Relativistic Total Energy
Recall an expression for a kinetic energy K of an object of mass m0 moving with speed u derived in the previous lecture Relativistic Kinetic Energy of this course:
| K = |
|
− m0·c² |
So, kinetic energy is represented as a difference between two other expressions.
Naturally, it's tempting to consider these other two expressions as representing some type of energy themselves.
Consider an equivalent to above formula
| m0·c² + K = |
|
Logically speaking, that something else must be some form of energy because, when we add two physical variables, they must be of the same kind and units.
Moreover, the right side of this equation that expresses something else multiplied by Lorentz γ-factor depends on the speed of object u and becomes equal to something else when speed equals to zero.
Obvious conclusion is that something else (m0·c²) represents some form of energy that is concentrated in an object at rest in a frame of reference associated with it, while the right side of a formula is the result of increasing this rest energy by an amount of kinetic energy (K) developed by this object because of its motion.
In other words, an expression
| E = |
|
Let's analyze this from the viewpoint of the Law of Conservation of Energy.
Assume an object is at rest in an inertial frame of reference associated with it, and its mass is M.
Assume further that at some moment of time this object splits in two equal parts flying in the opposite to each other directions with speed u.
The total energy of this system before a split was only the rest energy of our object, that is
Ebefore = M·c²
The total energy after a split is a sum of two equal amounts of energy possessed by two parts of our initial object.
If we assume that the rest mass of each part is M/2, we'll face an increase of the total energy of the system without any action from outside, which contradicts the Law of Conservation of Energy.
What follows is a conclusion that the mass of each of two parts must be less than M/2 to compensate an increase in kinetic energy of these parts.
Let's assume that the rest mass of each part of an initial object is m (less than M/2).
Than the rest energy of each part would be m·c² and kinetic energy of each part would be
| Kpart = |
|
− m·c² |
| Epart = |
|
| Eafter = |
|
Therefore,
| M·c² = |
|
2m/M = √1−u²/c²
What remains to be proven in this logical train of thoughts is that the total relativistic energy is invariant relatively to Lorentz transformations of coordinates from one inertial reference frame to another.
To accomplish that, we will relate the total relativistic energy to the relativistic momentum and use the corresponding results obtained for the relativistic momentum in UNIZOR.COM - Relativity 4 All - Conservation - Momentum lecture of this part of a course.
Recall the definition of the relativistic momentum
| p = |
|
The formula for the total relativistic energy discussed above is
| E = |
|
E2 − p2·c2 = m02·c4
or
E2 = p2·c2 + m02·c4
or
E2 = (p·c)2 + E02
where E0 is a rest energy the object has just because it has certan mass.
The above equality is called energy - momentum relation in Special Theory of Relativity.
In particular, for objects with no rest mass (like photons of light) with m0=0 the following equation between the total energy and momentum is held
E = p·c
As we see, the total relativistic energy of an object of a rest mass m0 can be expressed in terms of its relativistic momentum, which is invariant relative to Lorentz transformation, and constants that are independent of the frame where an object is observed (its rest mass and the speed of light in vacuum).
Therefore, the total relativistic energy is invariant relative to Lorentz transformation, as it should be.
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