Relativity & GPS Formula Calculations


  pg 5  

Scenario 3C:
Exact Satellite GPS Time Changes

However, another consideration must still be explored. How is it possible that Einstein managed to get his equations accurate when his reasoning has been seen to be flawed? If it is not time changing, but just the clock, then why are his formulae apparently being used?

One answer applicable to some scenarios could be that he simply measured how clocks on Earth moved at different rates depending on their height above sea level – and from there figured out his formulae. Another answer in other scenarios is that the calculations or even the formulae are not accurate at all.

There are many online claims that the GPS satellite adjustment would require a 38 microsecond difference per day for Relativity. This daily amount is said to be comprised of +45 microseconds for General Relativity and -7 microseconds for Special Relativity. Supposedly this would result in an adjustment for the Relativities of about 11km per day.

This is currently the top-ranked search-engine source for this claim:
(search “GPS relativity”)

However, consider this: If a satellite is moving at 4000 m/s and its clock is fast by 0.001 seconds in a day, then it will be accurate to within 4 meters in a day. But if the difference is 0.000038 seconds as Ohio State suggests then:

0.000038s x 4000m/s = 0.152m … (Calculation i)

About 15cm per day seems to be the correction that should be required for both Relativities for the moving satellite. My first point of corroboration for 15cm as the apparent correction for the satellite’s daily difference of 38 microseconds is here:

I’ll get back to how Ohio State (and most of the rest of the online sources) made that 11km error adjustment in a bit. But first let me clarify that a position on the Earth is determined by the GPS from the comparative proportional delay between the signals sent to the receiver on Earth from numerous satellites.

Only the proportions between the delays in the signals matter for this, and these proportions would all be affected by the same Relativistic proportion. (Careful, there are two types of proportion here.) So the proportions of the signals to each other would be the same regardless of Relativity. The Relativity proportions would cancel each other out!

How the GPS works is a process called ‘trilateration’, and is similar to triangulation. Here is a good online account of trilateration:

A simple way to think about this is to consider your self as a target within an equilateral triangle with satellites at each corner of the triangle. If a signal is sent from each satellite to the target, and the times each signal took are the same amount, then it follows that the target must be at the center of the triangle.

If we double or halve the time it takes for all of the signals, then nothing changes. We can also change the size of the triangle without getting a different result. The same proportional process works for any other position of the target in relation to any other triangle.

It also makes no difference what the clocks at each point on the triangle register. So long as the clocks all have the same wrong starting time, nothing changes. This is because the GPS compares the delays from all the signals to the target in order to figure the proportions. The clock on the target device itself has nothing to do with it.

Even if the velocity of the electromagnetic signal itself needed to be altered for Relativity, the result is unchanged. It’s a much simpler process than one at first thinks. In affect the calculation is merely one of geometric proportions. Of course it is not a 2-d triangular process, but a 3-d process instead. But I am just explaining it in its simplest terms. We have to just infer the extra dimension or read this link for a more precise definition:

There are a vast number of websites that claim all sorts of things, best you take plenty of time looking around at all the varieties of theories to be certain. It would be a good logical positivist idea to actually double-check the arithmetic too. You can read the description here:
There is no mention of Relativity at all in the accuracy of GPS.

How am I so sure that and are correct? After all, the internet is so full of so many claims about this. And the famous websites all say that GPS does use Relativity. Think on it logically for yourself:

If the clocks on the satellites were all somehow twice as slow as the clocks on the Earth, it would not result in any adjustment to how the position on the Earth is calculated because the geometric proportions between the delays of the signals from the satellites would be unchanged. has this to say:

…at present cannot easily perform tests of relativity with the system…

Several relativistic effects are too small to affect the system at current accuracy levels, but may become important as the system is improved; these include gravitational time delays, frequency shifts of clocks in satellites due to earth's quadrupole potential, and space curvature

But of course it may become feasible to actually measure the alleged affects of Relativity on satellites at some point in the future. A few sources suggest that such experiments are underway currently. So every piece of this analysis is still immanent. But if you do not believe that it is possible to see into the future, then such experiments can only show how a ‘clock’ changes due to the affects of gravity on the mechanism.


Scenario 3D:
The Eleven Kilometer Error

So from whence comes the 11km error adjustment which so many people on the internet are copy-pasting?

Clifford M. Will ( offers us such an explanation:

To achieve a navigation accuracy of 15 meters, time throughout the GPS system must be known to an accuracy of 50 nanoseconds, which simply corresponds to the time required for light to travel 15 meters. But at 38 microseconds per day, the relativistic offset in the rates of the satellite clocks is so large that, if left uncompensated, it would cause navigational errors that accumulate faster than 10 km per day!

Essentially this is the calculation that Clifford M. Will is using:

0.000038s x 300 000 000m/s = 11 400m … (ii)

This amount (Calculation ii) is wrong for a number of different reasons. The satellite is moving at 4000m/s; whilst the signal from satellite to Earth is traveling at the velocity of light. Those are two different entities moving at different velocities. So the adjustment cannot be the same. This is the formula for adjusting time according to velocity for Special Relativity:

Relativity GPS(iii)

This formula uses velocity by converting the 4000m/s of the satellite into a proportion of the velocity of light, so we need to determine ‘V’ for the formula above:

V = 4000m/s ÷ 300000000m/s = 0.000013 of C (Light)… (iv)

When we insert this amount of 0.000013 into Special Relativity (formula iii), then we get the answer of 0.000000000084 which is a proportional adjustment per second of time. So because we want to know the adjustment for one entire day we multiply this by 86400 seconds and get:

0.000000000084 x 86400s = 0.000007s … (v)

The adjustment is 7 microseconds per day for the satellite moving at 4000 m/s. (This one small part many people fortunately agree on, arithmetically speaking)

But the 7 microseconds has absolutely nothing to do with any object moving at the velocity of light – it is just the adjustment for time for the satellite’s velocity as applied to Special Relativity. The 7 microseconds is a portion of the 38 microseconds (45 – 7 = 38). It is part of a daily adjustment. So it is wrong to multiply the 38 microseconds by the velocity of light, (Calculation ii) because the amount of 0.000007s only concerns the 4000 m/s of the satellite per day. Also Clifford Will’s calculation which everyone is copy-pasting is clearly incorrect because the velocity of light of the signal only exists for 0.1 seconds not 86400 seconds. It only takes about 0.1 seconds for the signal to traverse from the satellite to the Earth.


Scenario 3E:
Double-check and Triple-check Everything

But wait a second. We have seen so many misconceptions thus far. It pains me to think that the initial calculations for Relativity may in fact not be correct. Where did I get those notions that the GPS satellite clock gains 0.000045 seconds per day because of General Relativity and loses 0.000007 seconds per day because of Special Relativity? Well, many websites all make these claims. So I’ll just tag it to the website with the best Google ranking:

The Ohio State Website combines the two adjustments for Relativity so that 45-7 microseconds equals 38 microseconds. We have corroborated earlier that the 7 microseconds seems to fit the formula for Special Relativity for the movement of the satellite. So let us plug the numbers into this formula for General Relativity:


The formula above (vi) uses ‘a’ for the acceleration of gravity, ‘H’ for the height, and ‘c’ is of course the velocity of light. So if we work those numbers into the General Relativity formula for time adjustment, then we get a proportion of about:

1.0000000022 … (vii)

This is the answer I get for an altitude 20 000km at 9.8 meters per second squared. This is a proportional adjustment for time because of General Relativity. The proportion is determined for a distance directly above a gravitational source for one second of time, so it must be multiplied by 86400 seconds for one full day. The result is 188 microseconds per day adjusted.

Do we include the radius of the Earth and make it about 27 000 km? I don’t think so at all because gravity is at its highest on the surface of the Earth. And if we considered two other points, one below the surface and one above the surface, then reflect on all three results we do not get the slowest time at the Earth’s surface. We should get the slowest time at the Earth’s surface because gravity at the surface is strongest.

This point seems quite clear because there can be no actual force of gravity at the center of the Earth. (You may need to think about that one a bit). No gravity at the center of the Earth equals the fastest time, not the slowest.

I see so many differing answers online, so I try the simplest formula:


Formulae vi and viii give the same proportion of 1.0000000022 which yields 188 microseconds per day. Hmm… Well 188 microseconds is not 45 microseconds!

Scenario 3F:
Correct the Corrections for the Satellite

So now that we have somewhat corrected the proportion for General Relativity, we must amend the results because Calculation i still yields the incorrect answer from the online copy-paste of 45 microseconds.

0.000188s x 4000m/s = 0.75m … (ix)

An adjustment of 75cm per day would be for General Relativity for the satellite moving at 4000m/s, whereas for Special Relativity the daily adjustment for the satellite in the other direction is:

-0.000007s x 4000m/s = -0.028m … (x)

So many discrepancies! You’ll have to do that entire math yourself to be certain!

Scenario 3G:
Time and the Velocity of Light

But how is time supposed to be adjusted for the signal between the satellite and the Earth for both the Relativities? (The previous scenario was just the satellite)

Let us look at General Relativity first. If time is faster for a satellite 30 000km away where gravity is less, then the duration for the signal sent to Earth could be differently measured. Ordinarily this signal should take 0.1 seconds to reach us. (The height is 20 000 km when the satellite is directly above us, so this fraction can vary. I use 30 000km for convenience as the signal often has to traverse this distance when not directly above the target, but its height is 20 000km above the surface gravity).

We need to use the proportion of 1.0000000022 (from formula vi or viii) which is multiplied by the 0.1 seconds. This 0.1s is how long it takes the signal to travel the convenient distance of 30 000km. We do not use the 188 microseconds here because that is the amount for an entire day.

So if we adjust the signal for General Relativity we get:

1.0000000022 x 300 000 000m/s x 0.1s = 30000000.066m … (xi)

So the adjustment for the signal is 33mm. We halve it from 66mm because gravity increases for the signal as that signal approaches the Earth. So the effect of General Relativity is on average half for the signal as it is for the satellite. The satellite has a constant amount of General Relativity acting on it because its height is constant. The height of the signal changes as it approaches the Earth.

Scenario 3H:
Special Relativity for the signal

Everything is still only an adjustment so far of less than a meter for the satellite, and the signals themselves. But I have saved the best calculation for last. What about adjusting the time for the signal from satellite to Earth for Special Relativity? This is where it gets really interesting. If you have studied the previous chapter then you will know the answer.
(See Chapter 27, Part 4, The Contraction of Space and Time Catastrophes)

If an object is moving at the velocity of light, then according to Special Relativity – time has stopped. I know this is ridiculous. But that is what Special Relativity tells us. Nothing that is traveling at the velocity of light is actually supposed to be moving at all according to Special Relativity. Consider this formula:

Relativity GPS (xii)

Formula xii is the same as Formula iii which gave us the 7 microsecond adjustment for Special Relativity for the satellite per day in Calculation v. In the formula if the velocity ‘V’ is equal to 1 (which is the velocity of light) then there is a division by zero and time is infinitely slowed.

t = infinity … (xiii)

It is not my theory, don’t blame me! I’m just trying to fix it. (You will need to go back to Chapter 27: Light and Spin, if you want to understand these references further).

We may have been tempted by all the previous calculations to consider that Relativity is still viable. But if we take those calculations at face value, then we should take the Special Relativity calculation for the signal at face value as well, and we get the contradiction that the signal does not actually move at all!...

  pg 5  


This is an extract summary
The full Chapter 28 is here:





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