How to determine real inertia without 4 vectors?

[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).