Time Dilation, Length Contraction and the Bending of Light

Abstract:

Fundamentals of light have been the origin of many abstract and unique phenomena in our universe, like time dilation, length contraction and gravitational lensing, to name a few. Multiple experiments, along with deductions from astronomers and scientists like A. Einstein, P. Fermat and more, have been able to provide a cohesive understanding of these happenings. This paper thus goes to review and put together the correlation and logical overlap of such topics, from the bending of light (refraction) in contrast to the “bending” of light in space.

Keywords: Bending of light, General Relativity, Eddington experiment, Huygens's Principle, Time Dilation, Length Contraction

Introduction:

The ground-work:

Do you think you can see an object that is directly behind a larger, more massive one? Even if you do, you would assume it’s a trick of the eye. Well, this is what Sir Arthur Eddington, in 1919, observed during an experiment about a solar eclipse on the coast of West Africa.

A few years earlier, back in 1916, when the revolutionary paper of Albert Einstein was published, titled The Theory of Relativity, many people of science saw there was no way to prove this at-the-time incomprehensible thought. Sir Eddington was not one of them. He was interested in Einstein’s theory given its various implementations in the field of science and astronomy, and Eddington decided to prove it. Sir Eddington and his fellow, Frank Watson Dyson along with other astronomers, travelled to the coasts of Africa and Brazil to determine the credibility of the revolutionary theory, realizing the solar eclipse of 1919 would be a perfect experiment.

This experiment helped Eddington show how crucial the bending of light could be measured because of the Sun's gravity, in accordance with the space-time mentioned in Einstein’s theory. This helped the theory as well since it predicted a deflection which was more than what Newton’s laws of gravitation could predict. The latter also mentions in his 1704 publication Optiks that bodies don’t affect light at a distance, and neither do they bend its rays. But what happened in that experiment?

The Sun would be directly in front of Hyades, a star cluster from the constellation of Taurus. And because of the light-bending effects of the Sun implied from Einstein’s theory, many stars along with Hyades would be visible for observation. The astronomers on this expedition would try to measure as many stars as they could, but due to unfortunate circumstances and wartime scenarios, both Eddington and Dyson partially failed in completing the tests. Their efforts still established Einstein as a crucial part of astronomy and physics and bring us to a fundamental question: how did they actually find this experiment useful? The crucial component in all that encapsulates this starts with light and one principle from Pierre de Fermat that defines the basics.

1. Fermat’s Law

To bind all properties of light together, Fermat’s principle came forth as a foundational base back in the 1630s. He stated, Light travels in a path that takes the least amount of time.

The basic principle(s) of a light wave is that it travels in a straight line and slows down in denser mediums. So in the case of a singular medium, light will travel in a line taking the least time to reach from point A to B. But then, considering two different mediums, we observe light breaks the above consideration. It bends at the boundary between those mediums. This was the fundamental principle that all other theories depended on, even till date.

Taking our case study of Sir Eddington and his experiment, given that light follows this principle, the objects behind the Sun should not have been visible at all, but this principle ends up being violated in the general case of refraction and, incidentally, gravitational lensing.

a. Newton’s Corpuscular Theory

Newton again back in the 1700s stated light is a ray of small corpuscles, or particles that always travel in a straight line. Then how would a stream of such particles find the shortest path in another medium?

When these corpuscles hit a surface at an angle, they return back from the surface at the same angle they hit due to an assumed elastic nature. That is, they reflect.

In reflection the angle of reflection is the same as the angle of incidence. This is because the particles having an elastic nature, create a repulsion between them and the surface. Opposite could be said for refraction, where these particles have an attraction towards the surface of the different media, which causes their speed to change (increase) and travel faster with a bent angle.

Incidentally, there were merits and demerits of this theory.

Experiments found out that the speed of light decreased in the denser medium, and the different colours of light were not because of the different sizes. It was also found the glass partially reflects and refracts light. It also failed to explain why interference and diffraction occurs.

But along with Newton, for questions that couldn’t be answered with particle nature, Christian Huygens proposed that this light could also have a wave-like nature.

b. Huygens's Principle

Imagine the light source as a single point and a “wavefront” as a spherical layer being emitted instead of being a stream of particles.

Now with this layer, if every point on that layer emits a wavefront of spherical form, those smaller “wavelets” having the same phase, when summed up or connected by their tangents, create the next wavefront. The following diagram visually represents the phenomenon.

Fig. 1: Huygens's picture of how a spherical wave propagates; each point on the wave front is a source of secondary wavelets that generate the new wave front.

2. Snell’s Law

To mathematically establish how much light actually bends in these regions, we can derive Snell’s law from the Fermat Principle, as explained in the Methodology section.

fig. 2: : Light ray hitting a boundary between the air and glass media with their refractive indices mentioned.

Without the derivation, we know Snell’s law states an equation that can help us get a relation between the angle of incidence and refraction, with the parameter of the refraction index.

Here, 𝜃1 and 𝜃2 are incident angle (i) and refraction angle (r) respectively.

n1 sin𝜃1 = n2 sin𝜃2

Snell’s law arises from the wave nature of light, where refraction of light is best explained mathematically.

Since energy must be preserved in all cases, even at the boundary of the mediums for refraction, the frequency of light must remain the same throughout as well. The speed only changes due to the resonance between the electrons of the material of the medium through which light is undergoing refraction. This equation helps us experimentally conclude as well that light is bending in a comprehensible medium, such as the atmosphere.

3. Why and how does light bend?

We have enough pieces of this map of how light can essentially travel around objects which can get to the crux of light bending. Take a look at the figure below. Do you see a correlation between Huygen’s and Fermat’s principle?

The light ray (from a distant source) travels towards the boundary of the different media. The wavefront(s) constituting the ray hit the boundary from first left to right, causing minor wavelets to be formed on the interface. Summoning up or joining these wavelets tangentially gives us the new wavefront, therefore giving us the “bending” of light.

fig. 3: A light wave hitting a boundary between two media with the wavelets creating a new wavefront.

This is how a light can bend in a medium like discussed above. The tangent, or the summation of the smaller wavelets, when the wave hits the boundary helps us find the “bend” that light went through. This is essentially light doing this by itself with no external factors or influence in any way.

In vacuum, like in space, it's easier to understand that light can propagate as a wave. The electromagnetic fields moving perpendicular to each other almost always undergo an internal diffraction. This diffraction is what causes light to slowly spread out and "bend." This effect is seen in light emitted from any source, e.g., lasers. The bending of light here is caused by internal diffraction, where if it meets an obstacle, it spreads out more and can reach around the object.

a. Diffraction of Light

A well-known example of diffraction is about a slit through a surface. Famously known as Young’s double slit experiment in case of two slits.

The duality of light, wave, and particle is what causes light to pass through the slits and interfere with itself. The light wave diffracts through the slits, and its particle nature is what the incident screen absorbs, hence light “falls” on it. Diffraction is also proved by Huygen’s principle mentioned above.

Fig. 4: The interference pattern is caused by the superposition of overlapping light waves originating from the two slits.

Though this happens only when the wave interacts with something as small or smaller than the wavelength of light. Which also means that light can undergo more evident diffraction if the wavelength of it increases.

Then what happens when light does, in fact, interact with an obstacle or an object in its way? It does undergo internal diffraction, which can be “altered” in some way depending on the kind of object it encounters.

If the object is made up of a conducting material, the EM fields in the light ray start to oscillate, and these oscillating electric currents create more light. This part of the light ray that hits a conductive material travels as a surface wave, therefore can be seen bending around the corner of the object by riding the curved surface of the object.

B. Space-time and light:

Space-time and the gravity of this situation

Light as a wave helped us understand that it diffracts, but the particle nature also has its advantages. Sometimes, in the case of matter or objects, this particle nature helps us find out how light behaves.

Light has no mass, but this is true only when it’s at rest. Instead, light, or photons, to be exact, have mass when they are moving at the speed of light. Which can be called relativistic mass.

Any object that has mass attracts other objects.

This is another fundamental theory given by Newton that formulated better answers about all matter around us. But how does this affect the case of a space-time field? Matter, or mass, to be exact, warps or creates sort of a dent in space-time. This is described in General Relativity, given by Einstein.

Postulates of Special Theory of Relativity

The laws of physics are applicable in all inertial frames of reference
The speed of light is same in all frames of reference

Postulates of General Theory of Relativity

The special theory of relativity controls local physics
There is no way to spot difference between gravity and acceleration

Greater gravitational attraction results from larger objects in space, such as stars, galaxies, etc., making their dent in space-time more noticeable. And thus, similarly happens with light. When it gets closer to such an object, space-time (acting as a lens) bends this ray of light. But if light is massless, then how would it get bent in space-time?

Like mentioned above, light has relativistic mass only when it's moving. Not just that, but GTR implies that the warping of space time, or the “gravity” of the dent, is proportional to the matter density than its "mass” which was termed so for the sake of a simpler explanation. Thus, the more dense an object is, the more the light ray(s) will be deflected from their path. Similarly, you could also say that this deflection happens due to refraction as well. When light enters the medium of an object’s atmosphere or surroundings, it undergoes refraction and hence gets bent.

5. Time Dilation and Length Contraction

Time Dilation:

Einstein created the theory of general relativity to understand accelerated frames and, with gravity, a suitable source of acceleration. Einstein's special relativity explained the concept of the slowing of a clock, which is observed by an observer who's in motion with respect to that clock. Einstein's special relativity theory states that time can pass at distinct rates in different reference frames. It is the experience of time passing slower for an observer who's in relative motion to the other observer. The difference within the lapsed time measured by clocks (different frames of reference) gives time dilation.

And from the above observations, time dilation can be defined as follows:

Clocks moving relative to an observer run more slowly compared to the clocks that are at rest relative to the observer. This slowing down of time is called time dilation.

Length Contraction:

Proper Length:

L0 (proper length) is the distance between two points measured by an observer who is at rest relative to both of the points.

Fig 5: The Earth-bound observer sees the object (μ) travel 2.01 km between clouds

If we measure the length of any object moving relative to our frame, we find its length L to be smaller (shorter) than the proper length L0. At relativistic speeds, close to or approaching the speed of light, distances measured differ when measured by different observers. Thus, according to Einstein’s Theory of Relativity, length contraction is the phenomenon in which the length of an object is measured to be shorter than its proper length measured in the rest frame.

The contracted Length L is given by:

How does this bend light again?

Light isn’t the only thing influenced by gravity.

Measuring the clock time for moving objects, as discussed earlier for Time Dilation, is dependent on the velocity of observers relative to each other in their frames of reference. But for objects at a much larger distance, the light rays also undergo a redshift because of the gravitational field present. More specifically, the light ray emitted and received at points on different gravitational potentials will undergo redshift (increase in wavelength, decrease in frequency), even without any relative motion between them.

Which sounds impractical to think about given that an observer receives fewer wave crests (redshifted) per unit time of the light ray. This does not technically happen; instead, the light undergoes time-dilation again, but this time it is due to gravity. Instead of light being received, we can think of clocks measuring the time of light travel at different gravitational potentials. Again, even though the clocks aren’t in motion relative to each other, they still measure different time values and run at different rates.

This gravitational time dilation also implies that we can measure the change in the speed of light, and to clarify such, we know that the speed of light can be changed because of a varying index of refraction. Which leads to an analogy that there is an “index of refraction,” which is affecting light when there is a gravitational field present. But let’s consider this is in vacuum (i.e., space), then there is no change in the “medium” for an index of refraction to be present. This means we can presume this index to be a position-dependent function.

Now to calculate the approximate deflection that the change in the refraction index causes, we can use the displacement of the light ray in the gravitational potentials, along with the time interval it takes. This gives us the speed of light (universal constant) according to the local proper time. Incidentally, the speed of light seen by an observer will change as the gravitational potential changes in space. Combining these changes in the gravitational potential and the speed of light, we can introduce a refractive index for space.

Furthermore, using Huygen’s construction of wavefronts in a transverse gravitational field, we can imagine a light ray creating wavelets at each time interval. Any change in the gravitational potential can cause a change in the wavefronts as well, i.e. wavefront bends. This change also causes a difference in the speed of light, which can help us derive the angle of bending in the overall calculations.

The same can be said for Length Contraction as well in a gravitational field. An object at a higher gravitational potential can be said to have a “longer” length than the one at a lower gravitational potential. Since this can only happen in the direction of which it is travelling, objects in space that have a larger gravitational influence can expand more of the moving object’s length.

7. Proving the theories of relativity & Gravitational Lensing

Time Dilation and Length Contraction are the effective consequences of Special Relativity’s postulates.

Taking STR and GTR postulates into consideration has helped astronomers and scientists calculate parameters that seemed beyond the scope of technology and theories of the time, as well as aided in discovering new and better solutions and concepts. One of such prominent effects that shaped the research in the field of astronomy and cosmology is Gravitational Lensing. It’s a rare case that can be observed by only the highest grade of telescopes due to the phenomenon being faint and distant.

Gravitational lensing occurs due to the object or celestial body curving spacetime, and incidentally, like discussed above, bending light that travels around it. This phenomenon affects light almost as if it's going through a lens. The result of this effect is that a distorted image of the light source can be observed as either a ring (Einstein ring) or a cross when seen through high-grade telescopes.

This is what Arthur Eddington and other astronomers observed and proved Einstein’s theories.

Methodology:

Derivation of Snell’s law from Fermat’s principle

The time to travel between the two points is the distance in each medium divided by the speed of light in that medium.

2. Time Dilation

To understand time dilation better, let’s take a simple example of a light clock. The clock functions when a beam of light is emitted from a source/clock and travels to a mirror, then is reflected back to the source. And let’s say that two observers A & B are recording the event.

The positions are represented in the image given below:

Fig. 6.(a): A is in motion & B is at rest both w.r.t ground.

For observer A, the light moves in a straight line, while ground and observer B are going left with velocity v. If the distance between the mirror and source is d, then the round-trip distance the light travels is 2d. Since the light travels with speed c, the one-way travel time tA as measured by observer A is:

tA = d/c

Will the time interval for B be the same? Let's see.

Since the mirror is moving to the right with velocity v, the path of the light will appear to be a diagonal line for B as shown in the image 1.(b) given below. So the light is travelling the same distance, but its speed stays the same (c), as stated in the special theory of relativity postulate.

Fig. 6.(b): Observation of motion of clock & path of light by observer B.

Now, let's find the time taken by light from B’s perspective by applying the Pythagorean theorem to the image above:

This is the time to go one way from the source to the mirror. The return time is the same, so the total time is 2tB. We can relate this back to the time in B's frame of reference:

The denominator is smaller than one, the time taken, so the time interval measured by observer B will be longer than that of A.

3. Length Contraction:

Derivation:

Consider a cosmic ray colliding with a nucleus within the higher atmosphere of Earth and producing a muon; then, to the Earth-bound observer, the speed relative to the muon is given via

V = Lo/Δt

because the object is in motion relative to this observer, the time (Δt) is relative to the Earth-bound observer, and the speed relative to the observer's velocity is given as

V = L/Δto

The moving observer observes the correct time Δto while he travels with the muon, and as a result, both the velocities are same; consequently,

Lo/Δt = L/Δto

As we know that Δt = γΔto, into the connection substituting this equation offers:

L=Loγ

ultimately, we get an equation relating the distances measured with the aid of other observers via substituting for γ

The shortening of the measured length, length of contraction (L), of an object moving relative to the observer’s frame of reference is given as:

L=Lo√(1−v2/c2)

These formulae and derivations are simplified for ease, but the depth of analysis proves much more in the grand scheme of all: time dilation, length contraction, and the consequence termed gravitational lensing. Serving as a revolutionary point, Einstein’s theories captured the newer angles in solving the queries and ideas that come with understanding space-time, Eddington paving the way for clear reasoning, and correspondingly, what we find out at our smaller and simpler levels, provided centuries ago, can help us connect the bigger dots centuries later. Light is a sensitive concept with multiple views to put together a cohesive answer about its nature, and its effects are no less complex. Through these ideas that we can comprehend, can we analogously get towards higher levels of estimations, calculations, and phenomena?

Conclusion:

The study of light and spacetime has modified our view of the universe. From Fermat’s principle to Einstein’s general theory of Relativity the journey has certainly been one of human curiosity. A key moment in that journey was 1919, when Sir Arthur Eddington observed in the course of a solar eclipse that light bends around massive objects, and that was the primary experimental evidence of spacetime curvature. That not only confirmed Einstein’s predictions but introduced gravitational lensing, the cornerstone of modern astrophysics.

Gravitational lensing has since become a key tool to uncover the universe’s secrets, like dark matter, distant galaxy properties, and the early universe. Time dilation, length contraction, and relativistic mass further explain the relationship of light, gravity, and spacetime and open up new areas of theoretical and applied physics.

The evolution of our understanding from wave theory of light to wave-particle duality within a relativistic framework indicates the collaborative and iterative nature of science. These discoveries not only assist us in understanding cosmic phenomena but additionally lead to technological and methodological advancements and lead us closer to the secrets of the universe. Through the lens of light we keep to illuminate the vast, unknown universe we call home.