On Jul 20, 6:26 am, jmfbahciv <See.ab...@aol.com> wrote: > >> > How do you define mass? How do you measure it with a ruler? > >> Mass is a measure of gravitational attraction... > > making GR compatible with QM. > You have claimed that everything can be described using only > space and time. So I've asked you to describe mass using > only those two entities so that a mass can be measured in > a lab.
Place a stationary shell around a spherical region of space and then let it go. Its volume will contract with an acceleration given by 4 pi G M, where M is the mass contained within the region. Once it gets moving, the equation for the volume contraction will also involve the first derivative of the volume contraction. Both cases are instances of the Raychaudhuri equation. The Raychaudhuri equation is present in both in relativistic and non-relativistic theory.
This is analogous to how charge is measured with a flux meter.
Mass can be defined solely in terms of the Lie group underlying the symmetries of space-time. This is the case both relativistically and non-relativistically.
In non-relativistic theory, each system has a state space that is a representation of the Galilei group, this group describing the symmetries of non-relativistic space-time. If the system is elementary the representation is irreducible. The mass of such a system is the central charge of the representation.
So, in non-relativistic theory, mass is defined solely in terms of the transformation properties of the system under the action of the Galilei group.
The notion of central charge was not widely known until the early to mid 20th century. So, classical physicists never even had the opportunity to recognize this important property, nor to make this definition. This proves, by the way, that the field of classical physics EVEN NOW is still in a state of continual evolution -- even if retroactively -- as more and more gaps and oversights from "classical" classical theory emerge. So we now have to distinguish "classical" classical theory from "modern" classical theory (and even classical non-classical theory from modern non-classical theory, since many of the new insights also get inherited by non-relativistic theory).
The other two invariants of the Galilei group (for the generic irreducible representation) are the one given by P^2 - 2mH (where m is the central charge, P the generator of spatial translations and H the generator of time translations), and W^2 where W = mJ + P x K (x denoting cross product), where J is the generator of spatial rotations, K the generator of Galilean boosts. These give you, respectively, the internal energy and internal angular momentum (i.e. spin) of the system.
In relativistic theory mass can also be defined solely in terms of the behavior of a system under transformation by the underlying space-time symmetry group.
There, the irreducible representations are classified as either translation-invariant or not (as they also are in the non-relativistic case). For the translation non-invariant systems, a further classification into "tardion", "luxon" and "tachyon" exists (for the non-relativistic case, luxon and tachyon combine into "synchron" -- something which was also absent from classical classical physics).
In all cases, the two invariants are P^2 - (1/c)^2 E^2 and W^2 - (1/ c)^2 W_0^2, where W_0 = P.J, where E is the generator of time translations. The distinction between tardion, luxon and tachyon rests solely on the sign of the first of these invariants.
Tardions have negative sign, so one can define the invariant m by m^2 = (1/c)^4 E^2 - (1/c)^2 P^2, taking the sign of m the same as the sign of E. That defines the mass of the system.
For tachyons, the invariant is positive, so one can only define the *impulse*, Pi by Pi^2 = P^2 - (1/c)^2 E^2. These systems represent an "instantaneous" transfer of impulse Pi across space (where "instantaneous" means, "instantaneous in at least one frame of reference"). The "synchrons" in non-relativistic theory share this feature with tachyons. So, the frame of reference for tachyons in which the transfer is instantaneous might be called the "synchron frame".
There is no meaningful definition for "mass" for tachyons. The "impulse" takes over that role.
Luxons fall into two classes, based on whether W has components perpendicular to P or not. If not, then W is parallel to P. I call these the "helions". The photon falls into this class. In so, then this leads to the representations known as the "continuous spin" representations. There are no fundamental systems known that fall into this class.
In both cases, there is no meaningful attribute "mass". Conventionally, it's just taken to be 0, since the Luxon is the m -> 0 limit of the tardion. It's also the Pi -> 0 limit of the tachyon, so its "impulse" can also be taken as 0.
Synchrons are the m -> 0 limit of tardions in the non-relativistic case. So their mass is 0. There is also a frame of reference in which H is 0. In non-relativistic theory, synchrons correspond to action-at- a-distance forces.