around that first person and they, the person outside the shell will age faster than the person inside the shell.

So making your space locally flat is not relevant at all, you could age slower than someone while being in a flat region (for instance if there is mass around you) or you could age faster than someone (if they are around some matter).

A rule of thumb is that near bodies you age slower, and that firing rockets makes you age slower. If you want to age faster than someone else, try to get them near bodies and try to get them to accelerate.

If someone tells you to place some mass near you to make you age faster, that is wrong, almost all regular sized masses in the solar system are weak fields which means the time dilation is the Newtonian gravity effect so placing mass near you just lowers you newtonian potential hence makes you age slower.

But let's get to the key. Time dilation is always relative to something. When a mass is placed closer to you, your clock runs slower than it used to. But in a flat region you and all the things around you age just as slowly, so you don't notice it until you compare yourself to other things that are far away.

There are two different notions of time dilation. Both refer to discrepancies between of time measurements between different observers that are either remote from one another and signal each other, or who compare clocks at a journey's endpoint after having made the journey beginning at the same point but taking different paths through spacetime. As such, both forms of time dilation involve nonzero measure regions of spacetime, and are thus nonlocal insofar that depends on the details of "shape" of spacetime over a noninfinitessimal region.

One arises in flat space and has nothing to do with curvature. The other notion arises from the deviation of spacetime from flatness, and could thus be said to be defined by curvature. However, it depends on the details of the curvature over a nonzero, noninfinitessimal region separating two remote observers. It too is therefore a nonlocal concept.

Curvature

Let's begin with curvature. It, unlike time dilation, is a local notion, and can be defined at a point as long as we know how to calculate the geodesics (straighest or minimum distance lines) between every pair of points in a neighborhood (i.e. the topological notion) no matter how small this neighborhood may be. Curvature measures how a vector, when parallel transported around small closed loop, differs from itself when it comes back to the beginning. If $X$ and $Y$ define an infinitessimal parallelogram, the curvature $R(X,\,Y)$ is a transformation that works out how a transported vector $Z$ changes when it traverses the parallelogram so that, in the traversal, $Z\mapsto(\mathrm{id}+R(X,\,Y))\,Z$ (think of $R(X,\,Y)$ as a matrix here transforming $Z$ so that $R(X,\,Y))\,Z$ is the deviation from independence of path for parallel transport).

Gravitational Time Dilation

Gravitational time dilation refers to how different observers with initially synchonized clocks will time things differently when they become remote from one another. The total dilation depends on the details of all the spacetime manifold that they and their signals visit in communicating with one another. It refers to the following thought experiment: (1) I synchonize my clock with that of observer B, and (2) B then travels to some far off place (e.g. I stay on Earth and B goes off and sits on one of the GPS satellites) (3) I send a modulated light signal to B encoding unit "ticks" of my clock. Result: when these signals reach B, their arrival will mark a slower tick relative to that of B's clock, even though the clocks were synchonized initially.

How much slower depends on the details of the region that contains the path that B took to remove themselves from me as well as the paths of our signals.

In the sense that if we found another region of spacetime with the same system of geodesics as the one just referred to, then this calculation will be the same. It doesn't matter where or when in spacetime this region is.

Flat Space Time Dilation

Nontrivial time dilation arises in flat Minkowski space, where there is no curvature. "Time dilation" refers a relativity of measurements i.e. it refers to how the proper time measured by different observers travelling through spacetime can be different depending on the path that they take through spacetime so that if they leave from one point in spacetime and later meet up again at another point but take different paths their clocks will differ:

O ye'll take the high road, and I'll take the low road, so our clocks will tell athwart when I 'gin meet thee

(Apologies for that, I am feeling particularly silly this morning). Proper time is to General and Special relativity what arclength is to Euclidean space. "Time Dilation" is therefore the analogue of the notion that the arclength along one side of e.g. a triangle is different from the sums of the lengths of the other two sides. This is the essence of the Twin Paradox: one twin takes a straight path, the other moves two other straight line segments that, together with the first twin's straight path, make up a triangle. The total arclength that they traverse will be different, although in Euclidean space the straight line twin's path has a smaller arclength than that of the doglegging twin, whereas in Minkowski space it is the other way around.

• GPS sattelites are not geosynchronous. They are at 20180 km. – JDługosz Aug 7 '15 at 18:11
• @JDługosz Thanks, corrected. If you see an obvious error of fact like that in an answer, it is quite alright - and indeed desirable - for you to edit it yourself. I almost always leave a note for the poster when I do this - firstly as a matter of courtesy - but in any case the poster is notified and can see the edit and may not understand why you have made the edit. – Selene Routley Aug 8 '15 at 11:36