How to determine real inertia without 4 vectors?

[DrStupid]

the second wrong in relativity

In the context if physics "wrong" means disproved by experiments. Which experiments do you mean?

there was no m and m0 in Newtonian physics

In classical mechanics m and mo are identical. This changes if Galilei transformation is replaced by Lorentz transformation. I can show you the corresponding derivations if you are really interested.

To me, the ratio of 4-force to 4-acceleration is the exact application of Newton's definition in a relativistic context.

Newton never published such a definition. He defined p=m·v and F=dp/dt and these formulas were intended for use with 3-vectors only. Using them with 4-vectors results in new formulas. Of course that doesn't mean that this new formulas are wrong or useless (in fact they are very useful in SR) but they are different from Newtons original formulas and so is the resulting new definition of mass.

 

[Bill_K]

To me, the ratio of 4-force to 4-acceleration is the exact application of Newton's definition in a relativistic context.

Newton never published such a definition. He defined p=m·v and F=dp/dt and these formulas were intended for use with 3-vectors only.

From Newton's Principia,

Law II. The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which the motive force is impress'd.

"Alteration of motion" is what we today would call acceleration, dv/dt. In other words, he said F = ma. Newton made no mention of momentum.

 

[atyy]

Hmm. In my mind, the only form of inertial mass in SR is invariant mass, which includes KE only to the extent it becomes part of the invariant mass of the system.

You mean like in the 4-vector analogue of "F=ma" as in the posts above?

 

[PAllen]

You mean like in the 4-vector analogue of "F=ma" as in the posts above?

Yes.

 

[atyy]

Yes.

How do you get from there to the Einstein equation G=T?

The ADM mass and Komar mass seem closer to the notion of gravitational mass as invariant mass. But the stress energy tensor seems closer to the notion of gravitational mass as energy.

 

[PAllen]

How do you get from there to the Einstein equation G=T?

The ADM mass and Komar mass seem closer to the notion of gravitational mass as invariant mass. But the stress energy tensor seems closer to the notion of gravitational mass as energy.

I don't know. I was only speaking to what plays a role of inertia in SR, not an approach for motivating stress energy tensor. If you don't use 4 vectors, how do you conclude which is the real inertia: longitudinal relativistic mass or transverse?

Maybe get to T via flows of 4-momentum, which brings energy and spatial momentum in together.

 

[DrStupid]

"Alteration of motion" is what we today would call acceleration, dv/dt.

That's simply wrong.

From Newton's Principia,

Definition II: The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.

or in the original:

Def. II: Quantitas motus est mensura ejusdem orta ex velocitate et quantitate materiae conjunctim.

Obviously Newton used the term "velocitate" for velocity. Therefore acceleration would have been "alteration of velocity" or "mutationem velocitate" in original. But he wrote "mutationem motus" and the quantitative definition of "motus" is given above. As "quantitate materiae" is Newtons term for mass, definition 2 means

motion = velocity * mass

That's what we today call momentum.

 

[DrStupid]

If you don't use 4 vectors, how do you conclude which is the real inertia: longitudinal relativistic mass or transverse?

If "real inertia" means M in F=M·a than it is

M = \frac{{m_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} \cdot \left( {I + \frac{{v \cdot v^T }}{{c^2 - v^2 }}} \right)

Longitudinal and transversal mass are the eigenvalues of M. Of course that is not what Newton called mass (or anybody else today).