A Proof Of E=γm₀c²: Energy-Mass Equivalence.

Find Einstein's original method of derivation of E=γm₀c² in §10 of:
"On The Electrodynamics Of Moving Bodies", by Albert Einstein, June 30, 1905. Annalen der Physik.
REMARKS↓
1. Another acceptable method of proof of E=mc² which assumes m=γm₀ without proof (enjoy!):
F=dp/dt=d(mv)/dt=d(m₀γv)/dt=m₀d(γv)/dt=m₀[γ(dv/dt)+v(dγ/dt)]. Where γ= 1/√(1-v²/c²).
Now dγ/dt=(dγ/dv)(dv/dt)=a(dγ/dv)=ac⁻¹(dγ/dβ)=ac⁻¹(d/dβ(1-β²)^⁻½)=ac⁻¹[⁻½(-2β)(1-β²)^⁻3/2]=(av/c²)(1-v²/c²)^⁻3/2, with β=v/c.
Therefore F=m₀[γ(dv/dt)+v(dγ/dt)]=m₀γa + m₀aγv²/c²[c²/(c²-v²)]=m₀γa[(c²-v²+v²)/(c²-v²)]=m₀γa[c²/(c²-v²)]=m₀γ³a♦
Hence, for an element of`work:
dW=Fdx=m₀γ³adx=m₀γ³(dv/dt)dx=m₀γ³vdv=m₀c²β(1-β²)^⁻3/2dβ. Therefore, K=W=m₀c² ∫β(1-β²)^⁻3/2dβ. Let u=1-β² so that du= -2βdβ or, -½du=βdβ. We get, K=W= -½m₀c²∫u^⁻3/2du = m₀γc² + constant = mc² + constant♦
NOTE: (dv/dt)dx=(dv/dt)(dx/dt)dt=(dx/dt)(dv/dt)dt=vdv.
2. The following journal articles are of historical significance:
a.) "DOES THE INERTIA OF A BODY DEPEND
UPON ITS ENERGY-CONTENT?",
By A. EINSTEIN
September 27, 1905 - Annalen der Physik.
b.) "ON THE ELECTRODYNAMICS OF MOVING
BODIES",
By A. EINSTEIN
June 30, 1905 - Annalen der Physik.
In a.) Einstein shows that light has mass equivalence, and
In b,) (see §10) Einstein proves E=mc² (with m=γm₀) for a slowly accelerated, continuous, electron beam.
3. The energetics of the Cockcroft-Walton experiment (1932):
⁷₃Li + ¹₁H → ⁴₂He + ⁴₂He + energy,
is regarded as the first experimental evidence of E=mc²= K + E₀. High energy protons (¹₁H) make inelastic collisions with lithium (⁷₃Li), producing helium (⁴₂He) and releasing energy. Where E₀ ≡ m₀c².
The motivation for the definition of the relativistic kinetic energy K≐ E - m₀c² follows from the splitting mc²=(mc²-m₀c²)+m₀c².
4. The method of proof used in the clip is a standard one - a variational problem.
The advantage in using this method of proof is that (unlike other methods) it clearly shows that beginning with the spacetime metric we obtain the Lagrangian function from which ALL of the information we need follows from.
The point is (which other methods do not answer), Einstein's famous equation (E₀=m₀c²) follows from the spacetime setting.
5. Corollary: E²=m₀²c⁴ + p²c².
6. When the Hamiltonian function H(x,p,t) is time independent
(i.e., H=H(x,p)) it is equal to the total energy of the system, i.e.,
H(x,p)=E or, simply, H=E.
7. In geometric units, E=m.

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Suppose that you're looking for, 1/√(1-x²) ~ a₀ + a₁x + a₂x² (x

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An authoritative historical account of *relativistic mass* can be found in, "Concepts of Mass in Contemporary Physics and Philosophy", by Max Jammer - Princeton University Press., Princeton, N.J., 2000.

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*The meaning of v/c

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*Corollary: E²=m₀²c⁴ + p²c²* . *Proof:* Beginning with E=mc² =γm₀c², E²=m²c⁴=γ²m₀²c⁴=m₀²c⁴/(1-β²), with β=v/c and γ=1/√(1-β²). Therefore, E²=m₀²c⁴/(1-β²)=m₀²c⁴{ ∑ᵤ₌₀ ∞ β²ᵘ} (*β

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Exercise: Substitute p = γmv into the RHS of E² = m²c⁴+p²c² to get E² = (γmc²)². Solution: E² = m²c⁴+p²c² = m²c⁴+(γmv)²c² = m²c²(c²+γ²v²). Now, γ⁻²=(1-v²/c²)=c⁻²(c²-v²) or, 1=c⁻²(c²-v²)γ² or, c²=(c²-v²)γ² or, γ²v²=γ²c²-c². Therefore, E² = m²c²(c²+γ²v²) = m²c²(γ²c²) = (γmc²)².

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♣ *The same concept when viewed in a different scale.* ♣ The Quantum Mechanical analysis of the free (of external fields), neutral, relativistic, Hydrogen atom shows that its energy eigenvalues are given by, E(n,k) = mc²{1 + F(α, n, k)}^(-½) -➊ With: F(α, n, k) = α²{n + √(k² - α²)}⁻² and, 1. α is the Sommerfeld (Arnold) fine structure constant, α=e²/ℏc, e being the electron charge, and 2. For the "reduced mass", μ= mM/(m+M) = m/(1+m/M) ~ m (m

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In the clip @0:04 i state "with A a constant (WHY?)". Answer: The constant A (a constant is always Lorentz invariant) is chosen so that the integrand A·ds has dimension of "action". What is the dimension of "action"? Well, for example, Planck's constant (h) is called the fundamental action, and the Planck-Einstein law is E=hf where f the frequency has dimension of reciprocal time. Therefore, the dimension of h (hence, "action") are energy*time.