Derivation of the Equation for Relativistic Mass

[pmb_phy]

If anyone knows, however, of a derivation of this equation using the Lorentz transformations and preferably no other assumptions that are specifically within the theory of Special Relativity, if such a derivation exists at all, that would be the greatest help.

I gave this to you in your PM but I think its okay if I post it now. Since I had a feeling that someone would want to follow this derivation someday I placed it on my web page here.

http://www.geocities.com/physics_world/sr/inertial_mass.htm

Now others can read it and ask any questions they might have about it. The Lorent transformation manifests itself through the velocity transformation relations.

Pete

 

[jhigbie]

Yes NanakiXIII, you are right. The literature usually skips over this derivation and the four-vector approach, while rigorous, usually leaves me cold. And, it is not the one that Einstein used.

I think, but I am not sure, that he used the idea of bouncing elastic balls, you know, one thrown from a moving train and the other one on the stationary platform.

But in the meantime, if you are comfortable with the idea of a Taylor's Series, you can expand "gamma" as a power series in "beta" and if you *start* with the idea that m/m₀ = gamma = 1 + (1/2)beta² + ... (higher powers) and this leads to the expression for kinetic energy:

(m-m₀)c² = (1/2)m₀v² + ... (higher terms),

So, it then becomes reasonable that (m-m₀)c² is in fact the kinetic energy and this of course makes us feel that this is probably the correct expression for relativistic mass. It does have the right form.

Einstein's approach to SR was quite novel. He *assumed* that a relativity principle applied not only to mechanical phenomena, but electromagnetic as well ... and that was a great leap of faith. It turns out that he was quite right, but he was the only one at the time to venture such an idea.

 

[stevmg]

By special relativity, we have time dilation and space contraction by this "gamma." Time is increased by division of the reference time by this gamma and distance is reduced by multiplication by this gamma. For f = ma, because the translation distance is shortened by this factor, the acceleration, compared to the inertial reference is shortened, but force is constant, so the mass must be increased to make up for the lost acceleration.

The derivation of this gamma is done by Einstein in his relativity book. It is an algebraic derivation with no calculus.

Once we know that relativistic mass changes with velocity, we show that a loss in velocity results in a loss in mass and pure energy being releasedf (delta m x c^2)

remember [(1 - v^2/c^2)]^-1/2 = (closely) 1 + (1/2)(v^2/c^2)

relativistic mass - inertial mass = inertial mass + (1/2)inertial mass x loss in velocity^2/c^2) - inertial mass = (1/2)inertial mass x loss in velocity^2/c^2)
delta mass x c^w = 1/2inertail mass x loss in velocity^2 = "mc^2" qed

Again, this requires that relativistic mass is increased from inertial mass by division by this gamma and it is the law of ther conservation of momentum or energy which requires mass to increase when linear acceleration id reduced by space contraction.

 

[PeterDonis]

Einstein's approach to SR was quite novel. He *assumed* that a relativity principle applied not only to mechanical phenomena, but electromagnetic as well ... and that was a great leap of faith. It turns out that he was quite right, but he was the only one at the time to venture such an idea.

PMFJI, but this isn't true. Lorentz derived the Lorentz transformation well before Einstein did, as did Larmor. Their derivations were specifically intended to find the transformation laws for the electromagnetic field that would be compatible with Maxwell's Equations. See the section on "Historical Developments" in the "[URL page on Maxwell's Equations[/URL]. What Einstein did was to assume that *mechanical* phenomena were governed by the Lorentz transformation, and work out the consequences; the consequences for electromagnetism were already well known.

 

[PeterDonis]

I think, but I am not sure, that he used the idea of bouncing elastic balls, you know, one thrown from a moving train and the other one on the stationary platform.

In the version of this thought experiment that's in his popular book on relativity, _Relativity: A Clear Explanation That Anyone Can Understand_, he used light flashes, emitted from each end of a moving train. I haven't been able to find an online reference that contains his description of the scenario.

 

[stevmg]

The flashing lights, etc. are in his book "Relativity"

 

[stevmg]

Remember that assumptions of constant c and the preservation of laws of physics in all inertial time frames leads to a mathematical "proof" of the Lorentz transforms. They are equivalent statements.

Many college textbooks actually derive the "relatvistic mass" equation by itself based on the Lorentz transforms.

 

[jesusfreak]

Energy-Momentum Relation:

E² = (m₀c²)² + (pc)²

Substitute mc² for E (energy-mass equivilence)

Substitute mv for p (relation between mass, velocity, and momentum)

it should look like this:

(mc²)² = (m₀c²)² + (mvc)²

Subtract (mvc)² over..

(mc²)² - (mvc)² = (m₀c²)²

m²c⁴ - m²v²c² = m₀²c⁴

Factor out (mc²) on the left..

(mc²) (c² - v²) = (m₀c²)²

Factor out a c² from the second set of parenthesis above...

(mc²) (c²) (1 - v² / c²) = m₀²c⁴

Take the square root of both sides...

mc² * √(1 - v² / c²) = m₀c²

"c²"s cancel out and then divide the √(1 - v² / c²) over..

m = m₀ / √(1 - v² / c²) [relativistic mass-invariant mass relation]

m: relativistic mass
m₀: rest mass (invariant mass)
v: velocity
c: the speed limit of the universe
p: momentum
E: energy

 

[gpeach]

pa₀+pb₀=pa₁+pb₁

γ(va₀).ma.va₀+γ(vb₀).mb.vb₀=γ(va₁).ma.va₁+γ(vb₁).mb.vb₁

This is the general expression for the relativistic momentum conservation with two particles, where 'ma' and 'mb' are the rest masses of the particles.
To understand the relativistic momentum definition, we need to see why this expression is valid for moving systems, upon the multiplication of the factor γ(v). So it is important to bear in mind the relation between time and proper time, as we can deduce from the basic experiment of a light ray in a moving mirror:

Δt²-Δx²/c²=Δζ² →

Δζ/Δt=√(1-v²/c²)=γ(v) →

γ(v)/Δt=1/Δζ (proper time)

Now, using the speed "diferential definition" (Δx/Δt), and applying the relation whith proper time above to the expression of momentum conservation along the 'x' coordinate, we get:

ma.Δxa₀/Δζa+mb.Δxb₀/Δζb=ma.Δxa₁/Δζa+mb.Δxb₁/Δζb

Applying the Lorentz transformation Δx=γ(V)(Δx'-V.Δt'), we obtain (canceling the commun term γ(V)):

ma.Δx'a₀/Δζa-ma.Δx'a₀V.Δt'/Δζa+mb.Δx'b₀/Δζb-mb.Δx'b₀V.Δt'/Δζb=ma.Δx'a₁/Δζa-ma.Δx'a₁V.Δt'/Δζa+mb.Δx'b₁/Δζb-mb.Δx'b₁V.Δt'/Δζb

Applying again the relation between proper time and time we have:

γ(v'a₀).ma.v'a₀+γ(v'b₀).mb.v'b₀-γ(v'a₀).ma.V-γ(v'b₀).mb.V=γ(v'a₁).ma.v'a₁+γ(v'b₁).mb.v'b₁-γ(v'a₁).ma.V-γ(v'b₁).mb.V

In this expression, if the equality

γ(v'a₀).ma+γ(v'b₀).mb=γ(v'a₁).ma+γ(v'b₁).mb

holds (this is the relatIvistic "condition" tor relativistic momentum conservation), then we have the momentum conservation:

γ(v'a₀).ma.v'a₀+γ(v'b₀).mb.v'b₀=γ(v'a₁).ma.v'a₁+γ(v'b₁).mb.v'b₁ →

p'a₀+p'b₀=p'a₁+p'b₁

With respect to the transversal components (x and y), the result is analogous and imediate...