Appendix II
August 9, 2012 • 11:45AM

Demystifying E=mc²

A Technical Addendum

The late 1800's was a rich period of back and forth between experiment and theory. With the sensorium being extended into the microcosm with improved instruments there was a rapid change from long held beliefs to new fundamentals of physics. The conceptions of matter and approaches to science was challenged and developed.

Let us briefly review the fundamentals of Special Relativity. At the time of Einstein's work two branches of science, mechanics and electrodynamics, had developed independently of each other. Newtonian mechanics was based on action at a distance happening instantaneously as opposed to electrodynamics which was based on continuous propagating action. When scientists were confronted with describing events where these two processes occurred in conjunction we ran into difficulties. A question that comes up is: does the velocity of light depend on the motion of the source of light? Is there an absolute velocity in reference to an absolute space or are all motions relative? If you look at a double star system, where two stars are rotating around a center point, you would be able to test this question. Assuming the velocity of the moving body is imparted to light it emits, let us say one of the stars orbits at 50 km per second and that its revolution takes two days. As it was moving away from us, the light that it emits would travel towards us at 300,000 minus 50 km per second (or 186282 – 31 in miles.) A day later, it would be on the opposite side of the central point, moving towards us. The light it would emit then, would come at us with a speed of 300,000 + 50 km per second. Even though the first light left a day earlier, the second ray of light from the same star could reach earth first given enough time, because it's traveling faster. If earth happened to be eight light years away from this constellation (the time required for the rays of light to catch up to each other) the two rays of light would be seen at the same time; we would see the same star on the left as well as on the right. This, however, has never been observed.


The Chandra X-ray image of Sirius A & B, a double star system located 8.6 light years from Earth, shows a bright source and a dim source.

Special Relativity takes this observed fact, that the speed of light is unaffected by the motion of its source or carrier, as its main principle, though stated somewhat differently. Regardless of your motion, you will always see the speed of light will not vary, and will appear the same as if you were stationary. Not only that, all the principles of nature will be invariant, no matter your motion.

Lorentz Transformation:

Is it possible to see the velocity of light as the same between two observers, one stationary and the other in uniform motion?
It is, if we give up our fixed notion of space and time.



Space and time are no longer absolute independent entities or fixed units of measure; but rather space-time is one single metric determined by outside bounding forces. Space and time are merely shadows cast by physical principles. The invariant principle of the speed of light is the container which shapes space-time. The unnatural Newtonian model of absolute space and time is crushed. No longer can there be space without time, nor time without space.

Space-time is contracted with motion, but by how much? This is where the Lorentz Transformation comes in.1For more on the relativity of time, see Einstein's book: Relativity: The Special and General Theory chapter IX. Also the author's video: The Genius of Einstein Each reference frame has its own space-time, different from other reference frames because of their relative motion. The Lorentz transformation is what allows us to convert one reference frames measurements to another.

For example:


Light clock animation: A flash of light bounces between two horizontal mirrors placed a certain distance apart from each other on a train. The tick and the tock of this clock will be when the light hits the bottom mirror and then the top mirror. Both the stationary observer on the embankment and the moving observer on the train should see the speed of light as the same speed. But the stationary observer sees the light travel a longer distance. For the observer on the train the light moves up and down; for the stationary observer it moves diagonally. We can use the Pythagorean theorem to see what the conversion between reference frames is. The height of the triangle is measured by the moving observer as the speed of light times the time the moving observer measures between the tick and the tock (t) [Distance traveled equals the speed of light multiplied by time.] The event of the light traveling from one mirror to the other for the stationary observer, apparently takes longer, because the flash of light travels a longer distance. The diagonal is measured by the distance traveled by the flash of light, the speed of light multiplied by the time the stationary observer measures between his tick and tock. The base of triangle is the distance traveled by the train; i.e. the velocity of the train multiplied by the stationary observer's time traversed(T.)

c= velocity of light
t=time traversed as measured by moving observer.
T= time traversed as measured by the stationary observer

c² T² = v² T² + c² t²

divide both sides by c²

T² = v²/c² T² + t²

Minus v²/c² T² from both sides and combine

(1 – v²/c²) T² = t²

Divide both sides by (1 – v²/c²) and square root both sides

T = t / √(1 – (v/c)²)

The one observer's time traversed is slightly contracted as compared to the other. Time is not an ultimate measure, but depends on your motion. Every reference frame has its own time. Time is relative. Giving up absolute time was not something easy for Einstein, as it was something all people took for granted.

The absurdity of math is shown by the fact that the Lorentz transformation formula was obtained from two different physical hypotheses. Lorentz initially came up with this formula to explain that the speed of light was changing with reference to our motion, but that the difference could not be detected because the physical distance traveled was being contracted by electrons being pressed together from running into the “ether.” Einstein uses the same formula to explain that the speed of light is not changing with reference to our motion, and ha did away with the ether completely. For one use of the equation, the speed of light is not constant; for another use of the same equation, the speed of light is constant. Two different physical explanations for the same math. As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality. --Einstein2Where the Lorentz transformation breaks down:
 
what if v = c? What if v = something greater than c? Both cases give you answers that the formula cannot handle. You cannot conclude anything from your mathematical results. New physical evidence is needed to further guide our use mathematics. It is easy to criticize a theory for where it breaks down mathematically, but until we push our technology to actually reach a physical paradox, until we are able to actually experiment with what physically occurs at the barrier of the speed of light (like what was done with the sound barrier or the coulomb barrier) it is just idle speculation for the sake of being novel, stuck in the unfalsifiable world of math. Only physical hypotheses are constructive.

Stretching the Boundaries of Special Relativity

How does Special Relativity differ from Newtonian mechanics? It is not enough for a new theory to be able to account for all of what the old theory covers already; there has to be something new that Special Relativity can explain that the Newtonian theory of yore cannot predict. If we can locate where a difference will turn up, we can find a place where Special Relativity is testable, and we can determine which theory is actually superior.

Before we get into the nuts and bolts of this comparison we have to tell a short story on the state of physics at the end of the 1800's.

The discovery of the electron in 1887 by J.J. Thompson, changed the whole playing field of physics. The speed and size of the electron was magnitudes different from what we had known up to that point. The fastest motions we had to account for were animal motions, train locomotives or chemical explosions such as gunpowder propelled bullets, cannon balls and TNT. The Atomic theory had only recently gained general acceptance and the hydrogen atom was thought of as the most fundamental (i.e. smallest unit) building block of nature. From early experiments with the cathode tube (a near-vacuum tube with an electric discharge inside) it was discovered that the rays of energy that glowed in the tube acted as if it had mechanical properties, as a body with inertia, that could be deflected by a magnetic field. It was hypothesized that the ray was not just waves of negative electric energy as light was thought of, but that it was made up of tiny corpuscles. This hypothesis was proven by experiments using permeable metals. Then the task was left to determine how small these particles were. From the ray's deflection in a magnetic and electric field, experimenters such as J.J. Thompson were able to measure the velocity of the particles and their charge to mass ratio. The determined velocity was much greater than that of any known body and was fast enough to be comparable to the speed of light. The charge to mass ratio came out as the same ratio no matter what type of gas was in the tube. The charge to mass ratio showed that either this new particle had a charge close to 2000 times that of hydrogen, or that its mass was only a 2000th part of a hydrogen atom. An experiment was devised to measure the charge of an electron directly using a Wilson expansion chamber (cloud chamber,) and from those results it was calculated that the constant mass of an electron was 1/1830th that of a hydrogen atom, the smallest mass of which was known to exist. Even with such a small mass, the electron was known to attach itself to matter or elements and profoundly modify their properties such as electrifying or ionization etc. With a constant mass calculated for the electron no matter what gas it was produced from, it was believed to be the new building block of elements. In the following years experiments were done to accelerate the electron to limit of our ability (to approach the speed of light.) The electron ray was run through a loop in a cathode tube where it would repeatedly pass through an electric field, receiving a greater impulse each time (much like adding a constant force to accelerate a body floating in space.) It was found that instead of a simple constant increase in velocity (constant force = constant acceleration.) Not only that, also the electron's mass and kinetic energy would change with its velocity. This was previously unknown phenomena in the domain of mechanical processes and material bodies.

Nuts N' Bolts

In the Newtonian theory of mechanics there's a very specific law for the determination of velocity according to multiple moving bodies; it's called the addition of velocities: va + ve = w. If a projectile leaves a moving object (lets say a rocket leaving a jet) a stationary observer on the ground would simply say that the rocket is going the speed as seen from the jet, plus the speed of the jet.
At high velocities this law differs dramatically from what happens according to The Theory of Special Relativity (which would not have been testable before the age of the electron.) Instead we use a formula which could be considered as a Lorentz transformation of the Lorentz transformation.

The Addition of Velocities according to the Lorentz transformation for multiple moving bodies is:

w = the velocity of the moving object according to a stationary observer, vₐ and vₑ are moving reference frames.
(I'll spare you the fairly simple proof of this equation which uses calculus.)
This transformation allows us to look at a moving body leaving an already moving body. Basically we have three reference frames: velocity according to the stationary observer (w,) and two other reference frames in motion relative to the first, each with a fixed relative velocity differing from each other. Take the case of headlights being turned on on a moving car (this mathematical experiment is the same as testing whether the speed of light is affected by the motion of its carrier.)

Vₑ = c (The speed of light) va= the speed of the car




w=c

The speed of light as viewed from both the stationary (w) and moving observer is the same constant (no matter the speed of the vehicle!)

Try this with speeds much less than the speed of light, some everyday speeds, and the result is the same as if we simply used the Newtonian addition of velocities (v + w), which is what we see in common experience.


Addition of Velocities: From our perspective, the man's speed is not simply the speed of the segway, but the speed of the segway plus the speed of the ship.

But the two theories have differing answers for higher speeds.

Acceleration

To approximate a situation of acceleration, something Special Relativity is not set up to handle, we will use this modified Lorentz transformation, where the two different moving reference frames will be two successive moments of the same moving body, to see how Special Relativity compares to Newtonian mechanics.

Acceleration according to Newtonian physics is defined as a constant force being applied. Imagine a satellite in space whose propulsion engine gives it a constant thrust. For example if you go from a speed of 0 to 200,000 meters per second, you will in the second second increase in speed by the same amount and so on arithmetically:

0 → A the rate of change is 200000, A → B rate of change is also 200000.

The speed at moment A is 200000, at B is 400000, at C is 600000.

So, according to Newtonian mechanics, no matter what the speed of the body is at, it takes the same amount of force to increase the same amount of speed. Inertia only depends on the mass of the body.

But in Special Relativity3See Einstein and Infeld's discussion of this in The Evolution of Physics -Relativity and Mechanics the measurements change according to the Lorentz transformation; the rate of change of speed does not stay constant with equal force applied, but continuously decreases.

where vₐ and vₑ both equal 200,000 meters per second.

[speed at the second moment.]

vₐ = 200,000 (constant force) and now vₑ = 276923.9 (current speed)

[speed at the third moment.]
and so on...

Look at the rate of change. From the first to the second to the third moment, how much are we increasing in speed? Our rate of change in speed is decreasing even though we are inputting the same amount of energy. What physically could be the cause for this increase in resistance? Where is the energy going?

It seems as though the increase of kinetic energy of a moving body increases its resistance. If two bodies have the same rest mass, the one with greater kinetic energy(motion) resists the action of an external force more strongly. In Special Relativity, not only does the resistance of a body depend on the mass of an object, but it also depends upon the velocity. This seemingly violates the conservation of energy. More force is required to go faster, and an apparently infinite force is required to go faster than the speed of light. All energy, something which use to be the opposite of matter, resists change in motion, as if it was matter. Energy behaves like mass.
(As it turns out, this was the best explanation available to account for what was happening with the electron.)
You can see where the first inclinations of mass being equivalent to energy came from.

Testing Special Relativity4See Einstein's 1905 paper, Does the Inertia of a Body depend upon its Energy-Content and Out of my Later Years, An Elementary Derivation of the Equivalence of Mass and Energy.

If the theory of Relativity states that all of nature's laws hold true no matter your reference frame (whether stationary or in uniform motion the principles of physics will appear the same) let us try out some laws we know, like the law of the conservation of energy, which seemed to break down at high velocities.

To prove Special Relativity is true, the law of the conservation of energy (as all laws) must hold.

Let us assume that a star that contains the energy Gₒ sends out two rays of light symmetrically in opposite directions. Let these rays have a total amount of energy E, so that the energy of radiation E/2 is emitted in each direction. We shall use Gₒ to denote the energy of the star with respect to a stationary system, and Hₒ for a system moving with a velocity “v” relative to the first system. Since the two rays start out symmetrically from the star, the star will experience no reaction and will remain at rest. The only difference between the two perspectives or systems, is the energy created from motion of the second system. Because the relative velocity between the two systems is constant, the difference in kinetic energy between the two states (the one before emission and the one after) should be 0.



The energy of the stationary system according to the conservation of energy, which requires that the amount of energy before and after the emission is equally accounted for, is:

Gₒ = G1 + ½ E + ½ E
Energy at initial state Energy subsequent after the event. Light energy in both directions
G1 + E

The energy of the same situation from a moving reference frame is:

{The Lorentz transformation converts for us what we would see for the light emission from the moving reference frame.}

To test the law of the conservation of energy in Special Relativity we will subtract the two accounts of energy for the reference frames from each other, hoping to find only the difference of a constant in kinetic energy, created from the constant velocity of the moving reference frame.


- Gₒ = G1 + E
____________

As Einstein says: The two differences of the form H – G occurring in this expression have simple physical significations. H and G are energy values of the same body referred to two systems of coordinates which are in motion relatively to each other, the body being at rest in one of the two systems. Thus it is clear that the difference H – G can differ from the kinetic energy K of the body, with respect to the other system only by an additive constant C, which depends on the choice of the arbitrary additive constants of the energies H and G.

(Hₒ- Gₒ) - ( H1- G1) =

The initial difference in energy between the two systems should be the same difference they have subsequently. The difference of their differences should be 0.

Kₒ Initial difference in kinetic energy, K1 subsequent difference.

Let us analyze this part

This is something slightly larger than 1, because of the large quantity of the speed of light squared in the denominator. 1, divided by something a little less than 1 (1 minus and extremely small faction) equals something a little greater than 1. This minus 1,

will be a tiny fraction; but it means there's something left over, instead of the expected 0.

The initial state has more kinetic energy than the subsequent state. There's been a loss of kinetic energy, even though the relative velocity has not changed. Kinetic energy (vis viva) is a measure of the power of motion, recognized as the form Leibniz gave it: ½ mv²=K. What is changing to effect K if v, the velocity, is constant?

Is the law of the conservation of energy incorrect or is Special Relativity wrong?
Do not get discouraged or settle for a quick answer. Let us push forward and explore this breaking point. We will use a few tools to transform the appearance of our result,which some may call a trick. ;)
Looking at alone, we will get rid of all divisions and turn it into a power:

Abra Cadabra!!!

Now, we will throw c (c²– v²)-1/2 into a series expansion. This is a tool which comes from Pascal's Triangle that turns binomials (something with two parts that's raised to a power) into a series. It looks generically like this:

(x+y) ͫ = x ͫ + mx ͫ ‾¹ y + [m(m-1)/2] x ͫ ‾² y² + …
So, now we will put our binomial into the expansion.
ignoring the smaller exponents.

Multiply by c

Multiply through.

Plug

back into
:

and distribute.

Kₒ - K1 = ½ [E/c²] (v²)

In this formulation, kinetic energy is equal to ½ energy divided by the speed of light squared, times the velocity squared. We are use to seeing kinetic energy written as ½ mv². But what's in the place of m?

m=E/ c²

Thus, although the star has remained at rest after the emission, the difference Kₒ - K1 is not equal to zero but to ½ E/c² v² (the initial state has more energy than the subsequent state.) In consequence of the amount of radiation E emitted by the star, its kinetic energy has become reduced, just as if its inertial mass has decreased by the amount ½ E/c². The loss of kinetic energy is due to a loss of mass which disappeared in the form of energy. Because the mass which energy represents was so small, it went undetected and was regarded as massless until the time of Einstein.

Being familiar with Einstein's thinking we know that what we just worked through is no how Einstein arrived at his discovery. After all the rigamarole, it's clear that Einstein knew where he was going. He didn't just stumble across E=mc² accidentally or simply follow the most logical steps in order to arrive at this profound conclusion. The math alone shows he knew what form he wanted and merely wrote his proof after the fact. He already knew where he had to go without ever being there before and without a precise map of how to get there. The idea already existed inside him and it was merely a process of tracing out a trail for others to arrive there too.

How did Einstein know how to set up his thought experiment? Einstein had the foresight of knowing what he was looking for without knowing it by name, leading him directly to the crux of the matter and allowing him to recognize it when he got there. A strong intention for a future goal determined his present actions. This experiment predicted something that no modification or extension of previous theories could have accomplished. This was accomplished by focusing on the physical paradoxes of the previous theories, instead of avoiding and trying to eliminate them. With this approach, Einstein was able to shake off the assumptions that gripped the generation that came before, discover a more perfected idea of nature which accounted for all the facts known before, resolved current paradoxes, and forecast physical states that only existed in potential.

E=MC² Application for Energy and Technology

Up until the time of Meitner, it was generally believed that firing a neutron at an atom of a heavy element would increase its mass, creating a new element. Through experiment, results showed that this hypothesis, along with the atom, were not so stable.

Following up an observation of Curie and Savitch, Hahn, and Strassmann found that a group of at least three radioactive bodies, formed from uranium under neutron bombardment, were chemically similar to barium and, therefore, presumably isotopic with radium. Further investigation, however showed that it was impossible to separate those bodies from barium, so that Hahn and Strassmann were forced to conclude that isotopes of barium [a smaller element] are formed as a consequence of the bombardment of uranium with neutrons.

At first sight, this result seems very hard to understand. The formation of elements much below uranium has been considered before, but was not rejected for physical reasons, so long as the chemical evidence was not entirely clear cut. The emission, within a short time, of a large number of charged particles may be regarded as excluded by the small penetrability of the 'Coulomb barrier,' indicated by Gamov's theory of alpha decay.

On the basis, however, of present ideas about the behavior of heavy nuclei, an entirely different and essentially classical picture of these new disintegration processes suggests itself. On account of their close packing and strong energy exchange, the particles in a heavy nucleus would be expected to move in a collective way which has some resemblance to the movement of a liquid drop. If the movement is made sufficiently violent by adding energy, such a drop may divide itself into two smaller drops. – Disintegration of Uranium by Neutrons: A New Type of Nuclear Reaction by Meitner and Frisch 1939.

Not only did these results reshape scientist's conception of processes of the nucleus, it challenged the principle underlying chemistry; the conservation of mass. The mass of the elements left over after bombardment did not equal the mass of the original element. Where did the missing mass go? It converted to energy according to Einstein's e=mc².

Moreover, the estimated difference between the mass of a uranium nucleus (plus an extra neutron) and the slightly smaller joint mass of the tow fragment nuclei meant (because of Einstein's mass-energy equivalence) the liberation of a large amount of energy; and the mutual repulsion of the two fragments would make them fly apart with just that amount of energy. Lise Meitner, a biography by O.R. Frisch


This was one of the first direct experimental proofs of Einstein's forecast that mass can be transformed into energy, close to 30 years after the discovery.

Fission:

The isotope of Uranium, U 235 is hit with a neutron and disintegrates into two smaller elements, Krypton and Barium (isotopes 92 and 141 respectively,) three neutrons, and energy.

U 235 + a neutron –> Kr 92 + Ba 141 + 3 neutrons
235.0439 + 1.0078 –> 91.9262 + 140.9144 + (3 x 1.0078)
236.0517 235.864

Mass defect?

Calculation of energy:
C = 3 x 1010 cm/sec

Even though fission came first in experimental results, fusion was the initial pursuit. It was believed that the entirety of the periodic table, all elements, were built up from the basic building block, Hydrogen. Therefore Helium would be made up of 4 Hydrogens. But the weight of four hydrogen did not add up to the weight of Helium. It was hypothesized by Eddington that the missing mass was the energy required to fuse these elements, as explained by Einstein's E=mc2.

Fusion:

From The internal constitution of the stars (1920) by Arthur Eddington

Study of the radiation and internal conditions of a star brings forward very pressingly a problem often debated: What is the source of the heat which the sun and stars are continually squandering? The answer given is almost unanimous – that it is obtained from the gravitational energy converted as the star steadily contracts. But almost as unanimously this answer is ignored in its practical consequences. Lord Kelvin showed that this hypothesis, due to Helmholtz, necessarily dates the birth of the sun about 20,000,000 years ago; and he made strenuous efforts to induce geologists and biologists to accommodate their demands to this timescale.
(One of the consequences of this hypothesis for the astronomers would be that most of the universe is much younger than our sun.)
From all sides – biology, geology, physics, astronomy – it would be objected that the suggested source of energy was hopelessly inadequate to provide the heat spent during the necessary time of evolution; and, so far as it is possible to interpret observational evidence confidently, the theory would be held to be definitely negative. Only the inertia of tradition keeps the contraction hypothesis alive – or rather, not alive, but an unburied corpse.

F.W. Aston's experiments seem to leave no room for doubt that all the elements are constituted out of hydrogen atoms bound together with negative electrons. The nucleus of the helium atom, for example, consists of 4 hydrogen atoms bound with 2 electrons. But Aston has further shown conclusively that the mass of the helium atom is less than the sum of the masses of the 4 hydrogen atoms which enter into it; and in this at any rate the chemists agree with him. There is a loss of mass in the synthesis amounting to about 1 part in 120, the atomic weight of hydrogen being 1.008 and that of helium just 4. Now mass can not be annihilated, and the deficit can only represent the mass of the electrical energy set free in the transmutation. We can therefore at once calculate the quantity of energy liberated when helium is made out of hydrogen. If 5 per cent of a star's mass consists initially of hydrogen atoms, which are gradually being combined to form more complex elements, the total heat liberated will more than suffice for our demands, and we need look no further for the source of a star's energy.

Mass of hydrogen: 1.008
If all elements are made up of Hydrogen, Helium would be 4 of them.

1.008
x 4 = 4.032

Helium's actual mass: 4.0026 grams.

Mass defect:
4.0320
-4.0026 = .0294

So where did the mass go? This is the energy of fusion.

Calculate energy:

Antimatter

The solutions to sustained fusion reactions will take an understanding of the shape of physical spacetime and its singularities, as opposed to brute Newtonian force. The question of antimatter is a step up from that challenge. Even though an anti-particle is much smaller than an entire atom, its mass is completely converted to energy.

The Scientific Monthly
January 1934
Discovery and early history of the positive electron
He (C.D. Anderson) observed the track of a particle which was proved to be positive; but this by itself is no sensational statement; there are various familiar kinds of positive corpuscles of much greater mass than electrons – alpha-particles and protons especially – and one must first of all inquire whether Anderson's particle could have been one of these. All these, however, are ruled out by the length and the appearance of the track. First, as to the appearance: As every one knows who has studied cloud-chamber pictures, the tracks of massive particle such as alpha-particles and protons are always much fatter and thicker than those of fast electrons. The massive particles produce many more ions per unit length of path, and therefore many more droplets of condensed water, than do the electrons.

Not only would it have been thicker, it would also have been much shorter; for the massive particle would have squandered its energy so rapidly in forming ions that its course would have come to an end in five millimeters, whereas the actual path on one side of the plate is five centimeters long.
Thus far, Pasadena and Cambridge had waited for the cosmic rays to furnish them with positive electrons; but now they began to find out how to produce these at will, and this in importance was not far behind the discovery itself. Chadwick and Blackett and Occhialini were the first to manufacture (if so crude a word may be permitted) the positive electron. Just outside the wall of the Wilson chamber they placed a piece of beryllium exposed to constant bombardment by the alpha-rays of polonium; this, as had been discovered only a little earlier, is a “source” from which both neutrons and high-frequency photons are continually being emitted;...A positive electron is supposed to come into being by virtue of a transmutation more drastic and amazing than any hitherto effected or imagined: it is supposed that a photon transmutes itself into a pair of electrons, one of each sign.

Such a transmutation – if it happens – leaves the net charge of the universe unchanged. If in addition it is to leave the net energy of the universe unaltered, we must invoke the principle of the equivalence of energy and mass: Einstein's relation,
E=mc2
where E stands for energy and m for mass and c for the speed of light in vacuo. The masses of two electrons at rest, when translated into units of energy according to Einstein's relation, amount to about one million electron-volts (1.02x106 is more nearly right, but the even million is a good round number perfectly satisfactory for these data.) The residue of the energy of the photon – its original energy, minus this million – then remains over; it might go into kinetic energy of the electrons, or into a new photon, or divide itself between these forms.

Here we witness, it may be, the disappearance of the last apparent barrier in physics: the barrier which seemed to separated the substance of electricity and matter form the substance of light.

Antimatter calculation:
mass of a proton: 1.67262158 x 10-24 grams

Multiply by 2 for the mass of a proton + an antiproton:
1.6725 x 10-24
x 2 = 3.345 x 10-24 grams No mass defect!

Energy calculation:

Footnotes

1For more on the relativity of time, see Einstein's book: Relativity: The Special and General Theory chapter IX. Also the author's video: The Genius of Einstein
2Where the Lorentz transformation breaks down:
what if v = c? What if v = something greater than c? Both cases give you answers that the formula cannot handle. You cannot conclude anything from your mathematical results. New physical evidence is needed to further guide our use mathematics. It is easy to criticize a theory for where it breaks down mathematically, but until we push our technology to actually reach a physical paradox, until we are able to actually experiment with what physically occurs at the barrier of the speed of light (like what was done with the sound barrier or the coulomb barrier) it is just idle speculation for the sake of being novel, stuck in the unfalsifiable world of math. Only physical hypotheses are constructive.
3See Einstein and Infeld's discussion of this in The Evolution of Physics -Relativity and Mechanics
4See Einstein's 1905 paper, Does the Inertia of a Body depend upon its Energy-Content and Out of my Later Years, An Elementary Derivation of the Equivalence of Mass and Energy.