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Special Relativity

Forget everything you thought you knew about the laws of the universe. It’s all wrong.

If you’ve heard about special relativity or learned about it before, this post will be a refresher into this amazing, buck-wild phenomenon. You’ll probably learn facets about it that you didn’t know before.

And if you’ve never learned about special relativity… prepare to have your entire world turned upside down, literally. This post will blow your mind. Essentially, everything you thought you knew about how the world works is wrong.

Don’t believe me? Read on.

What is Special Relativity

Before I explain its effects, I want to briefly tell you what special relativity is.

Special relativity is a scientific theory created by Albert Einstein. It explains space and time and how they interact.

Space here doesn’t just mean “outer space”. Space means the physical 3-dimensional world you interact in. Space is akin to distance in this context.

Special relativity upended the scientific world when it was discovered. It has transformed our understanding of physics, both from a theoretical/scientific perspective, as well as a practical perspective.

It has been mathematically proven and experimentally validated many times. It is as close to a scientific truth as virtually anything is.

It may sound like I’m belaboring the point about it being a correct model for the universe. But when you read on to understand what exactly this theory says about how the universe works and its implications, you won’t believe it.

As I said, it will blow your mind and upend everything you thought you knew. It is literally the most unintuitive thing I’ve ever learned about. Which is why I’m dwelling on the point of its correctness now, while I still have your trust as the reader.

Definition: The Speed of Light

If you’re not familiar, the speed of light is defined as the speed of light waves. This is the speed at which oscillations in the electromagnetic field propagate through space. Visible light (like light from a lamp or lightbulb) travels at this speed, as do radio waves, microwaves, signals to and from your WiFi router, etc.

The speed of light is often denoted mathematically with the letter constant c. This speed is roughly 3 x 108 meters per second, or 300 million meters per second. I’m defining it here because the speed of light is integral to everything about the theory of special relativity.

What are the Implications of Special Relativity

I’m going to start by sharing the implications of special relativity to you, as these implications are, in my opinion, the most mind-boggling part.

Time is Relative

Suppose that you and I are standing next to each other on a running track. We’re each holding super-accurate atomic clocks that we synchronize to show the same exact time.

Now, suppose that you sprint a lap as fast as you can around this track while holding your clock. When you finish sprinting your lap, you slow down and return back to your stationary, starting position, standing next to me, most likely full of lactic acid and breathing very heavily.

Our super-accurate atomic clocks that we synchronized before you ran your lap will show the exact same time now, right?… Right?

…Wrong. The time that has elapsed on your clock will be less than the time elapsed on mine. Time effectively moved slower for you because you were moving relative to me.

And it’s not just the time showing on your clock that is different. Because time effectively moved slower for you, you will have actually aged less than me.

This is not a joke. This is the way the world actually works.

Time is not Universal

Most of us intuitively assume that time is universal. We assume that no matter who you are, how fast you’re moving, or what you ate for breakfast, the passage of time is universal. It flows at the same speed for all of us.

It wasn’t until Albert schtein-drinking Einstein came along and proved us all wrong that we realized the one thing we thought was an absolute truth in the universe was wrong.

Einstein proved that time is relative to motion. Specifically, the faster you’re moving relative to a stationary frame of reference, the slower your time moves. Also, the slower you age.

If you could manage to travel the speed of light (which you can’t, it’s impossible for a particle with mass to achieve), time would cease passing at all.

Granted, special relativity only starts to significantly affect the passage of time as velocity gets up to what are considered “relativistic speeds” that are significant fractions of the speed of light.

In the example above, you would run around the track so slowly relative to the speed of light (even if you’re Michael Johnson) that the effect of time dilation would be almost negligible. However, it still would have an effect.

Airplane Experiment

The relativity of time has been experimentally validated, in a way similar to the hypothetical track example we just discussed.

In this experiment, two atomic clocks were synchronized. One clock remained stationary, while the other clock was placed on a commercial jet that flew around the world.

When the clocks were reunited, the time they showed differed in a way that was consistent with the predictions of special relativity. Specifically, less time had passed on the clock that had been on the jet than the stationary clock.

Length Contraction

Another implication of special relativity is that length “contracts” the faster you go, especially as you approach relativistic speeds.

Suppose you were standing at a railroad crossing, where a train is about to go by. Suppose I am on that train, holding a meter stick.

If this train was going fast enough, the meter stick that I was holding would actually look shorter than 1 meter from your perspective. The faster the velocity of the train, the shorter the meter stick would look to you. And for that matter, the entire length of the train would look shorter to you as well, by the same proportion.

If the train happened to be going at the speed of light (which again, is impossible, because the train has mass), you would not see the meter stick or the train at all. Their length would be 0.

Practical Considerations

As mind-boggling as these implications are, it can be easy to write them off as “scientifically interesting”, but not practically important. After all, the effects of time dilation and length contraction are relatively small at slower, more realistic real life velocities.

So is special relativity really something that matters to us, from a societal or engineering perspective?

It is true that most engineering applications can safely ignore special relativity’s effects and be as close enough to perfectly accurate as they need to be. One exception is GPS, or satellites.

They need to factor in the effects of special relativity. If they don’t, their calculations can be off by miles, due to gradual clock drift of the satellites. Additionally, space travel and anything astronomical obviously need to factor in special relativity to function correctly.

The Theory of Special Relativity

How in the world did Einstein come up with a theory as earth-shattering as this? And what does the theory actually say that leads to the groundbreaking implications described above?

Postulates

Special relativity is based on two fundamental postulates. From these two postulates alone, the entire theory, along with all of its consequent crazy implications, can be mathematically proven and derived!

Postulate #1: The Laws of Physics Behave the Same in All Frames of Reference

Essentially, what this means is that the laws of physics are the same for you sitting there reading this article as they are for you when you’re driving in your car. Which are the same as the laws that hold when you’re flying in an airplane orbiting the Earth.

The relativity of motion is directly tied to this postulate. Suppose you and I were the only people or things that existed in the universe. Imagine that I floated past you through space at roughly 10 mph. From your perspective, you were at rest while I floated by at 10 mph.

However, from my equally valid perspective, I was the one at rest, and you actually floated by me (in the opposite direction) at 10 mph. Who’s to say who’s right? Actually, we both are!

Motion is not absolute, it is relative. You can only describe an object’s motion as relative to the motion of some other object. We on Earth tend to choose a “stationary” frame of reference to be one who is at rest relative to the orbiting of the Earth.

The relativity of motion is why you can’t feel motion. Suppose you were sitting with your eyes closed in the passenger seat of your car in the parking lot. This would feel the same to you as if you were doing that while I was driving the car at 60 mph down the highway (minus bumps in the road and noises).

Postulate #2: The Speed of Light is Constant

Mathematician James Clerk Maxwell created a famous set of equations called Maxwell’s Equations in 1865. These equations revolutionized electricity and magnetism. One important implication of the Maxwell equations is that one should not be able to observe a light wave at rest.

More specifically, one cannot observe the electromagnetic field oscillate while not in motion. This implication seemed innocuous to most. But Albert Einstein thought about this implication and realized something profound using a thought experiment.

Einstein thought about what would happen if he could theoretically run alongside a beam of light, at the speed of light. He realized that if he could do this, then he in fact would be able to observe the electromagnetic field oscillate from a stationary perspective (stationary relative to the beam of light). This would violate Maxwell’s equations.

Furthermore, if he could do that, laboratory scientists should be able to simulate this and observe it in a lab. Which they hadn’t been able to do.

From thinking through this thought experiment, Einstein realized that there was only one possible explanation to reconcile this situation without violating Maxwell’s equations. The speed of light has to be the same for all observers and all reference frames.

In other words, the speed of light is universal.

Universal Speed of Light

This alone is actually a pretty unintuitive, mind-boggling conclusion.

Imagine that you are standing still, on the side of a highway. Now imagine that I’m in a car that passes by you on this highway, going 30 mph. (My car engine has a blown head gasket and can’t go fast, OK? Lay off 😉  )

Now suppose at the instant I am passing by you, another car blows by me in the fast lane, going 60 mph (while honking at me for going so slowly on a highway).

According to you, the speed of my car is 30 mph, and the speed of the car that passed me is 60 mph. But what are the speeds of the cars from my frame of reference (the frame of reference of the car going 30 mph)?

Well, if I treat my perspective as the rest frame, then the car that passed me is going 30 mph, since it is going 30 mph faster than I am. This makes sense, and doesn’t disagree with your conclusions.

From your 0 mph perspective rest frame, that car is going 60 mph. From my 30 mph perspective rest frame, that car is going (an additional) 30 mph faster than me, which I would calculate to be 60 mph if I factored in the fact that I was already going 30 mph. Great.

Now, imagine that the car that passed me is actually going at the speed of light, c, or 671 million mph. Here’s where things get weird.

You, from your stationary frame of reference, would calculate me to be going 30 mph, and the other car to be going the speed of light, or 671 million mph.

However, from my perspective, because the speed of light is universal, that car also needs to be going the speed of light relative to me and my frame of reference. That means that I would measure the car to be going 671 million mph faster than me.

So, if I factored in the fact that I was already going 30 mph, this means that we would disagree on the speed the car that passed me was going!

From Postulates to Special Relativity

So why does it matter that we disagree on the velocity of the car in the above example? It matters because it literally directly leads to the mind-boggling implications of special relativity previously discussed!

Velocity is simply distance over time. If we are disagreeing about the velocity of something based on which frame of reference we are in (stationary or in motion), then that means that we must either be disagreeing on distance (length) or time, or both!

It turns out we disagree on both, hence the implications of time dilation and length contraction previously discussed!!

Lorentz Transformations

From the 2 postulates of special relativity just described, not only can time dilation and length/space contraction be concretely mathematically proven, but they can be precisely quantified and derived!

Using only those 2 postulates, we can calculate exactly how much time and space are distorted by moving at a given velocity v. These equations are called the Lorentz Transformations.

They are as follows:

Time Dilation

T1 = T *  (1 – (v2 / c2)) 

In the above equation, T prime is the time gap between two events from a moving frame of reference. T is the time gap from a stationary frame of reference. C is the speed of light, and v is the velocity at which the moving frame of reference is moving relative to the stationary frame.

This equation tells us exactly how much the passage of time slows down for objects that are moving based on their velocity v relative to the stationary reference frame.

Length Contraction

L1 = L * (1 /  (1 – (v2 / c2))) 

In the above equation, L prime is the length of an object as measured in a moving frame of reference. L is the length of that object as measured from a stationary reference frame. V is the velocity at which the moving frame of reference is moving relative to the stationary frame. And c is the speed of light.

This equation tells us exactly how much an object’s length contracts if it is moving based on its velocity v relative to the stationary reference frame.

Pretty cool!

Other Special Relativity Implications

E = mc2

Even if you’d never heard of special relativity before reading this post, you’d likely heard of one of its implications: E = mc2. This equation can be derived as a consequence of special relativity. But what does it actually mean?

Well, in that equation, E means the total energy of a particle or object, m is the mass of that object, and c is the speed of light. So Energy = mass * (speed of light squared). This equation tells us that the total energy of a particle or object is equal to its mass times the speed of light squared.

But, it turns out E = mc2 is actually a specific version of the more general equation E2 = (p2 * c2) + (m2* c4).  In that equation, p is the momentum of the object, and all the other variables are the same.

This more general equation tells us that the total energy of a particle is its mass energy combined with its momentum energy.

E = mc2 is the special case where a particle or object is at rest. When an object is at rest, its momentum, p, is 0. So it has no momentum energy, and all of its energy comes from its mass energy. The equation reduces to E = mc2.

So E = mc2 gives you the total energy of an object or particle when it is at rest.

Massless Particles and Momentum

Conversely, for a particle or object that has no mass, m = 0. All of its energy comes from its momentum energy. E = pc.

One example of a particle that has no mass is a photon. A photon is essentially a particle of light. So light only has energy from its momentum.

But how does light even have momentum if it has no mass? For most objects, momentum is a consequence of their mass and velocity.

Well, unlike other objects, it turns out that particles that behave like waves, like light, can have momentum even though they have no mass. A wave transports momentum via its waving motion.

Imagine a wave of water. This is not water molecules moving along, but rather, it is the undulation of water molecules. This undulation propagating through space can exert force on another object (momentum) without actually transporting any mass.

In this same way, light has momentum despite having no mass.

Massless Particles and Motion

If a particle has no mass or motion, it does not exist. Therefore, a massless particle must be in motion to be said to exist. This is the case with light!

But remember Einstein’s thought experiment. If the velocity v of a massless particle was anything less than c, then one could choose a frame of reference wherein the massless particle is at rest relative to that frame of reference. And this is impossible, because then the particle would cease to exist from that perspective!

Therefore, massless particles must be in motion to exist, and they must be traveling at the velocity c, the “speed of light”. This is because, as we recall from one of the postulates of special relativity, if a particle is traveling at speed c in one reference frame, it is traveling at speed c in all reference frames.

Thus, by traveling at the speed of light, a massless particle avoids the paradox of existing in one frame of reference but ceasing to exist in another.

Photons (particles of light) are one example of massless particles, which is why c, or 300 million meters per second, is called the “speed of light”. But light isn’t actually the only massless particle. Gluons and gravitons are also massless, and thus, also travel at the speed of light!

Accelerating an Object to the Speed of Light

It should be very clear now that the speed of light is very special in the universe. We just discussed how massless particles must always be traveling at the speed of light in order to exist.

On the flip side, it turns out that particles with mass can actually not eclipse or even meet the speed of light. Any particle with mass will be going slower than the speed of light. Why is this the case?

Another interesting conclusion from the theory of special relativity is that it takes increasing amounts of energy to accelerate an object with mass faster and faster. As you get the object closer and closer to the speed of light, it takes exponentially more energy.

And it actually takes an infinite amount of energy to accelerate an object with mass all the way to the speed of light.

In this way, the speed of light can be considered to be a universal speed limit. All objects with mass travel below this speed limit. And massless objects travel at it.

The Twin Paradox

Imagine two twins. One twin gets on board a rocket ship that travels around at half the speed of light for a year. Let’s call that twin the traveling twin. The other twin stays stationary on Earth for that year. We’ll call that twin the stationary twin.

After the year has passed, the traveling twin and its rocket ship return to Earth.

Special relativity tells us that the traveling twin will have aged less than the stationary twin upon its return to Earth, since time for the traveling twin in motion slowed down.

But wait. Isn’t motion relative? This presents a potential paradox.

From the stationary twin on Earth’s perspective, the traveling twin hopped aboard a spaceship and went really fast for a year, then came back. So the traveling twin should have aged less.

But think about this scenario from the traveling twin’s perspective. From the frame of reference of the traveling twin, they stayed aboard a stationary rocket ship, while the Earth and everything on it, including the twin at rest, traveled really fast away from them and came back to them.

So wouldn’t special relativity tell us that from the traveling twin’s perspective, the stationary twin should have aged less? Which twin is right? Is there an objective answer to this paradox?

It turns out that yes, there is an objective answer to this paradox. When the traveling twin returns to Earth, it is the one that has objectively aged less than the stationary twin. But why is this the case?

Solving the Twin Paradox

The twin paradox can be explained because the traveling twin is the one that accelerated, then decelerated to come back to the stationary reference frame of Earth. This is what breaks the “symmetry” of the twin paradox.

Acceleration, unlike motion, is absolute as opposed to being relative. Acceleration can be felt. The relevant factor when considering the twin paradox is this: which twin actually underwent acceleration to get to the frame of reference where time and aging are actually measured?

In essence, there is no valid frame of reference where the traveling twin is stationary the whole time. So talking about aging from the traveling twin’s perspective is meaningless, in the sense that it’s not valid.

If you want to go even further down this rabbit hole, consider when the aging gap between the two perspectives gets reconciled.

Paradox Reconciliation

Suppose the traveling twin’s rocket ship was all windows, so the stationary twin could see inside it. Now suppose that the rocket ship flies really fast in circles around the stationary twin, instead of leaving Earth. So both twins can see each other aging in real time throughout the whole year of flight.

What would each twin see in this case?

From the stationary twin’s perspective, the traveling twin is in motion relative to it. So as the year of flight progresses, the stationary twin would watch the traveling twin age more slowly than them. And, sure enough, when the traveling twin decelerates back to rest and gets out of the spaceship, they will be younger than the stationary twin.

What does this scenario look like from the traveling twin’s perspective? Here’s where things get really weird (as if they weren’t weird enough already).

As the traveling twin is flying around the stationary twin, they would peer out their windows and see the stationary twin aging more slowly than them!

Why is this? Because from their frame of reference, the stationary twin is the one going really fast around them!

OK, something has to give here, because we already discussed how the traveling twin is the one that actually ends up aging less than the stationary twin upon its return to being stationary. 

Something does give. And it happens when the traveling twin decelerates back to rest after the year of flight is up.

So after one year, the traveling twin has had enough, and begins to slow their rocket ship to come back to rest. At the instant before they start slowing down, they peer out their window and see that the stationary twin has aged less than them.

However, as they start to slow down, they see the stationary twin start to age more rapidly! They continue to slow down more, and they see the stationary twin continue to age faster and faster!

By the time they’ve come to rest, the stationary twin has aged so fast that the stationary twin is now quite a bit older than them! Thus, consistency has been achieved between the two perspectives.

It’s just quite peculiar that while the rocket ship is in motion, each twin sees the other twin aging slower than them. It’s not until the actual deceleration phase of the experiment that the “paradox” gets broken, and the traveling twin, from their perspective, sees the stationary twin truly start to age more.

Conclusion

Welcome to the magical world of special relativity! I hope your brain hurts after reading this. Because I cannot even think about special relativity without my brain hurting.

I remember when I first learned about it, it literally seemed like all of the answers were against my intuition. The fact that time is relative, not an absolute constant in the universe, is so counterintuitive to everything that seems natural to us.

Kudos to Big Man Albert Einstein for having the creativity and imagination to even conjure up this thought, much less define it and prove it.

I hope I’ve peaked your curiosity into this subject. And I hope you don’t swear off my blog posts after reading this on account of it making you think too hard!

I promise not all of the posts are this mentally stimulating. Maybe next post will be about something simple. But then again, I am the Analytical Aspergian. So I can’t make any guarantees 🙂

Sources

Twin Paradox: http://www.owl232.net/papers/twinparadox.pdf

E=mc squared: https://www.wtamu.edu/~cbaird/sq/2014/04/01/light-has-no-mass-so-it-also-has-no-energy-according-to-einstein-but-how-can-sunlight-warm-the-earth-without-energy/

Lorentz transformation definition: http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html

Lorentz transformation derivation from a graphical perspective, Khan Academy: https://www.khanacademy.org/science/physics/special-relativity/lorentz-transformation/v/lorentz-transformation-derivation-part-1

Lorentz transformation from an intuitive perspective: https://oyc.yale.edu/sites/default/files/notes_relativity_6.pdf

Time dilation and length contraction: https://en.wikipedia.org/wiki/Lorentz_factor

Intuition of lorentz transformation: https://courses.lumenlearning.com/physics/chapter/28-2-simultaneity-and-time-dilation/

Electromagnetism: https://analyticalaspergian.wordpress.com/2021/05/24/the-magic-that-makes-the-world-work-electromagnetic-fourier-transforms/

Exercise Physiology: https://analyticalaspergian.wordpress.com/2021/07/13/sports-nutrition-and-supplementation-series-exercise-physiology/

Airplane experiment: https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experimentMaxwells Equations: https://en.wikipedia.org/wiki/Maxwell%27s_equations

Published by Analytical Aspergian

I am an Aspergian who loves logically analyzing the world around me. On this blog, I analyze anything that interests me, from economic design to electromagnetism to sports nutrition and recovery.

4 thoughts on “Special Relativity

  1. My brain hurts. I feel like I’m in kindergarten learning advanced math formulas. I appreciate your attempts to simplify the examples to help me understand the concepts. Nicely done.

    Like

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