Gravimetric Determination

PHY2061 Enriched Physics 2 Lecture Notes

Relativity 4

Relativity 4

Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.

Relativistic Momentum

Newton’s 2nd Law can be written in the form dp F= dt where the non-relativistic momentum of a body is p = mu where u = because of the Lorentz transformation equations,

dx . However, dt dx is measured differently in different dt inertial frames. Thus, Newton’s 2nd Law would not have the same form in different frames. We need a new definition of momentum to retain the definition of force as a change in momentum.

dx , where τ is the proper time in the object’s rest frame. Every observer dτ will agree on which frame is the rest frame. Also, since y ′ = y and z ′ = z , the transverse momentum (py and pz) will be invariant for a Lorentz transformation along the x axis. (This would not be the case if we did not use the proper time in the definition). We can rewrite this momentum definition as follows: dx dx dt Recall that momentum is a p=m =m vector quantity. Conservation of dτ dt dτ momentum, which still applies dt t =γτ ⇒ =γ From time dilation in Special Relativity, implies dτ that each component of 1 momentum is conserved. p = γu m u γu = 2 2 1− u / c

Suppose p = m

Note that u is the velocity of the object in a reference frame, not the velocity of a reference frame relative to another. In this definition of momentum, the mass m=m0 is the “rest mass”. That is, it is the mass of an object in its rest frame. Sometimes γ m is referred to as the “relativistic mass”, such that we can retain the Newtonian definition of momentum as p = mu . In this sense, the mass of an object grows as its velocity increases. But this convenient trick can be problematic. As we shall see, the kinetic energy, for example, is...