Clocks and Gravity: How Earth's Movement May Affect Timekeeping

[Newcopernicus]

If time is being warped by gravity/accelerated motion, does that mean that our clocks are inevitably slowed down since the Earth's constantly moving

 

[ShayanJ]

Things aren't that easy. I'm not an expert but I think it should be like this:
At first the different kinds of Earth's accelerated motions should be translated to equivalent gravitational fields. Then these fields should be added to gravitational fields due to other objects near Earth and also Earth's own gravitational field(GR isn't linear in metric so the meaning of "adding" should be taken with care.). Then we can say that Earth's time is slowed down relative to parts of space-time with weaker gravitational fields than the combined field we found for earth.

 

[cosmik debris]

The time shown on a clock on Earth is only slowed or fastened when viewed by an observer in a different place or state of motion, this is called a different reference frame. To anyone traveling with the clock it reads the proper time.

For example: a clock in a satellite in orbit will be viewed by an Earth bound person as running slow because of its orbital speed but running fast because it is higher in the gravitational potential. At a certain altitude these effects can cancel out. If you are on the satellite the clock reads proper time. It is the difference between frames of reference that determine whether the clock appears fast or slow.

 

[PeterDonis]

If time is being warped by gravity/accelerated motion, does that mean that our clocks are inevitably slowed down since the Earth's constantly moving

Slowed down compared to what? As cosmik debris points out, the idea of a clock being "slowed down" is a relative concept.

Also, to compare the rates of two clocks at all, there has to be some common point of reference between them. For example, to compare a clock on Earth with a clock in Earth orbit, you can use the time it takes for the satellite to complete one orbit, according to each clock; the periodic nature of the motion provides a common reference. But for a clock that is just flying away from the Earth, such as the clock on the Voyager spacecraft , where the relative motion isn't periodic, there is no common reference, so there's no way to compare the clocks in any absolute sense.

 

[PeterDonis]

the different kinds of Earth's accelerated motions should be translated to equivalent gravitational fields.

What motions do you have in mind, and how would you translate them?

(GR isn't linear in metric so the meaning of "adding" should be taken with care.)

Except when all the fields involved are weak, you can't add them the way you describe, precisely because of this nonlinearity. "Weak" fields are simply fields that are small enough that the nonlinear terms can be ignored.

 

[ghwellsjr]

If time is being warped by gravity/accelerated motion, does that mean that our clocks are inevitably slowed down since the Earth's constantly moving

Yes, atomic clocks at Greenwich near sea level run slower than identical atomic clock at Boulder, Colorado at an altitude of about a mile due to gravity.

 

[ShayanJ]

What motions do you have in mind, and how would you translate them?

Earth's rotation around sun and its own axis. And maybe the motion of solar system and milky way galaxy if they're accelerated.
I don't know how to do that, I just know in GR, accelerated motion can be ignored if we consider an appropriate gravitational field.

Except when all the fields involved are weak, you can't add them the way you describe, precisely because of this nonlinearity. "Weak" fields are simply fields that are small enough that the nonlinear terms can be ignored.

I just thought maybe there is an equivalent way of superimposing fields in nonlinear theories. Of course not the trivial addition but some kind of a more sophisticated kind of superposition. If that doesn't exist, then we should add all accelerations and find an appropriate gravitational field for that overall acceleration.

 

[PeterDonis]

Earth's rotation around sun and its own axis. And maybe the motion of solar system and milky way galaxy if they're accelerated.
I don't know how to do that, I just know in GR, accelerated motion can be ignored if we consider an appropriate gravitational field.

But all but one of the motions you describe involve zero acceleration; more precisely, they involve zero *proper* acceleration, which is the appropriate concept of acceleration when you're trying to "translate" an acceleration into being at rest in a gravitational field. The Earth orbiting the Sun, the solar system orbiting the center of the galaxy, and the galaxy moving through the universe, are all in free fall, with zero proper acceleration. See further comments below.

Earth's rotation about its own axis can indeed be considered as changing the effective gravitational field perceived by observers on Earth, and this does affect the "rate of time flow" among other things. For example, the Earth has an equatorial bulge due to its rotation; if you're standing on the Earth's equator, you're about 13 miles further from the Earth's center than if you're standing on one of the poles. However, clocks at both of those locations go at the same rate, because the Earth's rotation affects the shape of equipotential surfaces (surfaces of constant rate of time flow), making them into spheroids instead of spheres (the surface of the Earth, or more precisely the "geoid", the surface of mean sea level, is such an equipotential surface).

I just thought maybe there is an equivalent way of superimposing fields in nonlinear theories. Of course not the trivial addition but some kind of a more sophisticated kind of superposition.

Not really, unless you use the term "superposition" to mean just "combining however it turns out you have to combine". In the general case, the way to solve the nonlinear problem is to not split it up; instead of trying to solve for individual fields and then combine them, you just solve for the overall field directly. (By "solve" in these cases, though, we really mean "solve numerically on a computer"; for most situations we do not know exact analytical solutions.)

If that doesn't exist, then we should add all accelerations and find an appropriate gravitational field for that overall acceleration.

If you're talking only about proper accelerations, then yes, you could in principle do this, but in practice it never turns out to work that way, because there's never more than one source of proper acceleration, so to speak, that matters. For example, if you're standing on the surface of the Earth, the only proper acceleration you need to worry about is the proper acceleration caused by the Earth pushing up on you and preventing you from free-falling. Any other "acceleration" you have, such as orbiting the Sun, is not proper acceleration; as far as your orbit around the Sun is concerned, you are in free fall, just like the Earth.