Special Theory of Relativity
Let's start by having a look at one of the most famous equations of all time, one that looks so simple, yet is so powerful in describing the universe.
According to the special theory of relativity mass and energy are equivalent
Equation 41 - Special theory of relativity
Where E is the energy and m is the mass; the square of the speed of light, c2, is then just a conversion factor between two sets of units.
The Mass-energy equivalence is a result of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time components (the energy) to the space components (the momentum) in a non-trivial way.
The energy associated with the mass of the electron can be found using this equation. We know that the mass of an electron is given by me = 9.109x10-31 kg. We also know that the speed of light, c, is 2.998x108 ms-1.
We can substitute these values into equation 41 to give:
This is a very small amount of energy, and usually we specify small quantities of energy in electron volts (1eV = 1.6022x10-19 J).
The energy equivalent to the mass of the electron at rest is 511,000 eV or 511.0 keV. This is known as the rest mass energy.
Energy can be characterised as either kinetic or potential energy. The fundamental definition of the kinetic energy of a particle is given by the formula:
Equation 42 - Kinetic energy of a particle
Where v is the speed of the particle.
In cosmology, the gravitational interaction is the most important force, and so the most useful definition of the potential energy. For two-body problems, potential energy, U, has the form:
Equation 43 - Gravitational interaction
Where G is the Gravitational constant and has the value 6.673x10-11 Nm2 kg-2.
Potential energy in a two-body problem
Only differences in potential energy ever enter into calculations. In a two-body problem, we generally take U = 0 when the two particles are infinitely separated (this is why there is a minus sign in the above equation) but this is only a convention. If there is no force opposing the gravitation, the masses will move towards one another with increasing velocity.
At some point, the small mass m will have a velocity v relative to the larger mass M (it means we assume M to be at rest and measure the velocity at which m approaches M) and the distance will have reduced to r', as shown in the diagram above.
The total energy of a particle is defined as the kinetic energy plus the potential energy:
So in our case:
The energy and mass are conserved quantities, which means that they cannot be created from nothing or eliminated; they can only change form. Potential energy can, for example, be transformed into kinetic energy, and vice versa. Practically, it means for our two masses M and m that if there has been no intervention from any third body, the total energy in the system remains constant.
So how does it happen that, despite the gravitational attraction, the Moon does not fall on the Earth or the Sun does not fall immediately to the centre of our Galaxy?
This is due to dynamic equilibrium which is achieved by the orbital motion.
Orbital motion: an example of dynamical equilibrium in a two-body problem.
An orbiting body, at radius r, experiences the centripetal force Fc, which keeps it moving in a circle.
Equation 44 - Centripetal force
Where vc is the velocity of rotation and r is the radius measured from the centre of rotation and the rotating body.
Imagine a stone swung in a circle from a string and the centripetal force is provided by the tension in the string. Here the centripetal force is provided by gravitational attraction. If the centripetal force suddenly disappeared (the string is cut) then the orbiting body would fly off at a tangent. Gravitational attraction stops the body from flying off and keeps it moving in a circle.
The gravitational force is given by
Equation 45 - Gravitational force
The condition for dynamical equilibrium is then;
The above principles of Newtonian mechanics govern most of the dynamics in cosmological objects.
This post is part of the series Cosmology. Use the links below to advance to the next tutorial in the couse, or go back and see the previous in the tutorial series.
