How Do GPS Satellites Adjust for Both Special and General Relativity Effects?

[RandallB]

I understand time dilation in the Special Relativity.
As in the case of an orbiting Global Positioning Satellite (GPS)
Due to the speed of the GPS orbiting Earth an adjustment must be made for
time running slower than the clocks down here on earth.
A few calculations some code to speed up the onboard clock - bingo In-sync
with Earth’s Master GPS clock.

But how about the formula’s for calculating the effect of gravity?
And adjusting for General relativity as well? (They had to do it.)?
Once properly measured and the impact figured. An additional
adjustment to slow down the orbiting GPS clock due to the stronger
gravity field down on the surface of Earth making the Earth bound
GPS Master clock slower.

I believe the “Special Relativity increase” is smaller than the
“General Relativity decrease” required for the orbiting GPS.
As time in orbit runs fast relative to surface time.

First - what is the correct units fpr a Gravity field? other than “G”
Force / Mass? AKA, acceleration = distance per time squared?

Deep Space g= 0G = 0ft/sec/sec=9.8m/sec/sec
Earth Surface g= 1G = 32ft/sec/sec = 9.8m/sec/sec
Orbiting GPS g= <1G = <32ft/sec/sec = <9.8m/sec/sec

Once we have the units and values for: g for Earth Surface
and g’ (g prime) for the orbiting GPS satellites.
1)What is the formula to figure the faster time rate in the GPS satellite?
2)Is the formula easily derived as is the Special Relativity dilation factors?

With the proper tools, calculating the relative clock speeds of deep space,
other planets, and accelerating spaceships in next on the agenda.

The accelerating spaceship should be the hardest as I’ll need to
integrate the affect of the increase speed as well as account
for the gravitational affect of high acceleration.

Thanks RB

 

[pervect]

The formula for Gravitational time dilation in General relativity is given at:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

(The same value can be derived by inspecting the Schwarzschild metric and taking sqrt(g₀₀)).

The value of time dilation is not directly related to the gravitational field, but it is instead related in the weak field case to the Newtonian gravitational potential energy per unit mass. For convenience, I will use $$U = -\Phi$$ where $$\Phi$$ is the Newtonian potential per unit mass. Then:

U = GM/R

Time dilation = 1/sqrt(1-2*U/c^2)

Note that (energy/mass) / c^2 is dimensionless, as E=mc^2

Gravitational fields anywhere in the solar system can be considered to be "weak" for the purposes of calculating gravitational time dilation.

 

[RandallB]

Info at the link helped - PLUS binomial expansion helps a lot.

UNITS make good sense to me now as well.
g = 9.8 m/s^2, R = 6.38 x 10^6m (mean radius), and c= 3 x 10^8m/s.
As all the units cancel to a pure factor.

For the gravitational time dilation factor of: 1/(1-2gR/c^2)^0.5

Can we derive the formula from basic assumptions
As can be done with the "light beam clock" for SR.
Or is GR a bit to involved for a simple explanation like that.


Thanks RB

 

[pervect]

Yes, there is a way to derive the formula without getting too complex. The first step is to realize that gravitational red-shift and gravitational time dilation are two names for the same phenomenon. Given this insight, there are a number of approaches that can derive the amount of gravitational redshift - the URL I posted earlier takes one approach (look at the first entry on Gravitational red shift, they do a derivation there). One can also use energy conservation arguments to derive the amount of the red shift.

 

[RandallB]

GR Thought Experiment

pervect

Sorry maybe I don't get it but the Gravitational red shift detail on the earlier referenced Web Link seems to just apply the formulas not derive them.

I'm looking for a start on a good Thought Experiment that gets me to the formulas.

I figure I need to do some time differential calculations on the peak points of light waves used in the Inertial Reference frame used to do the SR bouncing light beam clock observations. Only using a very high acceleration rate rather that a constant velocity. To be honest that one seems a bit tough for me.

Instead can I just use a very large disk with blinking lights to keep track of time. One at the center axis and one in the Alternate Reference Frame attached to the edge of the spinning disk. The centrifugal acceleration on a vector pointed at the axis should be the relativistic equivalent of gravity pointed away from the axis shouldn't it?

Then calculate the velocity the reference frame is traveling in the circle to directly apply normal SR formulas to find Time dilation.

With the claim of equivalence will this give me a valid value for the gravitational time dilation?

Plus with the ability to send the light from the edge of the spinning disk directly from the edge of the spinning disk to my observation point above the disk.
And to send light up the vector to disk center and off a conical mirror up to the same observation point. This should show both sources of light from the edge to be red shifted verses the reference light from the Axis.
And by the same amount based only on the slower time at the disks edge. IE. The red shift is not directionally related to the direction of gravity.
Also, since an observer at the edge with the blinking light sees nothing to assume the axis base light is moving at all from their reference there are no SR changes to apply thus even the edge will see the axis light as blinking faster only Blue Shifted.

The more I look at this set up the better I like it. May get a little tricky when taken up to Black Hole accelerations.

So simple, I guess I'm concerned about why I haven't come across a site that goes over it. There are so many for SR!

Anything I'm missing or does it seem OK?

Thanks RB