The third law is just about the conservation of the momentum. It follows from the translational symmetry of the laws of physics and it holds for any description, whether or not it contains fields. The description in terms of a force, action at a distance, has the same force with the opposite sign acting on both objects. If there are fields, the fields may carry some momentum, too. In quantum field theory, the force comes from virtual particles – virtual quanta of the fields – and they again carry the right momentum so that it is always conserved. So the law always holds. Does it answer “How”?

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— Physics Stack Exchange

— Apr 6 ’11 at 12:17

— Lubos Motl

2013.08.23 Friday ACHK

The ergodicity requirement is that the ensemble average coincide with the time average. A sufficient condition for ergodicity is that the time evolution of the system is a mixing. (See also ergodic hypothesis.) Not all systems are ergodic. For instance, it is unknown at this time whether classical mechanical flows on a constant energy surface are ergodic in general.

Physically, when a system fails to be ergodic, we may infer that there is more macroscopically discoverable information available about the microscopic state of the system than what we first thought. In turn this may be used to create a better-conditioned ensemble.

— Wikipedia on Statistical ensemble (mathematical physics)

2013.08.22 Thursday ACHK

Penrose diagrams are frequently used to illustrate the space-time environment of black holes. Singularities are denoted by a spacelike boundary, unlike the timelike boundary found on conventional space-time diagrams. This is due to the interchanging of timelike and spacelike coordinates within the horizon of a black hole (since space is uni-directional within the horizon, just as time is uni-directional outside the horizon). The singularity is represented by a spacelike boundary to make it clear that once an object has passed the horizon it will inevitably hit the singularity even if it attempts to take evasive action.

— Wikipedia on Penrose diagram

2013.08.15 Thursday ACHK

Matrix representations

Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers \theta_1 and \theta_2. These Grassmann numbers can be represented by 4×4 matrices:

…

— Wikipedia on Grassmann number

2013.08.12 Monday ACHK

However, the Grassmann numbers can’t have particular real or complex values, not even infinite values, as I will discuss momentarily. Instead, they may be viewed as intrinsically “infinitesimal” in character. That’s why we don’t write the “plus minus infinite” limits on the definite integral.

…

The square of a Grassmann variable has to vanish because it’s equal to minus itself and zero is the only number that has this property. The vanishing of the square, θ^2=0, is also satisfied by “infinitesimal numbers” because we neglect (dx)^2 if we only calculate at the accuracy of dx. That’s why the Grassmann numbers are sometimes said to be “infinitesimal”.

— Celebrating Grassmann numbers

— Lubos Motl

2013.08.11 Sunday ACHK

Defining mass in general relativity: concepts and obstacles

Generalizing this definition to general relativity, however, is problematic; in fact, it turns out to be impossible to find a general definition for a system’s total mass (or energy).

The main reason for this is that “gravitational field energy” is not a part of the energy-momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein’s equation (and, as such, a consequence of these equations’ non-linearity).

While in certain situation it is possible to rewrite the equations so that part of the “gravitational energy” now stands alongside the other source terms in the form of the Stress-energy-momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.

— Wikipedia on Mass in general relativity

2013.08.06 Tuesday ACHK