7. Energy and momentum

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title: IL MONDO A VELOCITA' VICINE A QUELLA DELLA LUCE 
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flowchart TD
     B1([ energia ed impulso non sono grandezze separate])
     B1 --> B2([energia ed impulso sono le coordinate di un vettore nello spaziotempo, l'enermoto])
     B2 --> B3([la quantità di enermoto è la stessa per tutti gli osservatori.])

Energy and momentum.

In studying the motion of a body, in addition to its position in space at a certain instant of time, two important quantities are its energy and impulse, or momentum, the product of mass times velocity.

The rate of change of the impulse is equal to the force exerted, per Newton’s second law of dynamics.

Global symmetries and conservation principles

From the symmetries of space and time derive the principles of conservation of physical quantities:

  • the laws of physics are the same at every instant of time ====> invariance of physical laws by time translation ===> principle of conservation of energy

  • the laws of physics are the same at every point in space ==> invariance of physical laws by spatial translation ==> homogeneous space ===> principle of conservation of momentum (product of mass times velocity)

  • the laws of physics are the same in all directions of space

==> invariance of physical laws by rotation ==> isotropic space ==> principle of conservation of angular momentum (product of momentum times radius of rotation)

Energy and impulse of a body in motion are not independent

At velocities close to those of light, analogous to what happens for space and time, energy and impulse are not independent quantities but the four coordinates of an energy-impulse vector in a four-dimensional space (energy is the time coordinate, the three impulse components are the three spatial coordinates), called by some authors (Wheeler) *enermoto. *The length of the enermoto is an invariant quantity, identical for all observers in uniform relative motion between them.

Again starting from the fact that the speed of light is a limiting speed, we derive the relationships between energy and impulse (momentum) at speeds close to those of light, and the relationship between mass and energy.

It is also found that the total energy of a body is composed of three parts:

  • potential energy due to any external forces
  • kinetic energy dependent on the velocity of the body in that reference system
  • a basic energy, present even when the body is stationary relative to the observer, proportional to the rest mass. More precisely, it is equal to the rest mass multiplied by the square of the speed of light.

The last term, resting energy proportional to mass, makes it possible to explain the energy balance of many phenomena in the microscopic world, later discovered and studied by quantum mechanics, in which matter is transformed into energy or vice versa.

Principles of conservation and covariance

Instead of separate conservation principles we will have an energy-pulse conservation principle for the homogeneity of space-time, i.e., all points in space-time are equivalent and there are no privileged points. conservation principles related to the global symmetries of space-time are in addition to the covariance requirement, **the **invariance of physical laws by Lorentz transformations of coordinates.

Impulse and energy in special relativity

**Pulse in classical mechanics **Pulse p, or momentum, is is the product of a mass times its velocity:

the arrow above indicates that they are vectors, that is, they also have a direction and three spatial components, while the rest mass is a scalar, a number

**Kinetic energy in classical mechanics **Kinetic energy K is given by the formula

Law of Motion Newton’s law of motion, force equals mass times acceleration, F = ma, can be rewritten by shifting the cause (the force) to the right and the effect to the left, and substituting the definition of impulse as

The mass m here is the inertia, that is, the resistance to acceleration

Relativistic pulse

Impulse in special relativity must be redefined from classical physics, in which it is the product of mass and velocity, to account for time dilation. If:

  • p is the impulse of a body of rest mass m and velocity v
  • m is the rest mass of the body, in a reference system in which it is stationary
  • γ is the Lorentz factor for the relative velocity v

Pulse and velocity are 3D vectors, with three components along the Cartesian axes.

Relationship between impulse, energy, and mass for m > 0 and v < c

In special relativity total energy and impulse are related to each other like space and time:

  • are the four coordinates of a vector in a four-dimensional space (E/c, p_(x) , p_(y) , p_(z) )
  • energy divides by c to have homogeneous dimensions
  • (p_(x) , p_(y) , p_(z) ) are the components along the three Cartesian axes of a 3D vector of modulus

Total energy and pulse modulus are related by the formula

The first quantity on the left is the same for all observers in uniform relative motion, an invariant between inertial reference systems.

In four-dimensional spacetime, it is the square of the length of the energy-pulse vector.

Only the length of the 3D pulse vector can appear in the formula because the space is isotropic, it has no privileged direction.

Total energy and mass

From the previous relations for impulse and energy, we derive the formula for the total relativistic energy of a body in motion, free, that is, not subject to external forces and therefore with zero potential energy. If:

  • E is the total energy of a body, of rest mass m, moving with velocity v

  • m is the rest mass of the body, in a reference system in which it is stationary

  • γ is the Lorentz factor for the relative velocity v

  • c is the speed of light

Energy at rest Suppose a body of mass m is stationary for the observer, we know that γ = 1 if v = 0, so we derive for the energy E₀ of a body at rest the value:

So a body possesses energy even if it is stationary in the theory of special relativity, and indeed it is a very large amount of energy even if the mass m is small, because the square of the speed of light c² is a very large number.

Kinetic energy

Returning to the general case of a moving body with velocity v, using the previous formulas for total energy and energy at rest we can calculate the kinetic energy K

Kinetic energy and impulse for small velocities

For v = 0 or otherwise v <\ c, that is, in the case where the velocity is very small compared to that of light, γ can be approximated by 1.

By placing γ = 1 in the relativistic impulse formula we find the formula of classical mechanics

Again if the velocity v is small relative to the speed of light in the formula for kinetic energy, the linear approximation can be used

obtaining the limit for low velocities of kinetic energy, in which we eliminate the constant c

not coincidentally, it is the same formula in classical mechanics, into which special relativity falls in the case where the velocities involved are small compared to those of light.

Energy and momentum for a particle of zero mass

In the special case of particles always moving at the speed of light c the Lorentz gamma factor diverges to infinity, so it is not straightforward to use the previous formulas.

If m = 0, the particle always moves at the speed of light c, and the ratio of total energy to the modulus of the pulse is constant and equal to the speed of light

The last formula is useful for defining the pulse of a photon, a quantum of light of energy E equal to Planck’s constant multiplied by the frequency

Equation of motion of special relativity The law of motion, also known as Minkowski’s equation, in special relativity is

formally the same as classical mechanics, except that the pulse value is not the same, but for high speeds it is corrected with the Lorentz gamma factor.

why the sun and stars shine: energy from nuclear reactions

In stars, such as the Sun, the nuclei of lighter elements, such as hydrogen, combine in a nuclear fusion reaction to form heavier and more stable nuclei, such as helium, releasing energy in the process.

Nuclear fusion is the energy source of the Universe.

In these processes the sum of the masses of the initial reactants is slightly greater than that of the final products, e.g., by 0.5 percent, the difference in mass has been converted into energy. Special relativity with its relationship between mass and energy explains the production of energy and allows it to be calculated accurately. In fact, the amount of energy released in the process is the difference in mass multiplied by the square of the velocity. Another brilliant proof of the validity of the theory.

  1. A.Sommerfeld, in “Naturwissenschaft Rundschau”, 1, pg. 97. Article reprinted in “Gesammelte Schriften “, IV, pg. 640 (1968) quoted in John Wheeler, Edwin Taylor - “Spacetime Physics, “ ch. 3, “Same Law for All” (title definitely not chosen at random)︎