A Precise Mathematical Proof, for a Man standing at the Equator on EarthThese factors regarding the time-rate effects of Relativity are extremely minimal for us on the Earth, but they are easy to calculate. Consider a person standing at the Earth's Equator. To calculate the Time Dilation Effect of Special Relativity: A person there "orbits" at 1,674.366 102 kilometers per hour, "orbiting" the Earth every twenty four hour. This is 0.465 101 695 kilometers per second. The simple Time Dilation math formula (given below) then gives 0.999 999 999 998 796 560 (less than 1.000 and therefore a time slowing effect) as the Time Dilation time-rate (slowing) factor for that person standing at the Equator due to Special Relativity due to the rotational speed (NOT "experienced" by him but only as seen by a motionless observer).
For the time-rate Effect of General Relativity, for that person standing at the Equator of the Earth we have a 6 378 137 meter Earth radius and the downward centripetal acceleration there is 0.033 915 785 meters/sec2. This results in the Equivalency Factor (given below) being 1.000 000 000 001 203 440 which is then the General Relativity time-rate effect factor on the person standing at the Equator being 1.000 000 000 001 203 440 (more than 1.000 and therefore a time speeding effect) (due to acceleration).
Since BOTH Relativistic effects are always in effect, we must multiply these two factors to determine the NET Relativistic effect. That is: SR Time Dilation factor of 0.999 999 999 998 796 560 times the GR time-rate factor of 1.000 000 000 001 203 440 which results in the NET Relativistic time-rate effect being exactly 1.000 000 000 000 000 000 within experimental measurement error. This is a PRECISE MATHEMATICAL PROOF that the two Relativistic time-rate effects EXACTLY cancel each other out (for the person at the Equator)
Time is passing now, here on Earth, at a DIFFERENT rate than it is on the surface of the Moon! This is due to the gravitational effects of each of the masses of the Earth and Moon, an effect of what is called the Equivalency Principle of Einstein or of General Relativity. The difference in our rate of time passage on the Earth and Moon is an easily measurable fraction of a second per year faster! (using precise atomic clocks) The universally assumed idea that time is passing at exactly the same rate for everywhere in the Universe is certainly totally wrong! The nearby presence of a large gravitational mass, such as the Earth, causes time to pass slightly more quickly! An Einstein-style thought-experiment can prove that an universal assumption in modern Physics is completely wrong!
Since 2006, I have tried to get NASA or the ESA to include a Cesium clock in a soft-landing on the Moon. In 2018, the Japan Space Agency (JAXA) intends to launch another soft-lander to the surface of the Moon. Such a simple experiment, where identical Cesium clocks on the Earth and Moon might be continuously compared, will prove once and for all whether Einstein was right about his Equivalency Principle and therefore General Relativity. My calculations, based on Einstein, indicate that the Earth clock should count about 10,976 extra ticks every hour than the identical clock on the Moon, entirely due to the difference in our gravitational fields on the surface of the Earth and Moon, and on Einstein's General Relativity and his Equivalency Principle of gravitation and acceleration. Such a simple and inexpensive experiment would instantly confirm that Einstein was right and it would also demonstrate the logical flaws in the popular assumptions regarding Time Dilation, the Hafele-Keating experiment, the Twins Paradox, alleged Time Travel, wormholes, and other fields of modern Physics.Time Dilation certainly occurs during Special Relativity, which is when there is NO acceleration occurring, but during General Relativity, that is when acceleration applies, the effect is actually the opposite.
Interestingly, in virtually all situations, BOTH effects occur continuously and simultaneously! During some parts of the popular "Twins Paradox" (fantasy), both (competing) effects occur separately, and so they each need to be calculated separately! The popular part of that story is exclusively during the non-accelerating portion of the trip, where an Earth twin sees his spaceship twin seem to be aging and moving more slowly than himself, and that part of the story is valid, due to Special Relativity due to the DIFFERENTIAL (but constant) VELOCITY of the two. Unfortunately, the storytellers in 1960 neglected to realize that the spaceship twin (also not then accelerating) sees his Earth twin seem to be aging slower than himself! Oddly enough, they BOTH see that peculiar effect, at the exact same time!
Unfortunaterly, the storytellers totally neglected that during much of that
hypothetical trip, (BEFORE the storytellers began their story) the space
traveler had to massively ACCELERATE due to the rocket engine speeding him
up (from sitting having lunch on Earth with his twin before the trip began,
up to moving at a significant portion of the speed of light and then slowing
him down to be able (to then be in what Physicists call an Inertial Rest
Frame of Reference). THIS causes an entirely DIFFERENT and OPPOSITE time effect, that of
General Relativity. THIS effect can be calculated by a DIFFERENT (but
also simple) formula, called the Equivalency formula.
Special Relativity refers to the situation where an object is traveling at a constant velocity during a trip. We have long known that it is real and that it has the time dilation effects that Einstein had calculated for it. One of the first such experiments (around 1941) involved cosmic rays hitting molecules near the very top of the Earth's atmosphere at incredibly high impact force. The impact shatters the molecules into a lot of smaller pieces, some of which are muons. We know in a laboratory how long muons exist before decaying. It is around a half-millionth of a second, (0.000 002 197 second) into yet other particles. All scientists knew that even at the speed of light, an average muon could not quite travel half a mile (0.3999 mile or 2110 feet) before disappearing as it decayed into other particles.
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But the far more interesting subject is that regarding General Relativity. Where Special Relativity dealt with CONSTANT VELOCITY, in other words, NO ACCELERATION, General Relativity is about the many situations where (constant) accelerations occur.
One specific example of Einstein became very well known. In a thought experiment, he proposed two elevators. One of the elevators is in an extremely tall building on Earth, and he cut the cable supporting the elevator carriage, where the elevator would then fall downward with ever increasing velocity, but with constant acceleration. Einstein noted that an occupant of that freely falling elevator would be floating in the elevator, along with everything else there (at least, until it hit the bottom!). Einstein put his second, identical elevator carriage in deep outer space, where there was no gravity at all.
Einstein then stated that the occupants of the two would not be able to do any experiment to find anything to distinguish these two very different situations. Relativity. General Relativity. In the one case, there was no net acceleration detected in the elevator cage because there were two effects, one of them being the Earth's gravitational field and the other being the acceleration of the entire elevator in free fall, which always exactly matched. The combination resulted in no net acceleration effect INSIDE the elevator cage. This gave an environment inside the elevator cage which was just like the situation which is true in the deep outer space elevator cage.
Einstein's next rocket thought experiment showed a little more. This involved having two identical rockets. One is sitting on a launch pad, stationary, being subjected to the gravitational field of the Earth, which we describe as 9.8 meters/second2 or 32 feet/second2. The other space ship would again be in deep space, but now constantly accelerating due to ever-running rocket engines at exactly 32 feet per second per second. Einstein noted here as well that the two occupants could never do any experiment to tell which situation was really their case. Again, an acceleration due to motion cannot be distinguished from the effects of a gravitational field.
Einstein concluded from such thought experiments that an (inertial) acceleration due to motion must be exactly interchangeable, and indistinguishable, with an acceleration due to the gravity of a planet.
As it turns out, the equations for General Relativity are astoundingly complex, and NO ONE has yet fully solved them! Some assumptions were made and some simplified approaches were developed around 1960, and some new conclusions were presented after that. Because of those assumptions, which now appear to be obviously very wrong, virtually all Physicists today agree that there MUST Be the SAME TIME DILATION EFFECT which must exist in General Relativity, as is seen in Special Relativity. The Mathematical Proof given above and he following discussion will show that they are all certainly wrong in that assumption and conclusion.
This experiment continues for one year. The occupant of the rocket sitting on the Earth is agreed as having aged exactly one (Earth) year. What about the other occupant? It might first seem that he should also age exactly one year, but there is another effect which must be considered.
During that year, which is around 31,557,000 seconds, every one of which he could MEASURE that he accelerated exactly 9.8 m/sec2, he should certainly have had Log Book entries, the latest of which should show him now traveling at a cumulative 309,000 km/sec.
This is obviously not possible, as it would indicate that he would think that he had exceeded the velocity of light, which we know he could never experience! This shows that necessarily there is a relativistic effect regarding time and distance.
In fact, there is only ONE possibility which can explain this, and other similar problems. We know that his rocket engine thrust has not changed, so HE believes it is still creating the same acceleration every second, which could continue forever. The ONLY explanation is that the time interval which HE sees as one second is now actually taking a shorter real interval of time. What HE sees as 9.8 m/sec2, and will forever see, is now actually during a SHORTER real time interval, and so the ACTUAL acceleration has become less. If you think about that, the effect on him is that he now LIVES MORE seconds of time than he used to, or that he is now living life FASTER than he used to!
That is the OPPOSITE effect from Time Dilation!
He also detects another difference in his experience. The DISTANCE to a distant target destination star now appears (very slightly) FARTHER than it used to. This effect is not quite the same as the change of time rate effect.
Consider this thought-experiment to continue for 100 years. HIS Log Book might have initially seemed to have to now show a cumulative velocity of more than 100 times the speed of light. But his DAILY experience would always be that his rocket engine was constantly providing the 9.8 m/sec2 acceleration which provided him with an environment just as he had on Earth.
After that hundred years of accelerating, he would now ACTUALLY be traveling very close to the speed of light. That means that HIS EXPERIENCE would now be altered by more than a factor of 100 in time. HIS experience would be that hundred-year-long trip of constant acceleration, but a non-accelerating observer would see him travel for about ONE YEAR instead.
The Earth observer would see the traveler in Special Relativity to experience Time Dilation, that is, aging just a few weeks in a year of observing. During General Relativity, the traveler would be seen to experience exactly the opposite, such as aging a hundred years during a year of observation from Earth.
No other description could describe his ACTUAL PERSONAL EXPERIENCE, of having a rocket engine which could forever provide an acceleration of 9.8 m/sec2, and still not ever think that he would violate the speed of light.
When looked at from the Earth observer, similar but different effects are seen. Where the TRAVELER would never had had any sensation of change in the rate of time passage, the Earth observer would certainly see it. The Earth observer might see him APPEAR to travel (accelerate) for a year, that Earth observer would see the occupant AGE by 50 years during that year of acceleration! During that one year of observed trip, the rocket would have traveled less than one light-year in distance. Remember that the traveler would age 50 years during that acceleration, and the consequence is that the TRAVELER would perceive that he has traveled a far greater distance, on the order of 25 light-years.
There are four possible perspectives (only during the acceleration):
Note that BOTH see the other one to seem to have aged FASTER than himself! This is ONLY during the acceleration (and deceleration phase(s) of the trip.
During the middle part of the trip, BOTH would see the other as appearing to age MORE SLOWLY than himself, in the well-known effect of Time Dilation. The Earth observer would see the craft cruise at constant velocity for a LONG time, nearly a hundred years, but he would see the traveler age only a few days during that long period of observations! The traveler would believe that the cruising part of the trip was rather brief, just a few days in the exact center of the trip. He would see the Earth observer age those same few days and a few minutes more!
During the deceleration to slow down to arrive at the destination, the same effects as during acceleration occur.
The NET EFFECT of all this is that ALL PERCEPTIONS AGREE that he arrives just over 100 years after he left Earth! The Traveler would consider the trip to have been 50 years of acceleration, followed by a few days of constant velocity cruising, followed by 50 years of deceleration. The Earth observer would see the trip quite differently, maybe a year of extreme acceleration, followed by nearly 100 years of constant velocity cruising, and then a year of severe deceleration. He would have lived a total of just over 100 years in watching the trip. And the Traveler, looking back at Earth, would see the Earth observer age just OVER the fifty years he aged during the acceleration, followed by a few hours LESS aging during the brief cruising section, followed by seeing the Earth observer age just OVER the fifty years of his deceleration. In all cases, there is agreement that they both aged just over 100 years during the entire trip, and that both their calendars would have shown just over 100 years actually go by.
As is REQUIRED of Relativity, all perceptions have ultimate agreement!
No one seems to have ever done such an analytical examination of the Twins Paradox trip, from all four of the possible perspectives! If they had, they would quickly have realized that the premise of a Twins Paradox is really foolish and illogical!
My presentation which proves that the very popular Twins Paradox is not true and not even possible (Twins Paradox of Relativity Is Absolutely Wrong), includes the conclusion there that GR must have Time Speeding rather than Time Dilation, exactly the OPPOSITE effect of what nearly every Physicist now assumes to be the case!
It even includes the equation I derived from Einstein's equations where it shows that the time that we see passing on Earth is NOT identical to the rate of time passage in deep space, due to General Relativity effects due to the gravitation of the Earth. However, the difference is NOT something to get too excited about! Our clocks on Earth run FAST by about one part in 300 million, as compared to how they would run in deep space! That is only less than one second difference over a period of a hundred years, or a few minutes during the entire known history of mankind on Earth!
Still, for a Theoretical Physicist like myself, it is very troubling to learn that things that I was Taught in College were not precisely true! Because it indicates that there are whole new fields of Physics which must now be carefully examined to find out what the consequences are!
1 + (a * d) / c2.
c represents the speed of light, 299 792 458 m/s2. a represents the local acceleration due to a gravitational field. And d represents a distance, which is necessarily the radius distance in that gravitational field.
Note that the Equivalency Principle is a dimensionless parameter, [m/s2] * [m] / [m2/s2]
We might consider several situations:
EE is for the situation on the surface of the Earth due to the Earth's gravitational field.
MM is for the situation on the surface of the Moon due to the Moon's gravitational field.
SS is for the situation on the surface of the Sun due to the Sun's gravitational field.
ES is for the situation on the surface of the Earth due to the Sun's gravitational field, at our distance.
EM is for the situation on the surface of the Earth due to the Moon's gravitational field, at our distance.
So we have
|Deep Space||0.00 m/s2||.||1 + 0.00 * 10-10 or|
1.000 000 000 000
|EE||9.82 m/s2||6.378 * 106 m||1 + 6.96 * 10-10 or|
1.000 000 000 696
|MM||1.62 m/s2||1.738 * 106 m||1 + 0.31 * 10-10 or|
1.000 000 000 031
|SS||274.0 m/s2||6.96 * 108 m||1 + 21190 * 10-10 or|
1.000 002 119 000
|ES||0.000006 m/s2||149.598 * 108 m||1 + 0.89 * 10-17 or|
1.000 000 000 000 001
|EM||0.000033 m/s2||384.4 * 106 m||1 + 1.41 * 10-17 or|
1.000 000 000 000 001
We can see from the chart above that a given Earth year is different for each of these situations by this number of seconds.
|Situation||warping of time in seconds per Earth year|
|Deep Space||baseline 0.000 second|
|ES||~ 0.00000 second|
|EM||~ 0.00000 second|
For an extremely compact star, the effect becomes prominent. Consider a White Dwarf star which has ten times the mass of our Sun but has gravitationally collapsed into a ball with 20 mile radius. The surface gravitational field would be great, where the acceleration due to gravity would be around four billion times that on the surface of the Sun! That is around one trillion meters per second! The Equivalency Principle then must be about 1.2 on that surface, meaning that time should be passing around 20% faster there than on our Sun or here. An even more massive or smaller White Dwarf might have time passing on its surface at TWICE the rate that we consider 'universal'! The implications might be significant. What if the surface of that 'White' Dwarf happened to actually be much cooler and RED? If WE see time there was passing twice as fast, then we would see atoms which are ACTUALLY vibrating at RED frequencies would appear to us to be vibrating twice as fast, or in other words, seeming to have wavelengths that were half as long, In other words, in the white or blue portion of the spectrum. We have made MANY assumptions regarding White Dwarfs as to temperature and many other things which might be totally wrong! Just seeing a star which appears to be of a white or blue-white color might not always be trustworthy! Even the very COLORS of stars, which we have come to totally trust for data mining, might be sometimes very misleading, just due to effects of General Relativity!
C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago