Time Dilation



Image courtesy of physicsworld.com


A Brief Overview of Time Dilation with Speed

Time dilation is a term used often in physics to describe discrepancies between multiple clocks due to relativistic effects. The considerations I will show here are based on the behavior of light. Who cares, you might ask, it’s just light, how can that slow time down? Well, our very definition of time is based upon how light travels. In my undergraduate Atomic & Nuclear Physics class, our professor put it something like this (not verbatim, here):

“If we can build a clock that measures the time it takes for light pulses to bounce back and forth between two mirrors…

Image courtesy of https://www.pa.msu.edu

Then we can define that time as a second (or however long based upon the mirrors). Also, we could define that as X heartbeats per time. Now…

Image courtesy of https://www.pa.msu.edu.

But if we put this system in motion, the light has to travel a longer distance to come back to the mirrors (from the perspective of someone not in motion). Remember, the amount of time this light takes is linked to a heartbeat. So the longer it takes for the light, the longer it takes for the heart to beat”.

If we compare the path length h with the path length the light traverses, we get something called the Lorentz factor. The Lorentz factor gives us an idea of how severely time is dilating. The mathematical expression is listed below.

Image courtesy of www.stackoverflow.com

The only variable in this equation is v, the speed of the moving system of mirrors. Of course, it takes a lot of speed for one to get an enormous amount of time dilation. At just 50% light speed, the lorentz factor (γ) is only 1.15. So for every one second experienced by the observer, 1.15 seconds are experienced by the traveller. But, when we get to, oh, 99.99% light speed…the lorentz factor is 70.155. So every 1 second (day, week, year) for the traveller is accompanied by 70 times as much time experienced by the stationary observer, astounding! But good luck reaching such a speed…so far the only objects humans can accelerate to these speeds are light masses like protons. For a spacecraft: impossible with current technologies.

So what if one reached light speed? Hypothetically if one could do this, the lorentz factor would become 1/0, or infinity. So, infinite time dilation? Time standing still? What if I tell you that the lorentz factor also controls how much the mass of an object increases as it approaches light speed? So, infinite mass? Both of these considerations arise from a breakdown in our understanding. Is infinite mass possible? Well, that would require there to have been an infinite amount of mass in the universe from the get go. The physical meaning of infinity is kind of an enigma.

A great derivation for the Lorentz Factor is here:


Gravitational Time Dilation

Image courtesy of http://ffden-2.phys.uaf.edu/

As goofy as it appears, the image above is true. Two observers, at different distances from a gravitational source, will register a different “flow” of time as well. Many of you may have been made aware of this by the recent movie Interstellar, where it was referred to as “time slippage” and caused more than 20+ years of time difference in the storyline. Gravitational fields can cause time to slow down (to any observer outside the field). Remember, all things being relative, a person experiencing a slowing of time does not even notice, to them nothing is wrong. Well, nothing is wrong except the observer on the outside of the field is now aging faster.

For our purposes on Earth, the discrepancy in time between observers is tiny…nowhere near the amount from Interstellar. For example:

A graph showing time dilation effects on and near the Earth. Image courtesy of Wikipedia.

As stated previously: the closer one gets to light speed, the slower their clock will run with respect to a non-moving observer. Orbital satellites move very quickly with respect to the Earth’s surface, so their clocks are slowed down. But also (as the diagram shows), their clocks are sped up by their distance in Earth’s gravitational field. Keep these thoughts in mind, as an entire section of the site is devoted to relativity in the Global Positioning System.

A handy explanation as to an effect of time dilation is found below:


In the video, the explanation is based solely upon the frequency shifting of light. Remember, the behavior of light deals a lot with our definition of time. An observer at higher altitude will see a lower frequency light beam from the Earth.

The equation for time dilation due to a source of gravity is below:                                                                              

T =  T0 √(1-2GM/Rc2)

T represents the time experienced by an observer near a gravitational mass, T0 indicates “proper time”, time experienced by an observer at an infinite distance from the mass. G is the universal gravitational constant, M the mass of the object, R the distance from said object, and c is of course the speed of light. The term 2GM/Rc2 can never be greater than 1, as this would give us a square root of zero, which would lead to infinite time dilation.

Well, I should correct that last statement. A lot of our equations break down when we get close enough to black holes. In the instance of infinite time dilation, one would be inside the event horizon of the black hole, and no one knows what happens beyond that point. As a general rule: infinities in physics often mean that our understanding breaks down.



Tying Together

Time dilation with respect to speed and gravitation are two phenomenon of relativity. The first comes from Einstein’s special theory, and the second from his general theory. But both have been experimentally verified. It all comes down to how we perceive the behavior of light. For special relativity, it has to deal with path length for the light to travel. For general relativity, it deals with how many waves/second we count as the light stretches out climbing out of a gravitational field.


Light Cones