Time To Change E=mc² to E=mc²/2
There are hidden mistakes in the original and other derivations of the formula E = mc²
On multiple occasions we discussed that time dilation is enough to derive many Einstein’s results, and no space or metrics changes are needed. Check Newton-like Mechanics Replaces Einstein’s GR and Spacetime Surrogate for a quick start, or read my free Time Matters eBook (also free on Amazon). Time dilation, which is time unit (second, or sec for brevity) change, causes changes of time-dependent physical units like Joule = kg×m²/sec² and even time-dependent constants like Planck’s constant (measured in Joule×sec). Check details in chapters 20 and 35 of Time Matters, which expose these underappreciated dependencies.
Recently I checked the original and some other derivations for the formula E = mc², and all of them have time-related mistakes. Let’s start with the original proof.
Einstein derived E = mc² formula using a thought experiment (check https://www.youtube.com/watch?v=hW7DW9NIO9M for 2-minute video and https://www.youtube.com/watch?v=vopowgznsXw for a complete presentation):
Einstein compares the kinetic energy of an object after it radiates at least two photons (in opposite directions, not to change the speed of the object). And he calculates energy in two ways:
1) From a moving-at-speed-v-observer point of view: the initial object kinetic energy KE₁ minus energy E of the radiated photons multiplied by 1+v²/(2c²). Factor 1+v²/(2c²) is attributed to Doppler effect (see video and screenshot below);
2) From a static observer perspective, Einstein accounts energy –E of emitted photons first, then from the moving observer perspective, Einstein adds kinetic energy KE₂ of the object, which became a bit lighter after photons emitted (so KE₂<KE₁):
KE₁ – E×(1+v²/(2c²)) = –E + KE₂
But here is a mistake: in 2) energy of photons denoted as –E is in units of energy related to static time, but energy of the object denoted as +KE₂ is in units of energy related to dilated/slower time in the rocket. Einstein added these two values together despite them being in physically different units (only adding of unitless or in-the-same-unit numbers is correct, otherwise unit/number conversion is required).
There are other “simpler” and more popular “proofs” for E = mc², but all of them have mistakes. Here is another 2-minute “proof”, which we will correct:
(Check detailed explanation on relativistic mass correction here.)
Mass is explained here as a measure for inertial response by Newton’s 2nd law F = m×a. The bigger mass m is, the less is acceleration a for fixed force F. The top line in the image above claims that mass increases (and acceleration decreases) by time dilation factor D:
m = m₀×D, (Mistake❗),
where D = 1/sqrt(1–v²/c²) (which is Einstein's formula for relativistic time dilation). But in Universal Inverse-Square Laws without Singularities! we saw that object reaction (aka acceleration) in variable time is inversely related to D²:
a(1) = a(D)/D²,
where a(1) is observed-from-outside acceleration, and a(D) is acceleration in local-to-the-object time units. Thus, object in dilated time reacts to an external force by D²-times-weaker-than-anticipated reaction (anticipated reaction is reaction when there is no time dilation). It becomes more clear in modified-for-variable-time Newton’s 3rd law terms
Action = –Reaction×D² (see details in Time Matters chapters 56–58),
for example, the Sun is attracted by the Earth 1.000002² times weaker than the Earth is attracted by the Sun, because of time dilation for the Sun D=1.000002. In other words, time dilation around/in any object shields it from external effects. That is different from a symmetric case when time is shared between “actor” and “reactor”. Thus, inertia (countering external force) is increased by D², which is, usually and in this video of proof, attributed to mass (another word for inertia), but now we can correct it to:
m = m₀×D² = m₀/(1–v²/c²) = m₀×(1+ v²/c² + v⁴/c⁴ + …) = m₀ + m₀×v²/c² + …
Here we used the formula for geometric series 1/(1-x) = 1 + x + x² + …
For v << c we ignore smaller members and keeping
m = m₀ + m₀×v²/c².
Multiply by c²:
m×c² = m₀×c² + m₀×v².
Divide by 2:
m×c²/2 = m₀×c²/2 + m₀×v²/2.
Thus, m×c²/2 accumulates kinetic energy m₀×v²/2: when kinetic energy is added to m₀×c²/2, it still remains m×c²/2, with m₀ now changed to m. Therefore, m×c²/2, and not m×c², accumulates energy:
💥 E = m×c²/2 is the total energy of a body❗
Now, to the Interesting Relativistic Invariant:
- Potential + v²/2 = c²/2,
which, when multiplied by m, gives similar-to-classical-mechanics formula:
- Potential Energy + Kinetic Energy = m×c²/2 = E.
E = m×c²/2 is D² times bigger than potential energy at rest E₀ = m₀×c²/2, because m = m₀×D². Thus, there is no “E = E₀ energy conservation in a traditional sense”, but we have this relation instead:
- E = E₀×D²,
(which reminds me of Newton’s 3rd law adjustment by D²:
Action = –Reaction×D² from Time Matters chapters 56–58).
Does replacing formula E = m×c² with E = m×c²/2 cause major problems for physicists?
Actually, no. Recalibrating constants, units, recalculating results of experiments etc. happen all the time in physics. The primary usage of E=m×c² was for determining energy yield in fission and fusion reactions, when on the left side of the energy conservation formula physicists put entry element (like Uranium or Plutonium for fission, or Deuterium and Tritium for fusion) energies for masses M…, and on the right side they put produced-by-fission-or-fusion element energies for masses m… plus energy yield:
M₁×c² + M₂×c² + … = m₁×c² + m₂×c² + … + energy yield
And since masses of all participating elements/isotopes and elementary particles were determined beforehand, energy yield was calculated as:
energy yield = (Σ M – Σ m)×c².
Now, with the corrected formula for energy E = m×c²/2, we have to recalibrate
energy yield = (Σ M – Σ m)×c²/2.
Besides that, there could be other implications not directly related to the mass-energy formula. For example, for radiation/light energy (where Planck’s constant is used) some recalibration might be needed, since masses (let’s say of electrons in photo-electric effect) could be involved.
Physicists do adjustments/recalibrations all the time, they even do so-called “renormalization” of infinite theoretical values to finite/practical values.
How did such an energy difference go unnoticed?
Simplest example: It was estimated that of 64 kilograms of Uranium used in the bomb that exploded over Hiroshima, only 0.7 grams were converted into energy, the rest became radioactive waste. Could it be an inaccurate assessment? Could it be that 1.4 grams instead of 0.7 grams were converted into energy? Yes. Less than a gram missing of the initial 64 kilograms of Uranium or in after-fission various elements/isotopes and elementary particles is hard to measure. And gamma, x-ray, thermal radiation, mechanical energy of the blast etc.— all that is hard to account for accurately, impossible actually. And there were huge disastrous mistakes / miscalculations acknowledged, especially with H-bomb tests.
As for the simplest lab experiments, like electron-positron annihilation, there is still room for revision of measurements, constants, velocities, frequencies, relativistic time dilation effects, adjustments previously made and now taken for granted etc. No rush, just another refactorization by 0.5.
P.S. But now we know even more:
Since m = m₀×D², and energy is stored in E = m×c²/2 = m₀×D²×c²/2, where m₀ and c are constants, and only D varying on the right side, we can see clearly that energy E is accumulated in time dilation D by slowing time inside matter m₀.
