Another example of mixing up history with physics. If by mass one defines the norm of the energy-momentum 4-vector, it's a Lorentz scalar. And it can be shown that it is this quantity that reduces to the inertial mass, that appears in Newton's law, in the non-relativistic limit.

However that's, also, a historical account. 

The technical account that matters is that the mass is the charge-that can be shown to be Lorentz invariant-of the current that describes the response of a physical system to an external gravitational potential. In the case of gravity, the external potential is the metric tensor, the response is described by the energy-momentum tensor of matter and the mass is just the spatial integral of the time-time component of the energy-momentum tensor; just like the electric charge is the spatial integral of the time component of the electric current 4-vector.

And it turns out that this definition is consistent with the definition of the mass as given in the first paragraph.

Hi Moon,

Generally a quantity is describable by a tensor if is present an anisotropy (take the example of the index of refraction n in a birefringent material ), in the case of mass it would be a spatial anisotropy i.e. the motion is different along different directions. As far as experimental data is concerned no such significant deviation were observed and for isolated system  motion is isotropic.

Is mass a Scalar Quantity ?

Good question !

I think Yes in the same way as the Higgs Field , it make sense.

Kurt Wraae

 Another example of mixing up history with physics. If by mass one defines the norm of the energy-momentum 4-vector, it's a Lorentz scalar. And it can be shown that it is this quantity that reduces to the inertial mass, that appears in Newton's law, in the non-relativistic limit.

However that's, also, a historical account. 

The technical account that matters is that the mass is the charge-that can be shown to be Lorentz invariant-of the current that describes the response of a physical system to an external gravitational potential. In the case of gravity, the external potential is the metric tensor, the response is described by the energy-momentum tensor of matter and the mass is just the spatial integral of the time-time component of the energy-momentum tensor; just like the electric charge is the spatial integral of the time component of the electric current 4-vector.

And it turns out that this definition is consistent with the definition of the mass as given in the first paragraph.

Also in relativity mass does not change. It is the momentum that changes with velocity. So mass remains scalar quantity then. 

In old days people used to define relativistic mass in order to remain consistent with the previous Newtonian mechanics. But that is no more used nowadays. We always take rest mass as mass of the object and we redefine the momentum instead.

In general relativity for the light particle in Schwarcshield space-time in Newtonian limit it can be shown that there is neither transverse nor longitudinal gravitational mass, but only scalar mass exists.

The same conclusion can be drawn for the material particle

http://urss.ru/cgi-bin/db.pl?lang=Ru&blang=ru&page=Book&id=222186 .

Article Application of Lagrange mechanics for analysis of the light-...

Are you talking rest-mass or the mass/energy?

Notion of a mass tensor exists in the studies of crystal structures, though normally (at large) by the term "mass" one uses the notion of a scalar quantity which is one of the quadratic Casimir invariants of the Poincare group the other being the intrinsic spin.

In Solid State Physics, one often studies changes in group velocity due to changes in crystal momentum. This relation is achieved through a tensor. It's inverse is sometimes called the effective mass tensor, in analogy with the Newtonian definition for momentum of point particles p=mv.

Note that: Crystal momentum is a momentum-like vector associated with electrons in a crystal lattice. Group velocity is also the velocity of energy propagation in a crystal lattice.

 A (moving) body or particle with 3-momentum p and 3-velocity v has two different mass:

1) Inertial mass, or mass-energy, m: p = mv. This mass, m(v), is a number, but not a 4-scalar because p and v are not 4-vectors, they are coordinates (or components) of the 4-momentum.

2) Rest mass, or invariant mass, m0: m02 = m2 – p2/c2.   m0 is a 4-scalar.

Mass m is additive and satisfies the conservation law; mass m0 is not.

"What is mass?" Physics – Uspekhi, 43 1267–1270 (2000) 

 «Что есть масса?» УФН 170 № 12, 1363 (2000)

http://khrapkori.wmsite.ru/ftpgetfile.php?id=14&module=files

https://arxiv.org/abs/physics/0103051

https://arxiv.org/abs/physics/0103008

Radi, Formulas F=ma and F=dP/dt are not true for curvilinear space-time, since they do not take into account the transfer of energy and momentum to the gravitational field. In the general theory of relativity, the energy and momentum of matter are transferred to the gravitational field, so the same will occur for an individual particle.

Dear Vladimir, you are mistaken:

1) I do not use F and a at all

2) There are no concepts of gravitational field and gravitational energy in the general theory of relativity

mhm, let me just get this straight

What it means when i could say this  is a scalar or tensor ,simply scalar quantity doesn't change when u move from one point to another (no direction )so if ya bring ur mobile phone or whatever and moved it from ur table to ur bed does its mass change ? answer is no (Newtonian ).

this is a point of view ,another point moving this mass at particular speed or acceleration could increase the mass according to relativity ,oh yea Einstein got a point here ,but after that mobile phone stops moving it losses the mass it gained ah that's make sense ,otherwise i will find my phone heavier by time.

both are different approaches ,but there is another approach does this mass decay by time at longer range and we really could accelerate that decaying as in the atomic reactor

Nothing is absolute all are theories ,we have to stick to the most demonstrative one  , but don't forget as long as u follow ur expectation in a specific physics experiment, probably physics will beat u up. for example we all could say that the larger mass will hit the ground faster by our minds before we know the ugly truth which is all masses will hit the ground in the same time.

Mass, which is defined by P =  mv, is additive: m = m1 + m2 + ...

As already stated by others, the relativistic mass is a concept, that is not really used anymore, but only rest mass. The rest mass is always is scalar. Even the mass-energy equivalence does not change that fact. (Energy is also a scalar.)

Form your question, I infer that, what might have confused you is the existance of the stress tensor (also called mass-energy-momentum tensor) which is indeed a 4-times-4-matrix. However, only the time-time component is related to the mass. (The other components are related to quantities like momentum, shear, torson.)

So, in short, rest mass, i.e. what one usually understands by "mass", is always a scalar, even in both relativity theories.

Patrick Pöschke: "Energy is a scalar".

It is a fatal delusion! Energy is a component of the 4-vector

The mass of particle defines the fourth component of momentum and the rest mass defines the norm of the momentum.