# Time dilation issues

1. Oct 6, 2014

### Ryan Bruch

I really need an expert for these two:

1. How can we correct the clock of a satellite due to time dilation effect due to its motion relative to the ground when ground-based clocks can be equally considered in need of correction due to their motion relative to the satellite? Both equally valid relative views, according to special relativity, would mean there would be no singular, absolute time dilation to correct, and indeed, the need for correction should entirely cancel out, requiring no action whatsoever.

2. Since everything is relative in special relativity, it is equally valid to consider the Earth to be accelerating toward stationary particles in the upper atmosphere. In that case, time slows down for Earthbound observers. The particles then decay at their usual half-life pace in their stationary reference frame while only a fraction of these half-time passes for the speeding observers on Earth. Then, just as the speeding astronaut in the Twin Paradox returns to find a much older twin, the speeding Earthbound observers would encounter an extremely old population of cosmic ray particles, which means that they should have long since decayed, and should not have been detected.

2. Oct 7, 2014

### Matterwave

1) We don't necessarily have to "synchronize" the clocks (and if we did at an instant, they would quickly become de-synchronized afterwards anyways), we just have to realize that the clocks have differences between them, and account for these differences. Perhaps the easiest way (and since I'm not a GPS engineer, I don't know if this is the way that's used) is simply to make the clocks on satellites tick faster/slower than they should so that they match our clocks on the ground. This is not to say that the satellite's clocks are wrong, only that it's more convenient if we make them match ours on the ground. I don't understand why you think "the need for correction should entirely cancel out". Clocks in different situations tick at different rates; if we didn't take this into account, our GPS system would be way off.

2) I think you are getting twisted up in making too many jumps from one frame to another and back. A cosmic ray can be understood from its own point of view. In its own point of view it does decay at its regular time, and not at a dilated time. But in this frame, the distance it sees is contracted so it hits the same place on Earth when it decays. Where it decays physically (i.e. whether it hit a detector or not) is not something that can change based on point of view. From the point of view of us Earthlings, the time on the cosmic ray is dilated, and so they travel longer/farther than we would expect.

3. Oct 7, 2014

### ghwellsjr

This is interesting: In your first scenario, you have an accelerating clock (on the satellite) but you ignore its acceleration and treat it the same as the ground clock which is not accelerating and then in your second scenario, where the ground clock is also not accelerating, you treat it like the speeding twin who is accelerating.

You can only equate relatively moving clocks if they are not accelerating, that is, there will be a reciprocal relationship between two inertially moving clocks. In your first scenario, the ground clocks are considered inertial but the satellite clocks are constantly accelerating due to their change in direction (circular around the earth) so their rest frames are not equal or reciprocal. In fact, the satellite clock is much like the speeding astronaut in the Twin Paradox. Isn't that what makes the difference in the Twin Paradox? The speeding astronaut has to change direction (accelerate) in order to get back to his twin. Same thing for an orbiting satellite.

Then in your second scenario, there is a reciprocal relationship between the particles and the Earth so that in the inertial rest frame of each of them, it is the other one that is Time Dilated (and by the same factor). However, in this scenario, we don't care about the Time Dilation of the Earth in the particles' rest frame. We only care about the Time Dilation of the particles in the Earth rest frame as to why they manage to make it to the ground. There is also a reciprocal Length Contraction relationship between them so that in the Earth's rest frame, the speeding particles are Length Contracted but we don't care about that either, we only care about the Length Contraction of the atmosphere surrounding the Earth in the particles' rest frame as to why they can get to the ground in a short time--they don't have very far to go.

4. Oct 7, 2014

### Staff: Mentor

That is what is done with the GPS clocks; there is an extra correction applied to their clocks before timing signals are sent out, to slow down the clock rate to match ground clocks. The gory details are here:

http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html [Broken]

Last edited by a moderator: May 7, 2017
5. Oct 7, 2014

### Staff: Mentor

The correction is not to obtain any "absolute time dilation", it's to adjust for relative differences in clock rates. The relative differences don't cancel out.

A better question would be, "what are the actual relative differences in clock rate between clocks on Earth and satellite clocks?" Or, more importantly for this discussion, "are those differences symmetric?" The answer to the latter question is no, they are not symmetric. How is that possible? Because the relative motion involved is periodic: the satellite is in orbit about the Earth, so there is a natural point of comparison between the clocks: how much elapsed time does each clock read over one satellite orbit? If the answers for the two clocks are different, that difference is not "relative" in the sense of being frame-dependent; both observers will agree, for example, that the clocks on the GPS satellites (before any adjustments are applied--see my response to matterwave) read more elapsed time over one orbit than Earthbound clocks do over one GPS satellite orbit. If you don't like the term "absolute" to describe such a difference, you can use the term "invariant", which is often used in relativity to describe quantities on which all observers agree.

You're leaving out a key factor: relativity of simultaneity. Let me describe the scenario from both frames with that key factor put back in.

First, we have two key events, which we're going to describe from the point of view of both frames: event A, where a cosmic ray creates a muon (those are the particles in question) high in the Earth's atmosphere, and the muon is moving at relativistic speed down towards the Earth's surface; and event B, where the muon is detected at the Earth's surface. Those two events are invariants: all observers will agree that those two events take place, and that event B happens after event A. What different observers will *disagree* about is the distance between the two events, the time between the two events, and what other events A and B are simultaneous with.

In the Earth's rest frame, A and B happen some time $T$ apart, where $T$ is much longer than the half-life of muons at rest. Also, if we think about an observer on the surface of the Earth who is detecting the muons, there will be some event O on his worldline which is simultaneous with A, i.e., it happens a time $T$ before event B by the observer's clock. The distance the muon travels, in the Earth frame, is given by the distance between A and O.

In the muon's rest frame, A and B happen at the same place, i.e., zero distance apart, and they happen some time $\tau$ apart, where $\tau << T$ (and $\tau$ must also, of course, be less than the muon's half-life). In this frame, the observer on Earth, who detects the muon, is moving towards the muon at relativistic speed; so to the muon, this observer appears time dilated. That means that there will be some event M on the Earth observer's worldline which is simultaneous with event A; and the time on the Earth observer's clock between M and B will be much less than $\tau$.

This is where relativity of simultaneity comes in: on the Earth observer's worldline, event M happens much later than event O, so the Earth observer and the muon disagree, and disagree by a large margin, about which event is simultaneous with event A. But that's ok, because simultaneity is not a direct observable. The direct observables are the elapsed time on the muon's worldline between events A and B, and the elapsed times on the Earth observer's worldline between events O and B, and between events M and B. All observers agree on the values of these direct observables. They disagree about how to interpret what those values mean in terms of their rest frames, but again, that's ok, because their different interpretations don't change any direct observables.

6. Oct 7, 2014

### pervect

Staff Emeritus
There are important gravitational effects on the rate of clocks in satellites that prevent special relativity from being applicable and require general relativity, as special relativity can't handle gravity.

One can create scenarios where GR is not required by replacing the Earth with some reference point in free space far away from other matter, and imagining that the satellites are in a powered orbit. In this case it's pretty clear that the only inertial frame is that of the non-accelerating object and that the clocks in the powered orbit do not have an inertial frame.

If you don't distinguish the clocks by having one of them in a powered orbit, then you have the classic twin paradox. In this case, both clocks float around in free space but have different velocities. Each clock has an associated reference frame wherein the other clock runs slow.

If we wish, we can consider the effects of gravity, but this requires some basic knowledge of GR and the metric and I'm guessing at this point it would be better to make sure the predictions of SR are understood correctly first before getting into the full GR scenario.

So lets go back to the SR case where we have two clocks, no gravity, and the clocks are in relative motion. Call the two observers A and B. Do you understand how that in A's coordinate system, B's clock can run slow, and how in B's coordinate system, A's clock can run slow? I believe it would be counterproductive to proceed further until this simpler case is understood.