A few words on Length Contraction

 by Filippos Atlasis

In this article, I will be explaining an effect that occurs from Albert Einstein's famous Special Theory of Relativity: Length Contraction. My knowledge about this topic was primarily acquired while completing the Special theory of Relativity course offered by Standford university.

An introduction on length contraction

Length contraction is a concept which was firstly introduced by Albert Einstein as a part of his special theory of relativity. Einstein’s special theory of relativity was published as an extension paper to one of his other publications, “The Electrodynamics of Moving Bodies”, during the so-called “Miracle Year” in 1905. Although it might come as a surprise to many that Einstein never won a Nobel prize for either of his theories of relativity (special or general), his work can’t be undermined as it is proven correct time and time again in the years following its publication. Today, the effects of the special theory of relativity (including length contraction) are used in many aspects of modern life e.g Satellites which control the Global Navigation System.

What is Length contraction and When does it occur?

Length contraction is a consequence of time dilation. Length contraction states that the measurement of length is different for different observers that are moving with respect to each other. Measuring the length of a moving object will give a result of a shorter length to when the length of the object is measured whilst the object is at rest. All the effects of the special theory of relativity are applied when two objects move with respect to each other while maintaining a constant velocity/being in an inertial frame of reference. This means that if one or more of the objects accelerate/be in a non-inertial frame of reference, the special theory of relativity can’t be used to describe the effects on time and length. However, the effects of length contraction cannot be observed in our everyday world, because objects need to move with velocities close to that of light, and thus their effects are close to none. In most of our common everyday experiences, we can use the classical   Length=ΔTime*Velocity  equation to calculate a length. When an object’s velocity reaches significantly close to the speed of light, this classical equation won’t give a correct result as the effect of length contraction is significant enough to change our observations. As each object/person is in their own frame of reference, a person would always see the other’s length to contract because both of them would see themselves as being at rest and the other moving.

 

Do all measurements of “distance” (e.g width) contract?

One key concept about the special theory of relativity is the fact that several measurements of distance aren’t affected by length contraction; they remain invariant. In fact, the original name for the special theory of relativity was the “Theory of Invariance” because several quantities used in calculations remained constant. For example, the width of an object does not contract; there isn’t a width contraction effect. To justify this, imagine a train on tracks passing by a station platform. If the train could reach a speed very close to that of light, an observer standing on the station platform would see the length of the train contract as it moves past. However, if the observer saw the width of the train contract, suddenly the rails of the train would be off the tracks and the train would most likely crush. A person standing inside the train rather than observing it from the station platform would run into similar problems. For the person inside the train, the train is at rest and the rails are moving towards him at a speed very close to that of light. As the rails move towards him, he observes their length to contract. If he was to see their width contract as well, the rails would suddenly overextend off the tracks and again the train would most likely crush. Similarly to a width contraction effect, the person inside the train would observe an unfortunate event if the height of the object moving towards him contracted. Now, along with the tracks moving towards the person inside the train, imagine a tunnel coming towards the train at a speed very close to that of light. If the height of the tunnel was to contract, the train wouldn’t be able to fit through. On the other hand, the observer on the station would see the train easily fit through the tunnel because the height of the train would contract. In both cases, the observer on the platform and inside the train do not agree on the photo clock principle. This means that if a photo was to be taken at an instant moment in time, the two observers wouldn’t agree on the apparent position of the train.

 

How do objects contract?

In order to understand how objects contract, we have to imagine a situation where Person A is found inside a rocket travelling close to the speed of light and Person B is an observer outside of the rocket. The rocket of Person A is travelling at a velocity v. As it passes by Person B, the rocket accelerates slightly from v to v + Δv*[1]. To understand how objects contract it is essential to know the relativity of simultaneity and the concept of “leading clocks lag”. In short, if we place two imaginary clocks, one at the rear and one at the front of the rocket, Person B (the outside observer) would see the clock at the front of the ship fall behind the clock at the rear of the ship. Therefore, Person B would observe the rear part of the ship accelerate before the front of the ship. this slight acceleration at the rear of the ship would cause it to move before the rest of the ship and so start to compress towards the front. If the whole ship were to accelerate simultaneously for Person B, then each part of the ship would move together and so there would be no compression as the rear isn’t the part that is moving first. It is helpful to imagine that the particles that make up the ship would maintain fixed distances if the whole ship was seen to accelerate simultaneously, but in reality Person B observes the rear accelerate before every other part and so the particles found there “compress” towards the front.

How do you work out how much length contracts?

As mentioned above, the classical equation for calculating the length of an object is    Length=ΔTime*Velocity where an observer observes the length, time and velocity of an object. However, as the object travels very close to the speed of light, the observer wouldn’t see the time on his/her run in sync with the time on the clock of the object because of time dilation. In order for the observer to calculate the passage of time on the object compared to his/her clock, he/she would have to use the following Time Dilation equation:

Time (object)=1/γ  *  Time(observer)

where γ is known as the Lorentz factor. If we substitute this into the classical equation for length, we can derive the Length contraction equation:

 

 

Epilogue

It has to be noted that all the effects of special theory along with length contraction aren’t simply illusions to our eyes; they happen for real. For example, going back to the people in the train and train station, the person inside the train would indeed travel a shorter distance because the length of the tracks in front of him contract and the person on the train station would find out that the length of the train has contracted if he measured its length properly.

 

References

-         -  Course for the special theory of relativity offered by Stanford university.

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[1] Although I mentioned that the special theory of relativity is only used in inertial (non-accelerated) frames of reference, in reality, no object can reach a velocity v without accelerating to it first. This slight acceleration in the example above is needed to explain how length contraction effect occurs.

A Few Words on Time Dilation

 by Vaggelis Atlasis

Today I will be explaining one of the most well-known phenomenons of the special theory of relativity: time dilation. I learnt about this phenomenon while taking a course about the special theory of relativity on Coursera, by Stanford University. This information is based on what I learnt from lessons in Week 4 of the course.

What is a Light Clock?

 To help explain time dilation we will use a light clock (see Figure 1). A light clock is a clock comprised of two parallel mirrors a length L away from each other. They reflect a beam of light between them. For a full tick of the clock, the light beam starts at the bottom mirror, reaches the top one and is reflected back at the bottom mirror.

            A light clock; c indicates the speed of the light rays

The Situation

Bob, stationary, has a light clock at his possession. Alice is in a spaceship, moving at a constant velocity v, and has an identical clock that is synchronised to Bob’s. Both Bob and Alice perceive themselves to be stationary and their respective counterpart moving in a certain direction; Bob sees Alice and her clock moving at v with respect to him and Alice sees Bob and his clock (as well as the Earth) moving at -v­ with respect to her.

Bob on Earth, with his clock, observing Alice travel to the right in her spaceship, with her clock, with velocity v.

From Alice’s perspective, her clock is stationary and the time for a tick of her clock (Δtick_A) is equal to the distance travelled by light from the bottom mirror to the top one and back (2L) divided by its speed (c). However, from Bob’s perspective Alice’s clock is moving and the time for a tick of the moving clock (Δtick_B) will be different. As the light clock is constantly moving, by the time the light beam, travels the distance L, the top mirror will have moved some distance. Therefore, for the light to reach the top mirror it has to travel diagonally. Similarly to Δtick_A, Δtick_B will be equal to the distance travelled by light from the bottom to the top mirror and back divided by its speed. The distance travelled by light will not be 2L as light is travelling diagonally, not vertically. Instead, we will let this distance be 2D.

Alice's and Bob's clock from their respective frames of reference (perspective). For the moving clocks, snapshots have been taken at the points where light touches the mirrors (note how depending on the direction of travel the light rays travel in different directions).

Calculating the Relationship Between Δtick_A & Δtick_B (& Deriving the Lorentz Factor)

Alice's clock as Bob sees it. L represents the distance between the mirrors, D represents the distance light travels from the bottom to the top mirror (and vice versa). Finally, X represents the horizontal distance the mirrors travel within one tick of the clock.

We can calculate D using the Pythagorean theorem (a² + b² = c²). We know the vertical distance between the mirrors is L. The diagonal distance light travels from the bottom to the top mirror and vice versa is D, and the distance the bottom mirror moves in the time span of one tick of the clock (from Bob’s perspective), is X. 

Hence, we end up with

D² = L² + X².

By using the equation distance = velocity x time we find that

X = vΔtick_B,

where v is the velocity of the spaceship and Δtick_B is the time for one tick of Alice’s clock from Bob’s perspective. We can also rearrange the previous equation

Δtick_A = 2L/c into L = c(Δtick_A)/2

and

Δtick_B = 2D/c into D = c(Δtick_B)/2.

Finally, by substituting these equations into the Pythagorean theorem equation, we end up with:

(c(Δtick_A)/2)² + (vΔtick_B/2)²_­ = (c(Δtick_B)/2)²

We can then simplify and rearrange this equation to find a relationship between Δtick_A and Δtick_B.

c²Δtick_A²/4 + v²Δtick_B²/4 = c²Δtick_B²/4

c²Δtick_A² = c²Δtick_B² – v²Δtick_B²

c²Δtick_A² = Δtick_B²(c² – v²)

Δtick_B = √(c²ΔtickA²/(c² – v²))

Δtick_B = cΔtick_A/√(c²-v²)

Δtick_B = Δtick_A x c/√c²(1 – v²/c²)

Δtick_B = Δtick_A x c/c√(1 – v²/c²)

 

Δtick_B = 1/√(1 – v²/c²) x Δtick_A

 

       The  T1/√(1 – c²/v²) component of the equation  is known as the Lorentz Factor (γ). The velocity at which the “clock” travels with influences the Lorentz factor in the following ways:

-          For v = 0, Δtick_B = Δtick_A as none of the objects are moving and the Lorentz factor is equal to one.

-          For 0 < v < c, γ takes a finite value larger than 1. For example, when v = 0.5c, γ = 1,2 and when v = 0.999c, γ = 71. When v = 0.99999c, γ = 223.61

-          For v >= c, γ is either undefined or takes no real value. This shows us that no object with mass can reach the speed of light as well as that the speed of light is the natural speed limit.

Finding the Equation for Time Dilation

From the section above, we’ve concluded that

Δtick_moving = γΔtick_stationary .

By “moving” we mean an observer observing a moving clock and by “stationary” we mean an observer observing a stationary clock (Alice) from their perspective.  The equation effectively tells us that an observer measuring the time on a moving clock will observe every tick of the clock take longer than an identical one at rest. We can find the relationship between the time elapsed of a moving clock and a stationary one. Let’s say γ = 4. In this case, the duration of a single tick of the moving clock corresponds to the duration of 4 ticks of the stationary clock. If the duration of one tick is 5 seconds, the elapsed time on the moving clock will be 1 x 5 = 5 seconds and the elapsed time on the stationary clock will be 4 x 5 = 20.

Hence, Δt_moving = xΔt_stationary (t being time elapsed)

5 = 20x

x = ¼ = 1/γ

Therefore, we find that:

Δt_moving  = (1/γ)Δt_stationary.

Conclusion (Moving Clocks Run Slow)

Overall, we find that any “moving clock” (a moving object) in an observer’s perspective will run slower than an identical stationary clock, whether it is the observer’s or even if it is the same clock in another observer’s frame of reference where the clock is stationary. This is known as time dilation and is true for any moving object at a constant velocity, even if its effects are too small to notice.