Durango Bill's
Relativity and your GPS System
You might be thinking "What does Einstein's special and general theories of relativity have to do with a GPS system?" The answer is your GPS system would not work if the engineers who designed and built your GPS system had forgotten to include Einstein's equations in their design.
We take it for granted that your $100 (or more) GPS receiver will tell you where you are, nevermind which direction you should go in order that you will arrive at your intended destination. If you are hiking out in the wilderness, you depend on your GPS to find your way back to your car so you can get home. If you are taking a car trip, you depend on your GPS to show you which roads to take when you come to an intersection. If you are taking a transoceanic airplane trip and the only thing you can see is blue ocean water, you will be more comfortable if you understand that the pilot can simply press a button to eventually find an airport - specifically the airport that matches what is on your ticket.
To tell you where you are, your GPS receiver has to lock on to and receive information from at least four GPS satellites. It will use these 4 satellites to solve 4 equations - that have 4 unknowns. These unknowns are: your latitude, your longitude, your altitude, and the difference in time between the clock in your GPS receiver and the clocks in the 4 satellites. (It is the job of the GPS satellites to keep their clocks in sync.) The satellites in turn have to stay in sync with a ground control station. The bad news is that even if you started with identical clocks in your satellites and your ground control station, once you put the satellites in orbit, they will "tick" at different rates than the ground control station - and this is where Einstein's equations come into play..
Einstein devised 2 theories of relativity. Special relativity determines what happens to the relative clock speeds between 2 clocks that are in relative motion to each other. General relativity determines what happens to the relative clock speeds between 2 clocks that are at different heights/positions in a gravitational field. The effects of both can be calculated.
The amount that GPS satellite clocks have to be adjusted has been well discussed. For example see Real-World Relativity: The GPS Navigation System . The above article explains that because GPS satellites are moving in orbit around the earth, their clocks will run about 7 millionths of a second per day slower then an identical clock on the surface of the earth. Also, because the GPS satellites are far above the earth's surface, their clocks would run about 45 millionths of a second per day faster than identical clocks on the surface of the earth. This gives a net 45 - 7 = 38 millionths of a sec. gain per day. What isn't given is details of how fast the GPS satellites are moving, how high they are, how these numbers are calculated, etc.
The table below will show these results and give some idea how the data is calculated. The table also shows what adjustments would have to be made to your satellite clocks even if your GPS satellites were set up for some other altitude. (Actual GPS satellites orbit at about 20,186 kilometers above the earth's surface)
For the results shown below, the author has assumed:
The speed of light in a vacuum = C = 299,792.458 km/sec.
The earth is a non rotating sphere with a radius of 6375.4 km.
The acceleration of gravity at the surface of this sphere = 9.806650 meters/sec2
Velocity at a given distance above the earth's surface is the velocity needed for a satellite to have a circular orbit at that distance.
Clock Clock
Dist.from Dist.from Acceleration Circular Escape Slowdown in Slowdown in
Earth's Earth's of gravity Orbital Vel. at millionths of millionths of
Surface in Center in at this dist Velocity this ht. a sec./day a sec/day for
kilometers kilometers (in m/s^2) in m/s. in m/s. for Velocity escape vel.
-------------------------------------------------------------------------
0 6,375.4 9.806650 7.907.04 11,182.25 -30.05 -60.10
100 6,475.4 9.506099 7,845.75 11,095.57 -29.59 -59.18
200 6,575.4 9.219157 7,785.86 11,010.87 -29.14 -58.28
300 6.675.4 8.945013 7,727.32 10,928.09 -28.70 -57.40
400 6,775.4 8.682917 7,670.09 10,847.14 -28.28 -56.56
500 6,875.4 8.432175 7,614.10 10,767.97 -27.87 -55.73
600 6,975.4 8,192139 7,559.33 10,690.50 -27.47 -54.93
700 7,075.4 7.962209 7,505.72 10,614.69 -27.08 -54.16
800 7,175.4 7.741824 7,453.23 10,540.46 -26.70 -53.40
900 7,275.4 7.530465 7,401.83 10,467.77 -26.33 -52.67
1,000 7,375.4 7.327645 7,351.48 10,396.57 -25.98 -51.95
2,000 8,375.4 5.682304 6,898.66 9,756.18 -22.88 -45.75
3,000 9,375.4 4.534777 6,520.38 9,221.21 -20.44 -40.87
4,000 10,375.4 3,702762 6,198.20 8,765.57 -18.47 -36.93
5,000 11,375.4 3.080365 5.919.49 8.371.43 -16.84 -33.69
6,000 12,375.4 2.602658 5,675.29 8,026.07 -15.48 -30.96
7,000 13,375.4 2.228035 5,459.02 7,720.21 -14.32 -28.65
8,000 14,375.4 1.928838 5,265.72 7,446.85 -13.33 -26.66
9,000 15,375.4 1.686098 5.091.60 7,200.61 -12.46 -24.92
10,000 16,375.4 1.486455 4,933.69 6,977.29 -11.70 -23.40
20,000 26,375.4 0.572977 3,887.48 5,497.73 -7.26 -14.53
20,186 26,561.4 0.564981 3,873.85 5,478.44 -7.21 -14.43
30,000 36,375.4 0.301245 3,310.27 4,681.43 -5.27 -10.53
40,000 46,375.4 0.185336 2,931.73 4,146.09 -4.13 -8.26
50,000 56,375.4 0.125417 2,659.03 3,760.43 -3.40 -6.80
100,000 106,375.4 0.035225 1.935.74 2,737.55 -1.80 -3.60
384,400 390,775.4 0.002610 1.009.96 1,428.30 -0.49 -0.98
1,000,000 1,006,375.4 0.000394 629.34 890.03 -0.19 -0.38
1.0E7 1.0+E7 0.000004 199.59 282.26 -0.02 -0.04
1.0E8 1.0+E8 0.000000 63.13 89.28 -0.00 -0.00
1.0R9 1.0+E9 0.000000 19.96 28.23 -0.00 -0.00
Only earth is considered. Other astronomical objects are ignored.
Accuracy is dependent upon starting assumptions.
Precision errors may exist in clock slowdown calculations.
The total time speed-change in clocks in our actual GPS system (about 20,186 km. above the earth's surface) = Slowdown due to the velocity of the GPS satellites, plus the (negative) slowdown due to the escape velocity at the actual satellites' height, minus the (negative) slowdown due to escape velocity at the earth's surface. Mathematically this equals -7.21 +(negative)-14.43 -(a negative) -60.10 = -7.21 -14.43 + 60.10 = about +38.46 millionths of a sec. per day.
How to calculate the above table.
Dist. from earth's surface in kilometers. Distance that a satellite is above the surface of the hypothetical non rotating surface of a spherical planet earth.
Dist. from earth's center in kilometers. Just add 6,375.4 kilometers to the above.
Acceleration due to gravity at this dist. in m/s2. An acceleration due to gravity of 9.806650 meters per sec2. at the surface of a non-rotating earth (radius = 6,375.4 km.) is assumed.
The acceleration at any distance from the center of the earth is calculated by multiplying 9.806650 by: 6375.42 / Dist. from earth's center in kilometers2.
For example: The acceleration due to gravity at the GPS satellites' radius = 9.806650 x 6375.42 / 26561.42 = 0.564981 m/s2 .
Circular orbital velocity in Meters/sec. is calculated.
This can be calculated by multiplying the acceleration of gravity at that dist. by the dist to the center of the earth in meters (Note that we use meters and not km), and then take the square root of the result.
For our GPS satellites this calculation is:
Circular orbit vel. = Sqrt(0.564981 x 26561.4 x 1000.0) (Don't forget km. to m)
= 3873.85 m/sec.
Escape Vel. in Meters/sec.
This is the minimum vel. needed to escape the earth's gravity at this altitude
= vel. of a falling object if it fell from inf. to this point.
Just multiply the circular orbital vel. by Sqrt(2)
For our GPS satellites: Escape vel. = Sqrt(2) x 3,873.85 = 5,478.44 m/sec.
Clock slowdown due to motion (vel.) of satellite.
PerceivedRelativeTime = ClockAtRestInOuterSpace * Sqrt(1.0 - SpeedOfClockSat2/C2)
= ClockAtRestInOuterSpace * Sqrt(1.0 - (3873.852/2997924582))
(Watch for precision error)
Clock speed due to gravitational position (escape velocity) of satellites
ClockAtSatPos (Do for both earth suface and sat. height)
= ClockAtRestInOuterSpace * Sqrt(1.0 - EVatSat2/C2)
minus ClockAtRestInOuterSpace * Sqrt(1.0 - EVatGround2/C2)
(EV = Escape vel. at relevant location)
(Watch for precision error)
Faster
than the speed of light?
A second subject also falls into the realm of relativity. We might ask the questions: "If an object falls into a black hole, can it fall faster than the speed of light? Is there an 'Event Horizon' such that beyond this limit, no information from a falling object can escape?"
The answer to both questions is a resounding NO!
From Wikipedia: "Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (c). The special theory of relativity implies that only particles with zero rest mass (i.e., photons) may travel at the speed of light, and that nothing may travel faster."
"As of the 21st century, according to current scientific theories, matter is required to travel at slower-than-light (also STL or subluminal) speed with respect to the locally distorted spacetime region."
From the National Radio Astronomy Observatory
https://public.nrao.edu/ask/is-the-speed-of-light-changed-by-gravity/
Is the Speed of Light Changed by Gravity?
The short answer is no, the speed of light that you measure locally is unchanged by gravity.
Einstein's famous equation states E = MC2.
This is actually a condensed form of the equation since M (mass) is only implicitly given. If you expand the equation, than it would look like:
E = MC2/ sqrt (1-V2/C2).
(V = velocity of the mass involved and C = velocity of light.) Note what happens as V (Velocity) becomes closer to the speed of light. If "V" approaches the speed of light, then the denominator of the equation becomes infinitely small, and the M (mass) of the equation becomes infinitely large.
And this leads to another reason why a"Black Hole" can not generate an event horizon that would pull in a falling object at the speed of light. Assume an incoming hydrogen atom was sucked in until it reached the speed of light at an event horizon. According to E = MC2/ sqrt (1-V2/C2) the incoming hydrogen atom would have an infinite mass - and an infinite gravitational field. The infinite mass would pull in the rest of the universe as fast as its gravitational field could propagate outward. However, we are still all here which implies that an infinite mass has never existed.
If you concentrated all the energy in the known universe into the engine of a spaceship, you could not cause the spaceship to reach the speed of light. Even though "all the energy in the known universe" might be a very large number, it is still a finite number.
We can investigate this phenomenon via the equation of what happens when you add two or more large speeds.
Is the net result:
speed = speed 1 plus speed 2? (Assume both speeds go in the same direction.)
Nope. The approximation is reasonably OK at slow speeds, but at very high speeds you have to use:
Net result of addition of speeds = (U + V) / (1 + U*V/C2)
U = speed of first object
V = speed of 2nd object
C = speed of light = 299,792.458 km/Sec.
For example, if you fall into a gravitational field such that you would fall at 100,000 km/sec , and then fall further such that you would think that you would acquire an additional speed of another 100,000 km/sec., in reality your speed becomes: (100,000 + 100,000) / (1 + 100,000*100,000/299,792.4582= 179,975.07km/sec
Then, assume that you fall another distance thinking that you will gain another + 100,000 km/sec.. If you use the addition equation again, you will instead have reached 233,264.10 km/sec.
Repeating the process an arbitrary number of times produces the following result:
100,000.00
179,975.07
233,264.10
264,591.64
281,668.91
290,596.33
295,161.25
297,469.08
298,629.12
299,210.52
299,501.50
299,647.02
299,719.77
299,756.13
299,774.30
299,783.38
299,787.92
299,790.19
299,791.33
299,791.89
299,792.18
299,792.32
299,792.39
299,792.42
299,792.44
299,792.45
299,792.45
299,792.46
299,792.46
299,792.46 km/sec.
You never reach the speed of light You could use any other increment of speed increase and the result would be the same. In all cases, you almost reach the speed of light, but never quite make it. The only difference would be the "how quickly that you almost reach the speed of light but never reach it" would be the only variable.
As a complicating factor, let's assume that you can move either forward or backward at variable speeds within a "gravitational black hole" and shine a powerful flashlight out toward an observer spaceship that is somewhere out of harms way. How fast would the light beam be traveling at any (and all intermediate) point(s) while it traveled from you to the distant spaceship?
Again, using the equation, let U be the speed that light leaves the flashlight and V equals your speed (either forward or backward)
Speed of Your Flashlight beam
flashlight beam Speed Speed at observer
U V Calculated Speed
299,792.46 0.00 299,792.46
299,792.46 60.00 299,792.46
299,792.46 10,000.00 299,792.46
299,792.46 100,000.00 299,792.46
299,792.46 250,000.00 299,792.46
299,792.46 299,792.45 299,792.46
299,792.46 400,000.00 299,792.46 (Impossible but threw into the eq. for fun)
299,792.46 0.00 299,792.46 (Neg.means that you are traveling(backwards)
299,792.46 (60.00) 299,792.46
299,792.46 (10,000.00) 299,792.46
299,792.46 (100,000.00) 299,792.46
299,792.46 (250,000.00) 299,792.46
299,792.46 (299,792.45) 299,792.46
299,792.46 (400,000.00) 299,792.46 (Impossible but threw into the eq. for fun)
(Note. If you, the experimenter, tried this, and if you sent light signals at defined intervals according to your clock, there would be increasingly long periods in between your increasingly red shifted light signals reaching your (safe) observer, but your flashlight beam would always get there.)
In conclusion. No matter what speed you are traveling at, your flashlight beam always makes it out of the "black hole" to the observer. The conjecture that a "black hole" prevents information from escaping is erroneous.
Finally, consider the accretion disk of material falling into a "black hole". Something that is originating in the "black hole" is propagating outward to cause the inward falling material to fall toward the "black hole". The word "gravity" comes to mind. Thus the "black hole's" gravitational field is able to escape from the "black hole". So information is able to escape a black hole.
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