You may have heard that the mass of an object varies with its speed.
This can be confusing and not of much help in understanding relativity.
Here, the word "mass" refers to the rest mass of an object as it will nearly always be measured in its rest-frame.
It is a scalar quantity and is independent of the observer or its speed.
Read Section 37.7
We have mostly been concerned with inertial frames, i.e. constant velocity.
However, for non-constant velocities, we have to consider forces.
When
Relativistically, the first definition is correct if momentum is properly defined.
The classical definition, p = mv, has a problem. It changes in different inertial frames and so would not be conserved.
To be consistent with the First Postulate of Relativity, the Law of Momentum Conservation MUST be preserved.
The proper, relativistic, definition of momentum is p = gmv.
Note that at slow speeds, g = 1, and the Newtonian (classical) definition emerges.
Also note that this definition is not merely substituting the Lorentz velocity transform.
Something fundamental has changed here: Constant forces no longer produce constant accelerations!
If the net force F and velocity v are both pointing along the +x-axis, then by performing the derivative:
F = ma / ( 1 – b2 )3/2
or
a = (F/m)( 1 – b2 )3/2.
So, as the velocity increases, the acceleration caused by the force continuously decreases !
In fact, unless F is either parallel or perpendicular to v, the net force and the acceleration are not even parallel to each other!
When v = c ( or b = 1 ), the acceleration is zero no matter how large a force is applied.
It is then impossible to accelerate a non-zero rest-mass to a speed equal to or greater than c.
The speed of light is the ultimate speed limit!
Read Section 37.8
A body at rest (in a given, inertial, Reference Frame) has an energy E0 = mc2, the famous equation that everyone knows.
This relationship fundamentally means that there is no difference between energy and mass.
If the body is moving, the total energy is E = gmc2.
The kinetic energy is the difference between the total energy and the rest-mass energy, KE = E – E0 = (g – 1)mc2.
At speeds very much less than c, this reduces to the familiar KE = ½mv2.
(The binomial expansion of g is used to arrive at the low-speed approximation of KE, you should prove this to yourself)
We can relate the momentum to the total energy (kinetic + rest-energy) by taking the definition of momentum and writing it as
(p/mc)2
= b2 / ( 1
– b2 )
and the total energy and writing it as
(E/mc2)2
= 1 / ( 1 – b2 )
then subtracting these two relationships and rearranging to give
E2 - (pc)2
= (mc2)2 = E02
The right-hand-side is a scalar, and an invariant quantity; mc2 is the same number in every inertial reference frame. Although one observer may measure a different energy and different momentum than another, this relationship shows that the combination on the 1eft-hand-side of the equation is the same number to everyone. Thus, momentum and energy are intimately related in much the same way that space and time are.
E2 - (pc)2 = (mc2)2 = E02, an invariant;
(cDt)2 - (Ds)2 = (Dt)2 = {interval}2 , an invariant.
For a “particle” with zero rest-mass ( m = 0 ):
E = pc, for zero rest-mass.
This relationship will have profound implications for light (photons).
The value of the invariant space-time interval depends on what the separation of events in space and time is; the value of the invariant energy-momentum quantity is always the mass of the particle. The two quantities, space-time and energy-momentum can be written as a four component vector:
For space-time, the set (x, y, z, ict) are the components of a “four-vector”, which is the four-dimensional analog to the ordinary three-dimensional vector for position that you're accustomed. Note that i is the imaginary number ( i2 = -1 ) and that every component has dimension of length. This vector is used to represent any point in space-time and the interval between .
For energy-momentum, (ipx, ipy, ipz, E/c) are the components of a “four-vector”, which is the four-dimensional analog to the ordinary three-dimensional vector for momentum. Note that every component of the momentum-energy four-vector has units of momentum and that its magnitude is the rest-mass energy of the object.
These two four-vectors are transformed between frames of reference in exactly the same way. So, it's not too surprising that their magnitudes, the invariants {interval} or rest-mass, behave in the same way.
The name of the unit "electron-volt" tells you how much energy is represented by an eV, simply multiply the charge of an electron by one volt:
1 eV = 1.6 ´ 10-19 coulombs ´ 1 volt = 1.6 ´ 10-19 J.
Masses of particles are invariably expressed in electron-volts. When someone says, "The mass of a proton is 938.3 MeV", she means that the rest-mass
energy of a proton, mc2, is 938.3 MeV. Note that “MeV” means “mega-electron-volts” or 1 million eV or 106 eV
Special note, mass is not "converted" to anything; mass is the energy of system at rest. The total energy of an isolated system is conserved. If something happens to the system, you set the total energy of the system before equal to the total energy after. If the system is not isolated, but external work is performed on it, the total initial energy plus the net work performed is equal to the total final energy.
There is no separate mass conservation, it is ALL Conservation of Energy.
There are processes that occur
in which both momentum and energy are conserved (as they always should) but
that rest-mass of the system changes.
This happens when the object
itself changes during the “process” and is common in the nuclear
physics of radioactive decay.
Last Modified: 9 January 2006, GSI.