where m is the mass of the particle. Equation  gives the energy per unit mass of a particle that moves with constant speed.
OBJECTION: Baloney! Everyone knows that a free particle moves with constant speed along a straight path in space as observed in a free-float frame So as this motion proceeds, every possible expression that depends only on v = sit is also a constant of the
motion, for example the expression v'2, which is certainly not the correct expression for energy1 Your derivation proves nothing'
i RESPONSE You are almost right Any function of velocity v = s/t is indeed constant for the special case of a free particle in flat spacetime And if v is constant, so is tlx, as witnessed by Equation  But notice the priorities used in the derivation The Principle of Extremal Aging has highest priority, the expression for energy comes out of this principle Of all the quantities that remain constant because v is constant, the Principle of Extremal Aging picks out tlx = Elm as primary. (The following section shows that a similar analysis picks out the relativistic expression for momentum as a constant of the motion ) Chapter 3 contains a new and more general expression for energy in curved spacetime In that case the velocity is not constant—yet that more general expression for energy is correct and a constant of the motion nevertheless Our derivation of the expression for Elm in flat spacetime is thus a trial run for the derivation of the energy of a particle in the curved spacetime around a center of gravitational attraction
If the particle changes speed, then it changes energy. In that case it makes sense to talk about instantaneous speed and to use calculus notation. Let the pair of flash emissions in Figure 1 be separated by the incremental frame coordinates dt, ds, and incremental wristwatch time dx. The equation for E/m then becomes
Ordinarily we use the ratio E/m in equations, instead of E alone. Why? Because it emphasizes two important principles: (1) Only spacetime relations between events appear on one side of equations such as  and , reminding us that it is spacetime geometry that leads to these expressions, not some weird property of matter. (2) The ratio E/m has no units. Therefore, whoever uses these equations has total freedom in choosing the unit of E and m, as long as it is the same unit. The same unit in the numerator and denominator of  may be kilograms or the mass of the proton or million electron-volts. If you insist on using conventional units, such as joules for energy E and kilograms for mass m, then a conversion factor c2 intrudes into our simple equation:
Now view the particle from a reference frame in which the particle is at rest. In this rest frame there is zero distance s between sequential flash emissions. Equation  says that for s = 0 the frame time t and wristwatch time x have exactly the same value. For a particle at rest, then, equation  reduces to the most famous equation in all of physics:
Particle energy in special relativity
Rest energy famous formula r joules rest - mkgc
Note that equation  describes the rest energy of a particle. For a particle in motion, the energy is given by equation .
In equation , c has the defined value 2.99792458 x 108 meters/second. An equation of the same form is correct if E is measured in ergs, m in grams, and c in centimeters/second.
Fuller Explanations: Energy in flat spacetime: Spacetime Physics, Chapter 7, Momenergy.
6 Momentum in Special Relativity
The metric plus the Principle of Extremal Aging give us an expression for momentum
The relativistic expression for momentum is derived by a procedure analogous to the one used to derive the relativistic expression for energy. The figures look similar to Figures 2 and 3, but in this case the time t for the intermediate flash emission is fixed, while the position s for this event is varied right and left to yield an extremum for the total wristwatch time from the first flash to the third flash. (You carry out the derivation of momentum in the exercises at the end of this chapter.) The result is a second constant of the motion for a free particle:
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