A Moment of Genius – Einstein’s Theory of Special Relativity and E=mc2

Albert Einstein, one of the greatest scientists that ever existed in human history, revolutionized humankind’s perception on the universe and toppled its common sense with his Theory of Special Relativity, with which he derived the equation for converting mass to energy, E=mc2. It impacted science and the world beyond imaginable. But how did he do it? How did he just use mathematics to determine the relationship between two things that are deemed by most people uncomparable?

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In order to solve this mystery, two premises have to be made, known as special relativity postulates. One is the principle of relativity which constitutes three parts. First, there is no absolute motion or rest. Speed and velocity are relative terms, certain object is only moving when compared to another. Even if someone is sitting still, he is moving with the earth, the solar system and the milky way at a considerable amount of speed. When two frames of reference are moving at different velocities, they are both entitled to say that they remain rest while the other is moving. Second, special relativity only applies to inertial reference frames, meaning that the object can only be traveling at a constant velocity in a straight line. Third, physics is the same for all inertial frames of reference. No experiment can be done to determine one’s motion without a second frame, because relative to themselves, they are not moving.

1. Classical Physics

Imagine being on a train that is moving at a speed of 30 miles per hour and throwing a ball that is moving at a speed of 20 miles per hour relative to you in the same direction towards which the train is moving, a man on the platform measures the speed of the ball to be 50 miles per hour. This is intuitive, the speed of the ball being 20+30=50 mph. The same concepts apply when you turn around and throw the ball at the same speed but in the opposite direction of the moving train. The man on the platform will measure the speed of the ball to be -20+30=10 mph . Therefore we can simply derive the equation for the velocity (v) of an object in an inertial frame of reference relative to another inertial frame of reference to be v=v1+v2 where v1is the velocity of the object according to the first inertial frame of reference, which can be negative if v1 and v2 are in opposite directions; and v2being the velocity of the first frame of reference relative to the second frame of reference (the speed of the train to the man on platform).

2. Principle of Constancy of Speed of Light

The second postulate is the Principle of Constancy of Speed of Light. When everyone thought the equation v=v1+v2applied to all non-accelerating objects, the speed of light appears to be different. Back to the train thought experiment, but this time, instead of throwing a ball, you decide to shine a flashlight. If the train is moving at the same speed, 30 miles per hour, and the flashlight is shined along the direction of which the train is traveling, you would expect the man on the platform to measure the speed of the light to be the speed of light on the platform (c3108meters per second) plus 30 miles per hour. However, the man measures the speed of light to be exactly c. What if you shine the flashlight in the opposite direction? The result is, again, exactly c. This proves that the speed of light is definite, irrespective to the speed of the source of the light or the speed of the observer.

3. Lorentz Transformation

Let’s do a second thought experiment, this time, instead of being on a train, you are on a spaceship s’ and you are moving at the velocity of v relative to a man on another spaceship s. As s’ move past s, both of your clocks are synchronized and simultaneously the both of you shoot a light beam in the same direction. After a certain amount of time, t, has passed on the man’s clock and t’ has passed on your clock, the light beam would have traveled a distance of x from the man’s spaceship and x’ from your spaceship. Remember that you have been moving at a velocity of v all along, so after t’ has passed you are closer to the you light beam than the man is to his. Therefore x=x’+vt’ or x’=x-vt. However, the speed of light is constant so c=c, and from a newtonian physics standpoint the times passed are the same, hence t=t’. Also, x can be calculated as x=ct while x’=ct’. How could that be? This might seem counterintuitive because the speed of light doesn’t change, if it were any other object, its velocity will bev1+v2, which accounts for the difference in distances. This doesn’t make sense to the man or you so both of you allege each other made a mistake in measuring the time or the distance or both. We will call this mistake. The equations x=x’+vt’ and x’=x-vt become:

x=(x’+vt’)

x’=(x-vt)

In order to solve for , multiply the two equations together.

xx’=2(x’+vt’)(x-vt)

xx’=2(x’x+vt’x-x’vt-v2t’t)

From the equations x=ct and x’=ct’, equations as follow can be derived:

t=x/c

t’=x’/c

Substitute them in:

xx’=2(x’x+xvx’/c-x’vx/c-v2x’/cx/c)

Divide all terms by xx’:

1=2(1+v/c-v/c-v2/c2)

Therefore:

2=1/(1-v2/c2)

=1/1-v2/c2

And this is the famous Lorentz Transformation, where instead being a mistake in newtonian physics, it becomes a dilation factor for transforming coordinates between two inertial frames of reference accounting Einstein’s Special Relativity. Earlier equations can be further derived by applying the Lorentz Transformation:

x=x’+vt’1-v2/c2

x’=x-vt1-v2/c2

As seen by the equations, if v is really small compared to c , determining the square root would simply be 1 and the result is the same as regular newtonian physics. is only relevant if v is approaching the speed of light which is known as becoming relativistic.

Other than distances, the Lorentz Transformation can also be applied to times:

x’=(x-vt)

t=x/c

x’=(x-vx/c)

Divide both sides by c,

x’/c=(x/c-vx/c2)

t=x/c

t’=x’/c

t’=(t-vx/c2)

Similarly,

x=(x’+vt’)

t=(t’+vx’/c2)

Comparing t and t’, Einstein concluded that when speed approaches being relativistic, time measured by an observer will depend on his speed, in other word, time is no longer absolute, it is relative to speed in Einstein’s Special Relativity.

4. Time Dilation

Time is different for different frames of references. Einstein came up with the most daring hypothesis of the 20th century. However, he did not stop there. He further explored the concept known as time dilation.

Imagine a thought experiment where two mirrors are placed 1 meter apart facing each other in a spaceship s’, which is moving at a velocity v according to another spaceship s. if you are in the spaceship and shoot a laser perpendicular to one mirror, the laser will reflect back and forth c/21.5108times per second. A man on spaceship s observes this phenomenon, and sees that the light beam is reflected between the mirrors while the mirrors are moving. This means to him, the light beam is traveling between the mirrors in a diagonal path, which is longer than the perpendicular path you observe. Light travels at a definite speed, and for both observers the light beam reflects back and forth c/21.5108times per second, however, for him the light has traveled a greater distance. All of this contradicts to the Newtonian physics, but the only way to explain this phenomenon is that the two observers are experiencing time differently.

The length of the perpendicular path you observe is the speed of the light beam, c, times the time it takes according to you, t’. The length of the signal path the other person observes is the speed of the light beam, c, times the time it takes according to him, t. The distance you moved is the velocity, v, times the time he observes t. Therefore, we can derive the equations by using pythagorean theorem.

c2t’2+v2t2=c2t2

c2t’2=c2t2-v2t2

c2t’2=(c2-v2)t2

Divide it by c2

t’2=(1-v2c2)t2

Take the square root of both sides

t’=(1-v2c2)t

t=t’/1-v2/c2

Since nothing can exceed the speed of light, 0<1-v2/c2t . Therefore, Einstein concluded that a clock moving relative to an observer is running slow. This concept is called time dilation. However, motion is relative too, so if two observers are moving at different velocities, observer 1 and observer 2 are both entitled to say that the other person’s clock is running slow relative to his.

5. Length Contraction

Aside from time being dilated due to constant velocity, length of an object appears to be contracted in the direction towards which the object is moving. Imagine a particle traveling at the speed of light inside a spaceship, s’, which is again moving at a velocity of v relative to spaceship s. A man on spaceship s, sees it and after time t has passed, he measures the length traveled by the particle to be l. However, spaceship s’ measures the the length to be l’ after t’.

The speed of the particle can be expressed as:

c=l/t

c=l’/t’

Since c is always constant,

l/t=l’/t’

Substitute the previous equation t=t’/1-v2/c2

l/t’1-v2/c2=l’/t’

Cross-multiply:

t’l’1-v2/c2=lt’

Divide by t’

l’1-v2/c2=l

or

l’=l1-v2/c2

As stated previously 0<1-v2/c2l. This shows that length of a moving object measured by a relative stationary observer appears to be shorter compared to the length measured by an observer traveling with the object.

6. Four Vectors

Spacetime is a continuum of space and time described in Einstein’s Theory and General Relativity. It can be expressed by three dimensions of space with a fourth dimension of time, which can be written in a four vector system, named X,

X(x0, x1, x2, x3)

where x0represents time and x1, x2, x3 represent the three dimensions of space. However, x0is of a different dimension and represents time, while the other three represent distance. In order to convert x0into the same unit as the other components, x0=ct, making x0also a measure of distance. Also, a new term is created, =v/c. If we substitute in , x1for x and x0/c for t and so the equation x’=x-vt1-v2/c2 can be derived to:

x1’=x1-x01-2

And similarly, t’=(t-vx/c2) can be derived to:

x0’=x0-x11-2

For simplicity purposes, x2 and x3 are ignored, it is shown in one dimension of space and one dimension of time but they can be derived using the same way for x1.

A new term S2is created and S2=x02-x12. Then utilizing equations x0’=x0-x11-2 andx1’=x1-x01-2, the term x’02-x’12can be expressed as follow:

(xo-x12)2-(x1-x02)2

Multiply it all out and expand :

(x02-2x1x0+2×12-(x12-2x1x0+2×02))/(1-2)

Simplify:

(x02+2×12-x12-2×02)/(1-2)

(x02(1-2)-x12(1-2))/(1-2)

x02-x12

Therefore,

x’02-x’12=x02-x12

This means that S2is an invariant. No matter what frame of reference it is measured in, it is the same for every observer.

If we look at S2differently:

S2=x02-x12

S2=(ct)2-x12

S2=(ct)2-x12

Take out a c2t2:

S2=c2t2(1-x12c2t2)

Take the square root of both sides: (note:x/t=v)

S=ct(1-v2/c2)

Divide by c on both sides:

Sc= t(1-v2/c2)=

This is known as or the Proper Time, it is the time that is invariant, because both S2 and c are constant. It is the time recorded on the clock with which the observer is traveling. Every frame of reference has a relative time due to the difference in velocity, but every observer’s clock will have the same proper time.

7. The Four Momentum Vectors

In Newtonian physics, the an object momentum pis calculated as p=mvwhere m is the mass of the object and v is the velocity. This can also be expressed in a four vector system as follow:

P=m(X0, X1, X2, X3)

For simplicity purposes, only one dimension of space and one dimension of time is shown. The others can be derived by the same way.

p0=mx0/

p0=mct/

The space dimension can be represented by p1,

p1=mx1/

p1=mvt/

As formally derived

=t(1-v2/c2)

Therefore

t=1(1-v2/c2)

Hence:

p0=mc(1-v2/c2)

p1=mv(1-v2/c2)

According to the rule of binary expansion,

(1+x)n=1+nx+…

whereas the ellipsis represents the terms that are insignificant for our purposes. The equations can now be written as:

(1-v2c2)-12=1+v22c2+ …

p1=mv+mv32c2

This term consists the Newtonian momentum equation p=mv, the term mv32c2must be a relativistic correction when speed approaches the speed of light. When the object travels at a speed that’s relatively slow to the speed of light the term mv32c2will be nearly 0, having minimum effects on the equation, and the equation will produce a Newtonian physics momentum. Therefore p1is a momentum term. As to p0, it can be derived as follow:

p0=mc+mcv22c2

p0=mc+mv22c

Multiply both sides by c:

cp0=mc2+mv22

The term mv22 is a term used to calculate the the kinetic energy of an object of mass m traveling at a speed v. Therefore cp0must be an energy term (E). If the object is stationary, v=0,

E=mc2 +m022

E=mc2

the term mc2must be the energy of an object possess when it doesn’t have any kinetic energy, which can also be referred to the rest mass energy.

This is how the one of the greatest minds in the human history derived the equation that would revolutionize the perspective of humankind itself–E=mc2. From a scientific standpoint, Einstein’s Theory of Special Relativity is a huge breakthrough, it made humans reconsider what they saw and envision what they would see. It impacted all sciences, from the smallest, such as how an electron travels in molecular physics, to the biggest, such as stars and black holes in astrophysics. Aside from the impacts on these concepts that only scientists come in contact with, it has lots of more tangible effects on our day-to-day life. For example, the Global Positioning System, or GPS can not work precisely if it weren’t for Special Relativity. Although not traveling nearly as fast as light, GPS are moving about 10,000 kilometers per hour. If time dilation and length contraction were not accounted for, their clock will be 7 microseconds slower everyday relative to us and without a reconciled time and space, the accuracy of GPS will drift about 7 miles per day. Other than GPS, all words associated with nuclear, such as nuclear bomb, nuclear power plant or even nuclear fusion utilize the equation E=mc2. In fact, E=mc2is the fundamental of how any nuclear reaction works: converting mass into energy. Considering we are mostly composed of elements made in stars and stars are powered by nuclear fusion, Einstein’s Special Relativity proves our existence!

More importantly, according to a thought experiment by Stephen Hawking, if goldfish were to develop a system of physics through observations, they would get a wrong and curved version of our physics, because the lense they look through is a curved fish tank. Therefore to them, all lines are curved. If along cames a goldfish that resembles similarities to Albert Einstein, and came up with the term linear motion, it is so contradictory to their common sense that it would revolutionize their view of the world. How do we know that we are not in a fish tank or how do we know that our view on the world is not curved? We don’t, even with Einstein we still can not tell with absolute certainty; but, at least we know that Relativity has a higher chance than Newtonian physics at being the correct version. Other than the actual physical benefits that the Theory of Relativity provides us, from a philosophical standpoint, it is our best attempt to not just being a goldfish.

Works Cited

Time Dilation, www.emc2-explained.info/Time-Dilation/#.Wj1DGxhVZmA.

Four-Momentum. Wikipedia, Wikimedia Foundation, 21 Dec. 2017, en.wikipedia.org/wiki/Four-momentum.

Four-Vector. Wikipedia, Wikimedia Foundation, 21 Dec. 2017, en.wikipedia.org/wiki/Four-vector.

Galilean Transformation. Lorentz Transformation, hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html.

Length Contraction. Time Dilation/Length Contraction, hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html.

Lorentz Transformation. Wikipedia, Wikimedia Foundation, 21 Dec. 2017, en.wikipedia.org/wiki/Lorentz_transformation.

Michelson-Morley Experiment — from Eric Weisstein’s World of Physics. Scienceworld.wolfram.com, scienceworld.wolfram.com/physics/Michelson-MorleyExperiment.html.

Special and General Relativity. YouTube, YouTube, www.youtube.com/playlist?list=PLDB15F7E29A5F0426.

Special Relativity. Wikipedia, Wikimedia Foundation, 20 Dec. 2017, en.wikipedia.org/wiki/Special_relativity.

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A Moment of Genius - Einstein's Theory of Special Relativity and E=mc2. (2019, Jul 15). Retrieved September 25, 2022 , from
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