How to calculate the time dilation
Calculation of the gravitational time dilation
According to the general theory of relativity (GTR), physical systems that are exposed to gravity behave exactly like accelerated systems. The formula that I derived in the thought experiment on relativistic acceleration can be applied directly to the so-called gravitational redshift.
As gravitational redshift one describes the effect of the general theory of relativity that clocks run a little faster at high altitudes than clocks on the ground. This effect is known as a relativistic correction in the satellite navigation system GPS. In the calculation on the relativistic acceleration page we have now seen that the nose clock of a rocket has the length L. after accelerating to speed v around the time difference
In order to calculate the gravitational redshift for a satellite, one must also take into account that gravity decreases as one moves away from the earth. To stay in a simple example, I would prefer to assume a lower altitude at which the acceleration due to gravity (G= 9.81 m / s²) can be assumed to be constant.
If, for example, one compares the clock of an airplane that stays at an altitude of 30,000 feet for 15 hours with a clock on the ground, the result is the same time difference as with our rocket, if it is 30,000 feet long and for 15 hours with 9, Accelerated 81 m / s². This rocket would have a speed of 15 hours v= 529,740 m / s. You can now calculate the time difference between the two clocks using the above formula by calculating the speed v divides by the square of the speed of light and multiplied by the elevation of 30,000 feet. The result is a value of 53.9 nanoseconds.
A clock deviation of 53.9 nanoseconds in 15 hours can be measured well with atomic clocks. The example given here was actually carried out in the so-called Maryland Experiment 1975/76. For 15-hour flights at 30,000 feet, scientists from the University of Maryland have found a time deviation of 52.8 nanoseconds. The small deviation of my estimate from the experiment is due to the fact that the force of gravity is not exactly constant over this height. A closer analysis, done in Maryland, gave an accuracy of one-half percent.
Last change: August 31, 2007
© Joachim Schulz
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