Modern Physics/Acceleration, Force and Mass

Momentum

Classically, momentum is velocity multiplied by mass. We can use the same definition in relativity, and see where it takes us.

$\underset{\_}{p}=m_0\underset{\_}{u}=m_0\gamma (𝐯,c)$

You may sometimes see the product m₀γ which shows up here called the relativistic mass, but we will not be using this approach.

The mass, m₀, is generally called the rest mass, to distinguish it from the relativistic mass.

The spatial component of the four-momemntum is clearly the classical momentum, scaled by a factor of γ. At speeds much less than c this will be approximately 1.

The temporal component is m₀γc. To see what this means we can look at its value when v/c is small.

$m_0\gamma c=\frac{m_{0}c}{\sqrt{1-v^2/c^2}}=m_{0}c(1+\frac{1}{2}\frac{v^2}{c^2}+\frac{3}{8}\frac{v^4}{c^4}+\cdots )$

The first term in this expansion is a constant.

The second term is

$\frac{m_0{v}^2}{2c}$

which we recognise as being the classical kinetic energy, divided by c.

Now, adding a constant to the definition of kinetic energy makes no real difference, since all that matters are changes in energy, so we can identify this temporal component of relativistic momentum with the energy over c.

$\underset{\_}{p}=(\gamma 𝐩,E/c)$

We then have

$E=m_0{c}^2+\frac{1}{2}{m}_0{v}^2+\frac{3}{8}\frac{m_0{v}^4}{c^2}+\cdots$

Even at rest, the particle has a kinetic energy,

$E=m_0{c}^2,$

the most famous relativistic equation.

Force

Classically, we have

$𝐅=\frac{d𝐩}{dt}$

We can get the equivalent relativistic equation simply by replacing 3-vectors with 4-vectors and t with τ, giving

$\underset{\_}{F}=\frac{d\underset{\_}{p}}{d\tau }$

Provided the rest mass is constant, as it is for all simple systems, we can rewrite this as

$\underset{\_}{F}=m\underset{\_}{a}$

We already know about a so we can now write

$\underset{\_}{F}={\gamma }^4(𝐅,\frac{𝐅\cdot 𝐯}{c})$

The temporal component of this is essentially the power, the rate of change of energy with time, as might be expected from energy being the temporal component of momentum.