The distructive power of E=mc²

In the special theory of relativity, Einstein


stated that mass can be changed into energy, and energy to mass. Some people made use of this theory for distructive uses. Since the day when nuclear fission was discovered, people tried to create more powerful weapons using the mass-energy relation. For example, the atom bomb and hydrogen bomb. They uses the theory of nuclear fusion and fission to generate huge amount of energy.

So, does the E=mc² gives us only the power to destory?

That's a good question. The E=mc² gave us the power of distruction, but it also gave us the power to create. In nuclear power stations, nuclear fission is used. If einstein didn't gave us the mass-energy relation formula, nuclear power stations will not exist, and human will continue to depend on fossil fuel. Let's say that the mass-energy gives us both the power to destory and to build.

The mass-energy relation also helped explain the very begining of the universe. The first matter was created with energy. Everything in this world were once energy. Einstein learnt how god had created this world.

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The apple falls UP if the time is imaginary time

In the Newton laws, velocity is defined as (change in distance)/(change in time), and acceleration is defined as (change in velocity)/(change in time). Because velocity is a vector, it can change in two ways: a change in magnitude and/or a change in direction. According to Newton's Second law, 

where F is the resultant force acting on a body.


According to this definition, if we put (i)t into the formula, we get:

So, the force acted will be reversed. That means, in the world of imaginary time, an apple dropped will not fall DOWN, but UP.

Time Machines

In sci-fi movies or cartoons, the main characters can always travel back through time. We have discussed how time slows down before, however, we never know if time could be reversed. We can find the answer with the Time Dilation formula.

This time, instead of substituting v=c, we will substitute v=(√2)c. By doing that, we obtain:

Here iΔt is know as imaginary time. Imaginary time is different from regular time. If we imagine "regular time" as a horizontal line running between "past" in one direction and "future" in the other, then imaginary time would run perpendicular to this line as the imaginary numbers run perpendicular to the real numbers in the complex plane. However, imaginary time is not imaginary in the sense that it is unreal or made-up — it simply runs in a direction different from the type of time we experience. In essence, imaginary time is a way of looking at the time dimension as if it were a dimension of space: you can move forward and backward along imaginary time, just like you can move right and left in space.


To conclude, travelling back to past is not possible. There's no way something can travel faster than the speed of light. Even if a thing really did it, you still can't travel back to the past, as you will be travelling on a different "time line". So, we can just forget the dreams of reversing time.

Time Dilation

I discovered this clip in YouTube. The clip clearly explains how Time Dilation works. You should read my previous post again after watching this clip so as to understand the theory clearer.

E=mc²: Einstein explains his famous formula

This video should help you understand more easily about what the formula E=mc² predicts.

Why "c" is a Constant?

The special relativity stated that the speed of light will always be the same, no matter what. c=299792458 m/s when you are stationary, c=299792458 m/s when you are traveling at 99.5% the speed of light, c=299792458 m/s, no matter how fast you travel. But how did that happen?


It's the problem of velocity addition. For example, when you rune forward and throw a ball in the direction in which you are running, the ball moves faster than it would if you were standing still. If you can throw a ball 30 m/s when standing still, the same throw when you are running at 3 m/s gives the ball a speed of 33 m/s. You can summarize your ordinary experience in adding velocities as

where v is the velocity with respect to the ground, u' is the velocity of the ball with respect to you, and u is the velocity of the ball with respect to the ground. This is called the Galilean velocity addition formula. However, when it comes to Special Relativity, it doesn't work. The correct velocity addition formula, which was first given by Einstein, is

Now suppose one observer moves with velocity v relative to another observer. The first observer shines a light beam directly ahead of him at a velocity c measured in his frame. The speed of light as seen by the second observer can be found from the velocity addition formula given by Einstein.

That's why c is a constant.

Problem II

A free neutron is unstable and spontaneously decays into a proton and an electron. Th mass of the neutron is M_n=1.67495∙10^-27 kg, the mass of the proton is M_p=1.67265∙10^-27 kg, and the mass of the electron is M_e=9.1095∙10^-31 kg. How much energy is released when the neutron decays?


Click HERE for answer

The Time Dilation Curve & Spaceships in Star Wars

The curve on the left hand side is the time dilation curve. As you see, the time dilation rate expands unlimitedly when approaching the speed of light. This curve also applies to Relativistic Momentum, Relativistic Kinetic Energy, Relativistic Doppler Effect, etc. (Click the graph to enlarge it.)


Did you realize that when a spaceship in Star Wars went into the hyperspace (actually, hyperspace isn't something like what you see in Star Wars.), the length of the spaceship decreases, then disappeared in the space. Actually, this phenomenon can be explained by the Special theory of Relativity.


The effect used to explain that is Length Contraction, also known as the Lorentz Contraction. The theory stated that comparing to a stationary observer, length of an object appears to be shorter. The formula is as below:

Here is the curve of the length contraction.

Like other formulae in the Special Relativity, there is a singularity when the velocity reaches the speed of light. In this case, length observed shall reduce to 0. So, when the spaceship goes into hyperspace, it approaches the speed of light, and the the length observed will reduce for a certain rate. That's why the length of the spaceship reduces before it disappear.

Mass Energy Relation - The God formula

In the days before, we had discussed about the relationships between Pythagorean Theorem and the Special Relativity. But today I am not gonna talk about the Pythagorean Theorem. Instead, I will talk about something more interesting, the Mass-Energy Relation of Special Relativity.

The most dramatic example of energy released by a change in an object's mass is the release of energy in nuclear reactions, especially in fission and fusion. So, how did Einstein came up with his "God Formula" ? First, we have to understand that light carries momentum.

In Maxwell's electromagnetic theory, a prediction was made. The theory stated that when light strikes an object, it exerts a pressure. The presence of this pressure implies that light waves carry momentum. Moreover, the theory requires that this momentum be proportional to the energy carried by the light wave. With experiments supporting the theory, the result is that a wave carrying an amount of energy E has a momentum given by:

where c is the speed of light and p is the momentum carried by the light wave. We shall use this result and the principle of conservation of momentum to derive a relation between mass and energy.

Let's consider a thought experiment given by Einstein. Take a look at the frame below. 

There are two frames above us. At t=0, the box is at rest and emitted a light beam (orange). Remember that the box is suspended so that it's free to move without friction. According to the conservation of momentum, the box must recoil with an equal but opposite momentum, of magnitude:

Where v is the recoil velocity of the box and E is the energy carried by the light beam. 

The light beam requires a time t to reach the other end of the box, where it's completely absorbed. During the time  L/c, the box moves a distance x as it recoils. When the light beam is absorbed at the other end, the box stops and return to rest, in agreement of conservation of momentum.

Since no outside forces act on the box, its center mass must not move during this event, even though the position of the box obviously changes. The only way this can happen is for a shift of mass to occur within the box as it moves, maintaining the position of the center mass. Consequently, there must be an equivalent mass m associated with the light, of such magnitude that while the box of mass M moves a distance -x, it's movement about the center mass is compensated by the light moving a distance L:

Since we know the momentum of the box, and thus its velocity, we can use this information along with the time of flight of the light to evaluate the distance x that the box moves by the multiplying the speed v to give

The elapsed time t is the time it takes for the light to travel the distance L. Thus, we have:

Using this value for x and solving for the energy E, we get:

This is Einstein's Mass-Energy Relation.

Problem I

After that much reading, it's time for some brain exercises.


Do you still remember the twin-brother, Obama and McCain? Okay, here's the story. When Obama was half way to Mars, he discovered that his greatest enemy, Bush, had installed a bomb in the spaceship. The bomb was programed to explode 1 year later. The only way to stop the bomb from exploding is to send it to the space station. The distance between the spaceship and the station is 1.3 light-year measured from the stationary space station. The max speed of which the space ship can travel is 80% speed of light. So can Obama get back on time and stop the bomb from exploding?


Click HERE for answers.

Proof(s) of Time Dilation

Perhaps most of you may think "You gotta be joking me" when you first read the Time Dilation theory. Time Dilation have already been proven. Below is the experiment used to prove Time Dilation. (You don't have to read if you believe in it XDD)


The first measurement of time dilation for radioactive particles moving at high speed relative to the earth was made in 1941 by Bruno Rossi and D.B. Hall. We will describe a similar measurement made in 1963 by D.H. Frisch and J.H. Smith(D.H.Frisch and J.H. Smith, "Measurement of Relativistic Time Dilation using μ-Mesons," American Journal of Physics, May 1963, p.342). The experiment involves the detection of muons, which are subatomic particles that are produced in the upper atmosphere and that rain down toward the earth with a speed close to the speed of light. As they Travel downward, some of them spontaneously decay in flight. Consequently, the number arriving at a medium altitude - say on top of a mountain - is greater than the number that survive to reach sea level. In their experiment, Frisch and Smith counted the number of muons having a narrow range of speeds near the peak of Mount Washington, New Hampshire. Then they went down to Cambridge, Massachusetts, and counted the muons that survived the trip down to sea level. The probability that a muon will decay, and thus its mean life, is determined only by forces within the muon itself. Therefore, any dependence of their mean life on their speed relative to us is due only to relativity. 

Let's examine Frisch and Smith's findings in detail. The Mount Washington apparatus was set up to detect and stop muons with speed of 99.5% the speed of light. the time intervals between the arrival of a muon in the detecting apparatus and its subsequent decay were measured. Measurements from one run corresponded to a mean life of 2.2 μs. The average number of muons per hour arriving at this detector was 563. The time required for the muons to travel the 1907m from the elevation of the mountaintop to the elevation of the lab in Cambridge, measured with a clock stationary with respect to the lab, is:

After 6.4μs, only about 27 muons per hour would be expected to survive the trip down. But when detectors were set up in Cambridge to count muons of initial speed 0.995c, they observed an average rate of 408 muons per hour. We conclude that the muons decay more slowly when they are moving rapidly relative to us than when they are at rest relative to us.


How does the observed count of 408 muons per hour correspond to relativity predictions? According to our predictions, 408muons per hour correspond to an elapsed time of only 0.7μs. This means that time runs slower in the frame of reference of the moving muons than in the lab by a factor 0.7/6.4. We can check these results against the prediction  of the Time Dilation formula by rearranging that equation for v and substituting for Δt₀/Δt:

This calculated value for the speed of the muon in the lab frame is consistent with the value previously established for the experiment.


This experiment confirms time dilation and supports the special theory of relativity. Furthermore, it shows that relativistic effects can become large when the relative speed approach the speed of light. The effect observed here are general effects of relativity and are not limited to the special case of radioactive decay.

Dr. Edwin Jones and Dr. Richard Childers, "Relativity", Contemporary Collage Physics, 2001

Singularity of Special Relativity

In Einstein's Special Relativity, he stated that nothing can travel at or over the speed of light. According to his theory, all physics laws break down when something travels as fast as light or faster than it. In this post, I will explain why this happens.


It all happens in a factor called the Lorentz factor γ, that is:

Like I said before, if we sub v=c into the γ, we get a division by zero error. Just like Time Dilation, if we sub v=c into that formula, time expands unlimitedly. If we sub v=c into the formulae of Special Relativity, we get unreasonable values. For example, Length Contraction, Relativistic Momentum, Relativistic Kinetic Energy, Relativistic Doppler Effect, etc. Here are the Calculation steps.


Length Contraction:

According to Einstein's theory, if something travels at the speed of light, to an observer on that thing, length of everything shall reduce to 0. 


Relativistic Momentum:

According to Einstein's theory, if something travels at the speed of light, it carries unlimited amount of momentum.


Relativistic Kinetic Energy:

According to Einstein's theory, if something travels at the speed of light, it carries unlimited amount of kinetic energy.

In these conditions, normal physics laws break down. No one knows what happens when something travels at the speed of light. It is a singularity of Special Relativity. Therefore, Einstein stated that nothing can travel faster than light.

Division by Zero (Part III)

In my last post, I used the Time Dilation to prove the division by zero. However, some of the people who read my blog told me that they still don't really understand the proof. So, in this post, I will explain the theory again.

Firstly, what is Galilean Relativity? Take a look at the graph below.

In the graph, observer A in (a) is in a moving vehicle and is throwing a ball upward. The path in (a) is observed by observer A. However, to a stationary observer (observer B), the path observed is the path in (b). Einstein applied this theory to the light beam in Time Dilation. The frame A below is (a) in the above graph while B is (b) in the graph.

Frame A                                                             Frame B

Now we have a clearer image on Galilean Relativity, we can understand the Time Dilation of Special Relativity easier. I hope you still remember the formula of Time Dilation.

Now, sub v=c into the formula, we get:

We had discussed about the division by zero error before. Bhaskara Achārya defined that n/0 = . So, how did time go infinite? Take a look at the picture below.

In the picture, 

θ is the angle between the ground and the path of which the light beam travels. The relationship between 

θ and the velocity is as below:

Let's suppose the height of the device is 1m, the proper time is 1s and it travels at the speed of light. Sub those information into the formula of Time Dilation, we get:

Since v=c,

. Sub that into the formula, we get:

So, the angle between the path and the ground is 0. So, the light beam will never meet A', or we can say that it will take time to travel from A to A', then to B. That's why n/0 is equal to

.