Special Relativity - GPRA Project

Mass-Energy Equivalence

A long running dispute exists as to who discovered the mass-energy equivalence relation known famously as the equation `E=mc^2`. Many attribute this relation to Einstein, but that isn’t 100% correct. The mass energy relation was known empirically before Einstein through experimentation, and Henry Poincare derived the relation equation in 1900, 5 years prior to Einstein, in what is described as an “associated electromagnetic radiation energy with a “fictitious fluid” having momentum and mass”. Here is Poincare’s 1900 derivation:

`m_{em} = E_{em} / c^2`

The similarity couldn’t be more striking.

There is also an Italian scientist named Olinto De Pretto, an industrialist from Vicenza, who also published the equation `E=mc^2` in a scientific magazine called Atte in 1903, three years after Henry Poincare, and two years before Einstein.

So it can be said unequivocally that the mass-energy equivalence equation was not discovered by Einstein, but that doesn’t mean Einstein didn’t contribute to the relation. You see, his work with SRT gave context to the relation that didn’t exist prior, and more importantly, `E=mc^2` is derived as a consequence of SRT equation which further validates SRT framework against empirical mass-energy equivalence data. And that’s the truly amazing part of `E=mc^2` from SRT. It is that `E=mc^2` is not discovered in SRT, but is a derivation and validation of SRT.

The derivation of mass-energy equivalence is as follows in relation to SRT:

(1) `E_k = int F dot dx = int mv dot dv = 1/2 mv^2`

Where F is force at `[{kg} dot {m/s^2}]`:

(2) `F = ma`

Where `v` is velocity, and `m` is mass. The `x` symbolizes distance.

That first equation can also be expressed in the following what using some simple substitution as:

(3) `E_k = int F dot dx = int d(mv)/dt dot dx = int d(x)/dt dot d(mv) = int v dot d(mv)`

Where m is relativistic mass m’, just for clarity.

Follow simple integration by parts we get:

(4) `E_k = int ( v^2 dm + mv dv )` … keep this one in the back of your mind

Now from SRT mass-velocity relation, we get the following:

(5) `m = m_o / sqrt(1-v^2/c^2) = m_o c / sqrt(c^2 – v^2)`

(6) `c^2 – v^2 = ({m_o c}/ m)^2 = {{m_o}^2 c^2}/ m^2`

(7) `{m_o}^2 c^2 = m^2 (c^2 – v^2)`

(8) `{m_o}^2 c^2 = m^2 c^2 – m^2 v^2`

This equation alone can be used as the relativistic mass-energy-momentum conservation equation. We could stop here.

Now the key is to use dimension of the variables, because SRT is about changes to dimension that are measured essentially invariant from an observer in their own inertial frame.

Taking the dimension of mass only:

(9) `(m)(m) c^2 = (m)(m) c^2 – (m)(m)(v)(v)`

Remove the extra mass while moving the last term around:

(10) `m c^2 = m c^2 + mv(v)`

Though this equation (10) can be updated to look very closely like equation (4), the context of the terms involved have been lost. Which `m c^2` is relativistic?

To resolve this, we approach this from a slightly different direction using SRT’s geometric Pythagorean application (described in prior post) we get the same thing again:

(11) `c^2 = (c-v)^2 + v^2 = {c’}^2 + v^2`

Applying mass to this equation we get:

(12) `m c^2 = m (c-v)^2 + m v^2`

Notice in (11) and (12) that the terms on the right are object velocity dependent, while the term on the left is a representation of constant c.

This turns (12) into:

(13) `m c^2 = m (c’)^2 + m v^2`

(14) `m c^2 = m (c’)^2 + mv (v)`

Notice the resemblance to equation (10), but this time it gives a better context to which terms are constant and which are not.

The term on the left `m c^2` is using constant c, meaning that mass in that term cannot be constant. It is variable.

The first term on the right `m (c’)^2` is using relative c’, which is still measured as constant c in the moving inertial frame, but it also does mean that the associated mass is moving also, meaning that mass is also not constant. Essentially neither c’ and m in this term are not constant. They are both variable. Therefore taking the dimensions of both terms, where c’ = velocity = v, and m = relativistic mass, we can substitute these variables, which is done in equation (15).

The last term on the left `m v^2` has without doubt a variable velocity v, meaning that mass in that term is constant, that this mass is the initial inertial mass of the object at `v = 0 [m/s]`.

Now applying the dimension of variable terms, and defining the incremental variables to both sides, we get:

(15) `c^2 dm = v^2 dm + mv dv` … notice the relation to equation (4)

Using equation (4) and (15) we easily deduce that Kinetic energy `E_k` is related to the integral of `c^2 dm`:

(16) `E_k = int c^2 dm = mc^2 – m_o c^2 = m_o c^2 (1/sqrt(1-{v^2}/{c^2}) – 1)`

And from this, the total energy of an object is easily deduced to following using the mass relativistic difference between `m` and `m_o` as per the application of solving the integral:

(17) `E_t = mc^2 = E_k + m_o c^2 = {1/2}mv^2 + m_o c^2`

Where `E_k` is relativistic kinetic energy.

This equation (17) is the full mass-energy-momentum conservation relationship where `E=m_o c^2` is only part of. As you can see, there is much more physical context given.

If `v = 0` where the object is stationary (relatively in the context of SRT) and making `E_k = 0`, we get:

(18) `E_t = E_k + m_o c^2 = 0 + m_o c^2 = m_o c^2`

And there you have it. Even if the object isn’t moving, it still possesses immense internal energy. The total relativistic energy is a sum of kinetic and stationary inertial energy. To know the kinetic energy, total energy, or stationary inertial energy of a mass, you can use this equation.

Tests and Validations

Validation of `E=mc^2` is best done where quantum particles with mass are transmutated into particles without mass but possessing energy.

In pair-annihilation, two particles, electron and positron are merged and annihilate each other producing two photons.

`e + e^{+} to 2{gamma}`

The photon come out of this process with no mass (of course) but having a combined energy of

`0.511 [MeV] = (0.511×10^6)(1.6022×10^{-19} [J])`

`0.511 [MeV] = 8.18712253 × 10^{-14} [J]`

Which is equal to

`2E = 2m_e c^2 = 2 (9.1×10^{31}[kg]) c^2`

`2E = 2m_e c^2 = 8.17864224 × 10^{-14} [J]`

Again, essentially equal with a marginal difference of 0.1%.

The reverse creation pair-production process yields an electron and positron from one photon with the same energy matching the fermionic masses of the electron and positron.

References:

“Did Einstein Discover E=mc2?”, http://physicsworld.com/cws/article/news/2011/aug/23/did-einstein-discover-e-equals-mc-squared

“Einstein’s E=mc2 ‘was Italian’s idea'”, http://www.theguardian.com/world/1999/nov/11/rorycarroll

“Mass–energy equivalence”, https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence, retrieved March 2016

“E=mc2 passes tough MIT test”, http://news.mit.edu/2005/emc2

S. Rainville, J.K. Thompson, E.G. Myers, J.M. Brown, M.S. Dewey, E.G. Kessler Jr., R.D. Deslattes, H.G. Börner, M. Jentschel, P. Mutti, D.E. Pritchard. 2005. A direct test of E = mc2. Nature. Dec. 22, 2005. http://www.nature.com/nature/journal/v438/n7071/full/4381096a.html

Mechanistic Causes Behind Special Relativity

For most particle and subatomic physicists, Special Relativity is beyond doubt a highly predictive and accurate theoretical framework. To them, the empirical and very physical phenomena mapped mathematically by Special Relativity Theory (SRT) is very real and very measurable. But understanding the cause for these effects is not easily intuitive if all you have is data on the measured effects, which is what leads some to question these effects and SRT individually or collectively. I truly believe the reality of these effects, and that of the best modern predictive framework mapping them, Special Relativity (SRT).

This doesn’t mean that Special Relativity (SRT) is absolutely correct void of all problems. One major problem with Special and General Relativity is the disparity between gravity and quantum theory. Another is the in ability by General Relativity Theory (GRT) to predict the anomalous rotational velocity of spiral galaxies without the use of yet discovered hypothesized Dark Matter. Then there are the extremes of SRT and GRT that have truly never ever been experimentally verified, like having a quantum particle actually achieving the velocity of c in a particle accelerator, even though they’ve come close matching the Lorentz factor velocity curvature.

In my own research into fractal scaling cosmological (FSC) framework, what became very apparent to me when unavoidably formulating the explicitly defined spatial fractal quantum medium, were the underlining mechanistic causes for the very real phenomenological effects mapped geometrically, yet superficially (topologically), by the mathematical physical framework of Special Relativity Theory (SRT) developed by Einstein. These mechanistic causes can all be explained by matter objects interacting with a medium with very specific characteristics. Those characteristics are explicitly described in FSC, but superficially some of those characteristics held by the quantum medium (QM) are:

1) is a super-fluidic medium
2) is very rigid
3) has extremely quick malleability to restore itself into a state of least density distortion
4) is not made of any known tangible material and substance
5) is deeply entrenched inside macroscopic matter objects
6) and change to the density of this medium directly affects the relative dimensions of time and space (distance)

I am certain this space-time medium is related to the ill defined Higgs Field, but FSC’s quantum medium gives a very explicit, and more importantly, unavoidable description of the medium that is not only highly intuitive but also very predictive. Basically, this medium fills all of Universal space to varying densities measured indirectly as varying gradient gravitational fields. Essentially this medium is the “ether”, but not the classical versions of the ether, which is a very important distinction. It is the modern “new” ether to quote Einstein himself in response to a challenge by Lorentz.

Read the following to get Einstein’s actual views on the ether concept, which may surprise some readers. http://www-history.mcs.st-and.ac.uk/Extras/Einstein_ether.html

The problem is Einstein didn’t explicitly describe this medium in his Special Relativity Theory (SRT), though he did re-introduce such a concept explicitly described in his General Relativity Theory (GRT) many years later using continuum mechanics. I believe this, along with changes to fundamental dimensions, introduced a level of intangible non-intuitiveness in the theory that has bothered many over the decades. Simply, SRT was void of such an “etheric” medium in the derivation of the theory whose mathematically predictive results matched empirical data…or was it really devoid of the “ether”? You see, Einstein used the Principle of Relativity and Constancy of Light in any inertial frame of reference, without the use of a mechanical ether concept, to essentially re-derive the work of Hendrik Lorentz in the Lorentz Ether Theory (LET). In fact, the Lorentz transforms are explicitly used by Einstein in SRT. Therefore, indirectly Einstein is indeed referencing an “ether” medium if only through the works of Lorentz in LET. What Einstein ingeniously did was map the effects, though in that time it’s better said he predicatively mapped these effects, independent of any assumed cause for them in order to match the empirical anomalous data of the time. It is essentially what Newton did with his gravity theory many years prior. It is called famously “Hypotheses non fingo“, or “I frame no hypotheses.“ What this means is that Einstein, like Newton, framed no assumption to the causes of the effects measured empirically. In this case it was the Michelson-Morley Experiment (MMX) and Maxwell’s Electromagnetic framework, with all it’s supporting empirical evidence, that Einstein had to map into a mathematically predictive framework…and he did so successfully, even though it appeared highly non-intuitively, especially because the transformation on the dimension of time itself which was new and perplexing to some.

Now Einstein did make some assumptions outside of a possible underlining cause for these phenomenological effects, which makes me strongly question whether he truly was devoid of the “ether” concept in formulating SRT. I truly believe Einstein used an “ether” model as a conceptual tool in his derivations. Why? Indirectly, in the era he lived in, the “ether” was a mainstream concept. The fact he referenced Lorentz’s LET work, he understood the latest contemporary “ether” concepts. Also, his assumption of the Constancy of Light involved the concept that the Speed of Light (SoL) was independent of the light’s source speed (the photon emitter). This effect is analogically and empirically observed with waves in fluidic mediums, like water. This is coincidentally truly something very “etheric” to assume independent of using a fluidic medium concept.

The point of all this is that the phenomenological effects mapped mathematically by SRT, and GRT also, can be explained with a very unique “ether” model, a model in which distortions in the density of this “ether” medium changes the dimension of time and distance relatively as observed inside and outside those density distortions relatively. This is what my own research into FSC is exploring using FSC’s quantum medium which I believe sheds good light on the gravity vs. quantum theory disparity in modern physics.

The best way to conceptualize the physical phenomena mapped by SRT is to envision the Doppler Effect using a spacecraft racing against a light beam emitted from the craft itself, where both are traveling in a smooth and even space-time continuum spatial “medium” density (quantum medium in FSC). The diagram depicts a snapshot of this race at a constant arbitrary time t1 using three different velocities for the spacecraft. (This diagram is not geometrically correct, meaning the angles and line lengths are not accurate.)

At v=0, no Doppler Effect is measurable.

At v >> 0, a measurable Doppler Effect is present.

And at v = c, the Doppler Effect has reached its “luminous” Mach limit.

What is important to see here is that the geometric Pythagorean representation of competing velocities also represents an aerodynamic/hydrodynamic-like compression of the medium in front of the traveling matter object. The region in front of the craft is called the bow shock, which is compressed medium due to the craft pushing through the medium using propelling force. This compressed medium is then essentially denser compared to the medium’s density if the craft was at v = 0. It is also denser simply because the object is traveling through the medium, and equally the medium is traveling through the object, at a faster velocity. Essentially the pressure and density of the medium on the object’s material structure is proportional to the velocity of the object through the medium. As an aside, this increased pressure and density of the quantum medium due to relative velocity is essentially the mechanistic cause for scaling in my fractal scaling cosmological (FSC) framework of macroscopic objects. Due to this increased density, the mechanistic cause for time dilation gives rise to time dilation and slows time down by a specific velocity dependent functional factor as to have the observer in the craft to still measure the beam of light as traveling at velocity c away from the craft, even though another more stationary observer measures a luminous Doppler Effect between the craft and the leading beam of light, meaning c – v is still measured as c by the moving observer in the craft. Mind boggling isn’t it?…or is it really. Keep in mind the mystery of these effects dissipates as you come to fully realize the mechanistic cause behind the entire dynamic (keep reading).

`c’ = c – v = sqrt(c^2 – v^2)`

In particle accelerators, electric force pushes charged quantum particles against and through this medium. This pushing through the medium can have measurable effects matching the effects mapped by SRT, such as mass increase (MI) and length contraction (LC). But it was time dilation (TD) that really set apart the effects mapped by SRT and those mapped by LET and normal aerodynamics / hydrodynamics. This is very important to note and realize. Why? Because the medium we’re dealing with here is not made of normal known matter like water or air, but is made of the stuff that makes up quantum particles (a literal quantum medium as per FSC). I call it quite justly the quantum medium. In FSC, this medium has an important self-similar fractal property, which gives significant theoretical insight into its composition. This medium is a very highly entrenched medium permeating through all macroscopic material matter objects, to varying degrees, including clocks and human brains. Meaning that an increase in the density of this space-time medium will have a direct effect on the measured and experienced passage of time at the quantum scale up (at the least).

Length Contraction (LC)

Length contraction in particle accelerators is simply an artifact of the medium’s drag experienced on charged quantum particles being propelled by very strong electric force fields in the accelerator, which utilize the particle’s charge to propel it. Essentially the artificially induced electric force propellant pushes the particle against and through the quantum medium whose drag coefficient increases as the particle reaches c. Basically, the electric force propellant is crushing the propelled object against the quantum medium, against the space-time continuum, which essentially flattens it as v approaches c.

At v = c,

`F_{applied} – F_{drag} = 0`

`l’ = l sqrt{1 – v^2/c^2}`

Mass Increase (MI)

Mass increase is caused by the same effect that causes length contraction. Essentially the increasing quantum medium drag makes the object harder to accelerate, even though the applied force to propel it stays the same.

`F_{applied} = m a_{measured}`

`m = F_{applied} / a_{measured}`

The applied force is constant, and the measured acceleration of the particle is fairly easy to obtain in a particle accelerator, but the problem is as the particle reaches c, the measured acceleration falls out of sync with the force applied to accelerate it. This disparity is measured as a difficulty to move the particle due to an assumed increase in mass, though it’s better to call this calculated mass “apparent” mass.

`m’ = m / sqrt{1 – v^2/c^2}`

Time Dilation (TD)

Like I said before, time dilation is really what sets these space-time continuum, quantum medium (qm), effects apart from other known and more tangible types of mediums (ex. water or air). In this application of a very unique version of aerodynamics/hydrodynamics, Time Dilation (TD) is caused because of an increase in the density of the quantum medium (qm) as the object’s velocity v reaches c, where the object starts experiencing an increasing drag due to a aerodynamic /hydrodynamic shock bow compression wave caused by the moving object in the region of the object itself (see diagram above). This increase in density makes all quantum particles in the object more difficult to move, due to an apparent increase in their mass. This includes the electrons in a digital clock and the electrons in a human brain. This includes the atoms in mechanical devices and the atoms in a human body. This increased quantum medium (qm) density directly affects our perception of time mechanically and neurologically.

`density_{qm} = function(v,c)`

`pressure_{qm} = function(v,c)`

`t’ = t / sqrt{1 – v^2/c^2}`

Understanding the mechanistic cause behind Time Dilation (TD) essentially demystifies the effect and time itself.

What is Time Dilation?

Time dilation can be difficult to understand, which is why I’ll dedicate this post to time dilation which is a very real physical phenomena, and not some invention by Einstein. Einstein, and others before him like Lorentz, conceptualized that time wasn’t constant, more so Einstein where Lorentz didn’t fully agree or understand what the mathematics was fully describing. Einstein solidified the concept with his work, which lead experimenters later on to validate his predictions that it existed in the manner he described, without Einstein giving an underlining mechanistic cause, just a superficial geometric description of the effect. Where Einstein’s work stops short (incomplete), my Fractal Scaling Cosmology (FSC) framework explores and explicitly describes a mechanistic cause for time dilation, along with many other things (see https://www.gpofr.com/distortion-time-dilation-experiment/).

What is time dilation? Time dilation is a compensating attenuation on the “normal” passage of time to compensate for the observer’s frame velocity in reference to c, so that the the observer in any frame will always measure c. The velocity of light traveling between different observers, who are themselves traveling at different velocities, travels at a constant c in the same space-time density (quantum medium density as per FSC).

The following diagram is in relation to my own work but touches on SRT and common c in all frames.

I’ll be approaching the topic of time dilation in a slightly different manner mathematically. I personally love a Pythagorean application to derive modified x and t dimensions that mimics the Doppler Effect on the experience of dimension itself. The Pythagorean approach to time dilation gives a view conceptualization of the actual stationary dimension vs. the experienced dimension involved in a simple geometric presentation.

From the Pythagorean diagram above, notice carefully at v = 0 for a stationary observer, c’ = c. This means that our reference to time is through the reference to a constant c, the only true reference. The observer’s velocity doesn’t change the value of c, that is because c in any given space-time quantum medium “material” density (as per my work in FSC) doesn’t change in a non-dispersive medium (source velocity of light source doesn’t add or negate the value of c), and for this to be true along with the observer’s velocity not to affect the measurement of c, that means that the observer’s experience of time must have changed to compensate for his velocity. This doesn’t hold between two different space-time quantum medium “material” densities (as per FSC), but for Special Relativity (SRT) in the same quantum medium density (SRT fits within FSC, as LET fits within SRT). Essentially x’ and t’ are modified versions of x and t dimensions (actually x distance is invariant), and are the versions experienced by a moving observer. Each observer at different velocities in the same space-time quantum medium “material” density will experience different x and t dimensions.

Formulating Pythagorean diagram above:

`c^2 = (x/t)^2 + (c’)^2`

Where c’ is still measured as c within the same frame, because of a modification to x or t dimensions.

`c^2 = (x/t)^2 + ({x’}/{t’})^2`

Since we assume that distance is invariant, therefore x’ doesn’t exist, only x exists.

`c^2 = (x/t)^2 + (x/{t’})^2`
`(x/{t’})^2 = c^2 – (x/t)^2`
`(1/{t’})^2 = (c/x)^2 – (1/t)^2`
`t’ = 1/(sqrt((c/x)^2 – (1/t)^2))`
`{t’}/x = 1/(sqrt((c)^2 – (x/t)^2))`
`{t’}/x = 1/(sqrt(c^2 – v^2))`
`t’ (c/x) = 1/(sqrt(1 – v^2/c^2))` … look familiar?

Since `c/x` derives a time component, since it’s using c and not c’, the time component is considered “proper” time.

`1/t_c = c/x`
`{t’} / t_c = 1/(sqrt(1 – v^2/c^2))`

Finally

`t’ = t_c/(sqrt(1 – v^2/c^2))`

Now how does this visually translate into a scenario of 3 observers measuring the speed of light for a beam of light from their respective inertial frames (in a non-dispersive perfect fluid quantum medium)?

Here c’ is unique to each observer’s velocity in relation to the beam of light, and is derived by:

`c’ = c sqrt(1 – v^2/c^2)`

The calculated value of c’ is still experienced by all observers in their respective velocities as c, and this is because time is experienced differently for each observer in their respective inertial frames due to their respective velocity’s affect on the time dimension.

`t’ = t / sqrt(1 – v^2/c^2)`

So what causes this time dilation mechanistically? From my Fractal Scaling Cosmology (FSC) framework, please read https://www.gpofr.com/distortion-time-dilation-experiment/.

But we’re not finished with SRT. How do the respective observers experience/measure the other observers? This is not a very difficult question to answer. What you have to keep in mind is that the beam of light traveling at c is the common reference in all observer inertial frames.

From Observer A, Observer C is moving away at near the speed of light trailing the beam of light, and Observer C is aging exceptionally slow (clock apparatus moving exceptionally slow in the inertial frame of Observer C). The trailing part is important here. The distance from Observer C and the leading photons in the beam will keep growing because Observer C is traveling at 0.99c. This is good to keep in mind in this scenario.

Now it gets interesting from the perspective of Observer C, and this is where many can get very confused. From Observer C, Observer A appears to be moving away at a velocity close to c, but instead of Observer A aging exceptionally slowly (clock apparatus moving exceptionally slow in the inertial frame of Observer A) from the perspective of Observer C, Observer A is actually aging exceptionally fast yet moving away from Observer C also exceptionally fast. This might appear to contradict SRT at first, but it really doesn’t, because in relation to the beam of light traveling through the space-time medium, through all three inertial frames for each observer, it is Observer C who is actually traveling near the speed of the beam of light, and not Observer A. This becomes even more emphasized when you consider that the space-time medium is a non-dispersive medium, where the source velocity of light does not affect the velocity of the photons (as per FSC), meaning it doesn’t matter which inertial frame is emitting the beam of light (as per SRT). It is very important to cautiously remember, and fully understand, that the beam is common to all frames. Also the distance from Observer A and the beam source will essentially stay the same from the perspective of Observer C in this scenario depicted in the diagram above.

Again from my work in Fractal Scaling Cosmology Framework (FSC), the following diagram depicts that in different space-time quantum medium “material” densities, each respective region has its own value for c, yet each observer will still experience the normal value of c in each respective region because of the medium’s density affect the dimensions distance and time, but not across two different regions. Between two different space-time quantum medium densities, light will experience refraction.

Relative Constancy of the Speed of Light in Different Density Quantum Medium Space-Time Continuum Regions of the Universe