A Scientific Conceptualization of an Incredible Phenomenon: Time Dilation

An exploration of Time Dilation, a concept first posited by Albert Einstein in 1905 as part of his new theory of special relativity.

Oscar Petrov
11 min readMar 3, 2022


This paper deliberates on the theory and actuality of a mind-boggling phenomenon referred to as Time Dilation, as well as its underlying effects. The paper begins with a brief introduction to the history of physics, in which it discusses the deviation from the regularity of determinism in Newtonian Physics (also, Classical Physics) — which brought primary emphasis on macroscopic situations — during the 20th century, to what became now known as (Einstein’s) Modern Physics — which is principally composed of relativity and quantum mechanics. Lastly, the paper explores in great depth the effects of Time Dilation and its significance in physicists’ understanding of the physical and natural world. The body of the paper works to elucidate some of the counterintuitive facets that surface when quantum phenomena and relativistic effects are jointly considered. If it is of worth to the reader, do take notice of this famous quotation:

“Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”

—Neils Bohr (1885–1962)


At the start of the 20th century physicists began to adopt a different understanding for how the world works, espousing the idea that physics could be thought of as a space-time manifold, an empty “stage” in which events take place. Long before the turn of this century, the Newtonian physics of the classical era had dominated the status quo, always necessitating itself when a macroscopic situation came about — although doing so with impressive accuracy. But such classical mechanics was no longer viable in a changing world, for it was only a mere “approximation to the truth.” For once physicists started probing lesser distances but with higher energy masses, there were many flaws and discrepancies that could not be explained by such Classical Physics; Some of the most notable of these, included “Blackbody Radiation,” the Photoelectric Effect, the Hydrogen Atom, and Compton Scattering. It was such incompleteness regarding many of the classical (c. 16–1900s) theories which began to necessitate a newer and more novel approach to the world of physics (i.e., in some aspects of the field, at the very least).

In his 1905 paper On the Electrodynamics of Moving Bodies, Albert Einstein reforms this status quo with his newly developed Theory of Special Relativity. This theory was a major breakthrough in the world of physics, for it was responsible in establishing an interconnection between space and time (space-time), as well as its relation to inertial objects. Albert Einstein’s findings allowed him to conclude that when objects near the constant speed of light, time dilates and length contracts, an incredible discovery in the world of physics. Einstein based the reasoning of his paper on two main postulates. The relativity principle and the constancy of the speed of light. The former dictated that the laws of physics are the same for all inertial (non-accelerating) frames of reference (observers); and the latter stated that the speed of light within a vacuum is an invariant (unchanging) — a physical constant with a value of 299,792,458 meters per second — and is entirely independent of the relative motion of an observer (or source). (Also of importance to note is that the term “special” denotes that these postulates hold true for inertial reference frames only.) Such postulates at the turn of the century — having been essentially repurposed from that of previous physicists, Newton and Galileo, respectively — ultimately changed the way that we humans understood the physical world.


In October 1971, American astronomer R.E. Keating and physicist J.C. Hafele conducted an experiment to test the effects of time dilation.

They brought four cesium-beam atomic clocks aboard commercial jet flights and flew the clocks around the world twice — one clock traveled eastward and the other westward — in order to test a hypothesis first posited by Einstein’s special relativity. Aiming to measure the relativistic effects of moving at such high speeds on the atomic clocks (which were incredibly accurate devices) with this experiment, Keating and Hafele sought to analyze differences between the moving clocks on the commercial airliners with that of the stationary clocks at the United States Naval Observatory.

They predicted that the clock flying eastward would lose 40 nanoseconds, whereas the clock flying westward would gain 275 nanoseconds. To compare this to human-like terms, “1 nanosecond is to one second as one second is to almost 32 years.”

“During October, 1971, four cesium atomic beam clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein’s theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip … Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanosecond during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock “paradox” with macroscopic clocks.” (J.C. Hafele and R. E. Keating, (1972))

The results of the experiment denoted that, in comparison to the stationary clocks at the observatory, the eastward-flying clock lost 59ns while the westward gained 273ns. While there was a “discussion” regarding the “accuracy and therefore reliability” of the results, this experiment is nevertheless often regarded as one of the first experiments to yield physical proof of time dilation, and its effects.


In his theory of special relativity, Einstein explains how space and time affect inertial objects that are moving in a straight line (in a vacuum, of course). He expounds on the concept that that time is relative for all observers, and how it passes at different rates for objects that are in motion in comparison with objects at rest. He proposes the following time dilation equation (which will be derived and explained, shortly):

Time Dilation formula.

The above equation quantifies time dilation, by explaining the mathematical relationship between time, velocity, and the constant speed of light (with t-naught denoting the proper time, and t denoting the improper, and dilated, time). Something of noting importance is the fact that time dilation has a more substantial effect at speeds close to the speed of light, c.

The mathematics of time dilation effect is explained in the following example. Looking at Figure 1 below, we denote that the observer on the left (i.e., on Earth) is stationary and the observer on the right (in the rocket) is moving relative to the stationary observer.

Figure 1 (above): illustrates the concept of Time Dilation. Source.

The stationary observer has a light clock, a device that beams pulses of light up to a mirror and then reflects the light back down to a detector to denote 1 “tick” of the clock. The time it takes to travel up and down the mirror is t-naught. Since the speed of light is ‘c,’ the distance that the beam of light travels would be equivalent to:

In other words, we can say that the proper time denoted by one “tick” of the clock is given by:

Equation 2.

The moving observer also has a light clock, which also sends light up to a mirror which then reflects the light back down to a detector to denote 1 “tick” of the clock. From the perspective of the observer on Earth, it is evident that the pulses in the moving observer’s light clock have to travel a greater distance compared to that of the stationary observer — as the light clock is moving with a velocity, v•∆t, in the x direction. Since the speed of light is constant, and the moving observer’s light beams must travel a greater distance when viewed by the stationary observer, it is reasonable to conclude that the beam of light would also take a longer time to be detected, for d = s • t. This is in accordance with the fundamental theorem of Pythagoras (which states that the hypotenuse of a right triangle is always the longest side), and it is further depicted back in the diagram in Figure 1 with red-dotted lines. One “tick” of the clock for the moving observer, relative to the stationary observer, would be quantified by the following equation:

Equation 3.

Quantifying the distance that the light must travel for the moving observer relative to the stationary observer, we obtain, through applying Pythagoras’ Theorem:

Equation 4.

Where the quantity, ct,(above) represents the length, L, that the light travels. By substituting equation 5 into equation 3, we are able to obtain:

Equation 5.

Rearranging the equation for proper time to isolate “d” yields:

Equation 6.

Now substituting equation 6 into the equation 5, we get:

Equation 7.

We can now square both sides, rearrange, and simplify the terms to make the observed time the subject of the formula:

Equation 8.

Simplifying further, we get:

Equation 9.

Some algebraic manipulation of equation 9 yields the following relationship:

Equation 10.

Finally, in making ∆t the subject of the formula, we conclude with the following time dilation formula:

Time Dilation formula.

The time dilation equation, which represents the amount of time between the ticks of two separate moving clocks, can also be analyzed in respect to the Lorentz factor, or γ, in the following equation:

Equation 11.

Where the Lorentz Factor is equal to:

Equation 12.


The following concerns the Lorentz Factor, as shown in Equation 12, which represents the factor by which proper and improper time differ.

As seen in the table in Figure 2, time dilation does affect us in our everyday, but at such an inconceivable rate. Selecting a speed just under the fastest speed that humans ever reached (24,791 miles per hour, or 11083 meters per second during NASA’s Apollo 10 mission in 1969) yields a dilated time value of 1.00000000056, a difference on the order of -10, or a ten-billionth.

Figure 2: shows an interpersonal connection regarding the effects of time dilation on humans.

This goes to show the imperative axiom that unless the velocity of the object in motion (i.e., relative to an inertial observer) is a considerable fraction of the speed of light, the dilation effect on that object is practically negligible as γ would surmount to be nearly 1, and thus proper and improper time will be differ negligibly.

Through this, it is clear why time dilation plays such a minor role in the lives of humans in today’s age. But while these effects are inconceivably small, they are nonetheless real and verifiable, as previously explained in experiments utilizing atomic clocks.


Erroneously referred to by many physicists as “nothing but an illusion,” Time Dilation is, in fact, a real phenomenon that has been continuously supported via many experiments. Not only is it key to understanding this natural phenomenon for helping some of science’s greatest mysteries — especially that which relate to Relativity and Simultaneity which had been “plaguing other scientists for many years” — but it also allows us to correct some of science’s greatest misinterpretations.

Einstein’s Special Theory of Relativity ultimately accomplished a number of things. It established a relationship between space and time, otherwise known as space-time; It repurposed Maxwell’s equations for electromagnetism in a different manner; It declared that Newtonian theory was useful in analyzing macroscopic events, in which the Lorentz factor was surely negligible (or incredibly close to 1, due to the velocity of the object being minute in comparison to the speed of light), but that it wasn’t entirely accurate; and lastly, but not least, it corrected some of the most important misunderstandings of the earlier centuries.

It is for this reason that Newtonian theory and laws — which is evidently an extremely accurate approximation for macroscopic events — are gradually being replaced by ideas of this new general relativity, although I would estimate that such Newtonian laws would persist for yet a long time as they are more than accurate enough in the scope of our own lives.

The defining aspect of the body of this paper serves as a discussion of physics, as opposed to an explanation. It serves to demonstrate and evaluate the effects of a newly conceptualized phenomenon. The work seeks to propagate a new perspective of the world in which we live in, to propose an alternate convention to the world of physics. One might compare it to the often communed analogy regarding time, in which humans perceive Time to be a river, flowing down a gradient and in one direction, stopping and slowing down for no one, and “sweeping everything and anyone along with it.”


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Oscar Petrov

A curious manifestation of billions of exploding neurons. I like to think about brains + the universe. Also passionate about ethics, philosophy + human rights.