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## Derivation of relativistic mass

I have taught special relativity for many years, but every time I teach it I present the result that mass changes as a function of velocity as a consequence of the modified version of Newton’s 2nd law.

As almost everyone knows, Newton’s 2nd law says that

$F=ma$

where $F$ is the force applied, $m$ is the mass, and $a$ is the acceleration felt by the body. In Newtonian mechanics, mass is invariant, but a consequence of special relativity is that nothing can travel faster than the speed of light $c$. This raises the conundrum of why can’t we keep applying a force to a body of mass $m$, causing it to continue accelerating and to ultimately increase its velocity to one greater than the speed of light?

The answer is that Newton’s 2nd law is incomplete. Einstein showed that mass is also a function of velocity, and so we should write

$m = \gamma m_{0} \text{ (1) }$

Where $\gamma = \frac{ 1 }{ \sqrt{ (1 - V^{2}/c^{2}) } }$ is the so-called Lorentz factor and$m_{0}$ is the rest mass (also known as the invariant mass or gravitational mass), the mass an object has when it is at rest relative to the observer. Hence we can argue that, as we approach the speed of light, the applied force goes into changing the mass of the body, rather than accelerating it, leading to a modified version of Newton’s 2nd law

$F = \gamma m_{0} a$

where both velocity and/or mass change as a force is applied. But, because of the fact that $\gamma \approx 1$ until $V \approx c/2$ (see Figure 1), very little increase in mass occurs until $V$ has reached appreciable values.

The variation of $\gamma$ (the Lorentz factor) as a function of the speed $V$. Until $V \approx c/2, \; \gamma$ is very close to unity

However, I have always found this an inadequate explanation of the relativistic mass, as it does not derive it but rather argues for its necessity. So, as I’m teaching special relativity again this year, I decided a few weeks ago to see if I could find a way of deriving it from a simple argument. After several weeks of hunting around I think I have found a derivation which is robust and easy to understand. But, in my searching I came across several “derivations” which were nothing more than circular arguments, and also some derivations which were simply incorrect.

## Two balls colliding

The best explanation that I have found to derive the relativistic mass is to use the scenario of two balls colliding. Although it would be possible, in theory to have the balls moving in any direction, we are going to make things a lot easier by having the balls moving in the y-direction, but with the two reference frames $S \text{ and } S^{\prime}$ moving relative to each other with a velocity $V$ in the x-direction. Also, the balls are going to have the same rest mass, $m_{0}$, as measured in their respective frames $S$ and $S^{\prime}$ (the rest mass of each ball can be measured by each observer in their respective reference frames when they are at rest in their respective frames).

The blue ball $B$ moves solely in the y-direction in reference frame $S$, and the red ball $R$ moves solely in the y$^{\prime}$-direction in reference frame $S^{\prime}$. Ball $B$ starts by moving in the positive y-direction in reference frame $S$ with a velocity $u_{0}$, and ball $R$ starts moving in the negative y$^{\prime}$-direction in reference frame $S^{\prime}$ with a velocity $-u_{0}$ in frame $S^{\prime}$.

Reference frame $S^{\prime}$ is moving relative to frame $S$ at a velocity $V$ in the positive x-direction. So, as seen in $S$, the motion of ball $R$ appears as shown in the left of Figure 2. That is, it appears in $S$ to move both in the negative y-direction and the positive x-direction, and so follows the path shown by the red arrow pointing downwards and to the right.

At some moment the two balls collide. After the collision, as seen in $S$, ball $B$ will move vertically downwards in the negative y-direction, with a velocity $-u_{0}$. Ball $R$ moves upwards (positive y-direction) and to the right (positive x-direction), as shown by the red arrow in the diagram on the left of Figure 1.

In reference frame $S^{\prime}$ the motions of balls $B$ and $R$ looks like the diagram on the right of Figure 1. In $S^{\prime}$, it is ball $R$ which moves vertically, and ball $B$ which moves in both the $x^{\prime}$ and $y^{\prime}$ directions.

Two balls colliding. Ball $B$ (in blue) moves solely in the y-direciton as seen in frame $S$, ball $R$ (in red) moves solely in the y-direction in frame $S^{\prime}$.

## The velocity of ball $R$ in $S$

To calculate the velocity of ball $R$ as seen in $S$, we have to use the Lorentz transformations for velocity. As we showed in this blog here, if we have an object moving with a velocity $u^{\prime}$ in $S^{\prime}$ which is moving relative to $S$ with a velocity $V$, then the velocity $u$ in frame $S$ is given by

$u = \frac{ u^{\prime} + V }{ \left( 1 + \frac{ u^{\prime}V }{ c^{2} } \right) } \text{ (2) }$

This equation is true when the velocity is in the x$^{\prime}$-direction, and the frames are moving relative to each other in the x-direction. So we are going to re-write Equ. (2) as

$u_{x} = \frac{ u^{\prime}_{x} + V }{ \left( 1 + \frac{ u^{\prime}_{x}V }{ c^{2} } \right) } \text{ (3) }$

However, if the velocity of an object is in the y$^{\prime}$-direction, rather than the x$^{\prime}$-direction, then we need a different expression. We can derive it from going back to our equations for the Lorentz transformations

The Lorentz transformations

This time we write

$dy = dy^{\prime}$

and

$dt = \gamma \left( dt^{\prime} + \frac{ dx^{\prime}V }{ c^{2} } \right)$

So

$\frac{ dy }{ dt } = \frac{ dy^{\prime} }{ \gamma \left( dt^{\prime} + \frac{ dx^{\prime}V }{ c^{2} } \right) }$

Dividing each term in the right-hand side by $dt^{\prime}$, we get

$\frac{ dy }{ dt } = \frac{ dy^{\prime}/dt^{\prime} }{ \gamma \left( dt^{\prime}/dt^{\prime} + \frac{ dx^{\prime}V }{ dt^{\prime}c^{2} } \right) }$

$u_{y} = \frac{ u^{\prime}_{y} }{ \gamma \left( 1 + \frac{ u^{\prime}_{x}V }{ c^{2} } \right) } \text{ (4) }$

Equations (3) and (4) allow us to work out the components of ball $R$’s velocities $u_{x}$ in the x-direction and $u_{y}$ in the y-direction in frame $S$.

$u(R)_{x} = \frac{ 0 + V }{ \left( 1 + \frac{ 0 \cdot V }{ c^{2} } \right) } = V \text{ (5) }$

$u(R)_{y} = \frac{ -u_{0} }{ \gamma \left( 1 + \frac{ 0 \cdot V }{ c^{2} } \right) } = \frac{ -u_{0} }{ \gamma } \text{ (6) }$

After the collision, the velocity of ball $B$ becomes $u(B) = -u_{0}$. What about ball $R$?

We can see that $u(R)_{x}$ will not change, and $u(R)_{y}$ after the collision will be $- \frac{ + u_{0} }{ \gamma }$.

## The momentum before and after the collision

We are now going to look at the momentum of balls $B$ and $R$ before and after the collision, as seen in frame $S$. We will start off by assuming that the mass is constant for both balls, that is that $m=m_{0}$ for both balls, despite the two reference frames moving relative to each other.

If we do this, we can write that the momentum in the x-direction before the collision is given by

$(p(B)_{x} + p(R)_{x})_{i} = 0 + m_{0}V = m_{0}V$

The momentum after the collision in the x-direction is given by

$(p(B)_{x} + p(R)_{x})_{f} = 0 + m_{0}V = m_{0}V$

So, momentum is conserved in the x-direction. But, what about in the y-direction? Before the collision, the momentum is given by

$(p(B)_{y} + p(R)_{y})_{i} = + m_{0}u_{0} + m_{0} \left( \frac{ -u_{0} }{ \gamma } \right) =m_{0}u_{0} - \frac{ m_{0}u_{0} }{ \gamma }$

After the collision, the momentum in the y-direction is given by

$(p(B)_{y} + p(R)_{y})_{f} = m_{0}(-u_{0}) + m_{0} \left( \frac{ +u_{0} }{ \gamma } \right) = -m_{0}u_{0} + \frac{ m_{0}u_{0} }{ \gamma }$.

If we assume that momentum is conserved, we can write

$m_{0}u_{0} - \frac{ m_{0}u_{0} }{ \gamma } = -m_{0}u_{0} + \frac{ m_{0}u_{0} }{ \gamma } \rightarrow 2m_{0}u_{0} = \frac{ 2m_{0}u_{0} }{ \gamma } \rightarrow \gamma = 1$

So, if we assume that the mass of both ball $B$ and ball $R$ in frame $S$ is $m_{0}$, the momentum in the y-direction is only conserved if $\gamma =1$. But, $\gamma$ is only equal to unity when the relative velocity $V$ between the two frames is zero; in other words when the two frames are not moving relative to each other! If $V \neq 0$ and mass is constant, momentum will not be conserved.

In physics, the conservation of momentum is considered a law, it is believed to always hold. In order for momentum to be conserved, we can qualitatively see that the mass of ball $R$ needs to be greater than the mass of ball $B$ as seen in frame $S$, as the speed of ball $R$ in the y-direction in frame $S, |u(R)_{y}| = u_{0} / \gamma < u_{0}$.

## Allowing the mass to change

We have just shown above that, if we assume both masses are invariant, momentum will only be conserved in the y-direction in the trivial case where the two frames are stationary relative to each other. So, let us now assume that, if $V \neq 0$, we have to allow the masses to change.

We will assume that mass is a function of speed. For ball $B$, the momentum in the x-direction is still zero, both before and after the collision. For ball $R$, we will now write the momentum in the x-direction, both before and after the collision, as

$p(R)_{x} = m(R) u(R)_{x} = m(R) V$

What about in the y-direction? For ball $B$, before the collision we can write

$p(B)_{y} = m(B) u(B)_{y} = m(B) u_{0}$

Where $m(B)$ is the mass of ball $B$ in frame $S$ which is affected by its velocity in frame $S$, which is $u_{0}$.

For ball $R$ as seen in frame $S$ we can write that the momentum in the y-direction before the collision is given by

$p(R)_{y} = m(R) u(R)_{y} = m(R) \cdot \left( \frac{ - u_{0} }{ \gamma } \right) = \frac{ -m(B)u_{0} }{ \gamma }$

Where $\gamma = 1/(1-V^{2}/c^{2})$, the Lorentz factor due to the relative velocity $V$ between $S$ and $S^{\prime}$.

After the collision, we can write the momentum for ball $B$ in the y-direction as being

$p(B)_{y} = m(B) u(B)_{y} = -m(B) u_{0}$

And, for ball $R$ we can write

$p(R)_{y} = m(R) u(R)_{y} = \frac{ +m(R)u_{0} }{ \gamma }$

Equating the momentum in the y-direction before and after the collision, we have

$m(B) u_{0} - \left( \frac{ m(R) u_{0} }{ \gamma } \right) = -m(B) u_{0} + \left( \frac{ m(R) u_{0} }{ \gamma } \right)$

$\rightarrow 2m(B) u_{0} = 2 \left( \frac{ m(R) u_{0} }{ \gamma } \right) \rightarrow m(A) =\frac{ m(R) }{ \gamma }$

For ball $B$, we will write

$m(B) = \gamma_{B} m_{0}$

where

$\gamma_{B} = \frac{ 1 }{ \sqrt( 1 - u^{2}(B)/c^{2} ) } = \frac{ 1 }{ \sqrt( 1 - u_{0}^{2}/c^{2} ) }$

(that is, $\gamma_{B}$ depends on the speed of ball $B$ in frame $S$, and that speed is $u_{0}$).

So, the momentum of ball $B$ in the y-direction is given by

$p(B)_{y} = m(B)u(B)_{y} \rightarrow \boxed {p(B)_{y} = \frac{ m_{0} u_{0} }{ \sqrt( 1 - u_{0}^{2}/c^{2} ) } \text{ (7) } }$

For ball $R$, we will write

$m(R) = \gamma_{R} m_{0}$

Where $\gamma_{R} = 1/\sqrt{ (1 - u^{2}(R)/c^{2} ) }$ depends on the speed $u(R)$ of ball $R$ as seen in frame $S$. (Note: the mass does not depend on just the y-component of ball $R$‘s speed (as is often incorrectly stated), it depends on its total speed).

To calculate the value of $u(R)$ we note that it is made up of the x-component $u(R)_{x}$ and the y-component $u(R)_{y}$. But, $u(R)_{x} = V$, and we showed above that $u(R)_{y} = -u_{0}/ \gamma$, where this $\gamma = 1/\sqrt{ (1 - V^{2}/c^{2}) }$.

Using Pythagoras to calculate $u(R)$, we have

$u(R)^{2} = V^{2} + u_{0}^{2}/\gamma^{2} = V^{2} + u_{0}^{2}(1 -V^{2}/c^{2})$

so

$u(R)^{2} = u_{0}^{2} + V^{2}( 1 - u_{0}^{2}/c^{2} )$

Using this value of $u(R)$ we can write

$\gamma_{R} = \frac{ 1 }{ \sqrt{ ( 1 - u(R)^{2}/c^{2} )} } = \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} ) } }$

But, the terms $( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} )$ can be factorised as

$( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} ) = (1 - u_{0}^{2}/c^{2})(1 - V^{2}/c^{2})$

And so we can write

$\gamma_{R} = \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } \sqrt{ (1 - V^{2}/c^{2}) } }$

But, $1/\sqrt{ (1 - V^{2}/c^{2}) } = \gamma$, so we can write

$\gamma_{R} = \gamma \cdot \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } }$

This means that we can write the momentum for ball $R$ in the y-direction as

$p(R)_{y} = m(R) u(R)_{y} = \gamma_{R} m_{0} u(R)_{y}$

$p(R)_{y} = \gamma \cdot \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } } \cdot m_{0} \cdot \frac{ u_{0} }{ \gamma }$

$\boxed{ p(R)_{y} = \frac{ m_{0}u_{0} }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } } \text{ (8) } }$

Comparing this to Equ. (7), the equation for $p(B)_{y}$, we can see that they are equal, as required.

So, we have proved that, to conserve momentum, we need mass to be a function of speed, and specifically that

$\boxed{ m = \frac{ m_{0} }{ \sqrt{ (1 - u^{2}/c^{2}) } } }$

Where $u$ is the speed of the ball in a particular direction in frame $S$.

## How to add velocities in special relativity

As I mentioned in this blogpost, in special relativity any observer will measure the speed of light in a vacuum to be $c$, irrespective of whether the observer is moving towards or away from the source of light. We can think of the speed of light as a cosmic speed limit, nothing can travel faster than it.

But, let us suppose that we have two reference frames $S$ and $S^{\prime}$ moving relative to each other with a speed of $v=0.9c$, 90% of the speed of light. Surely, if someone in frame $S^{\prime}$ fires a high-speed bullet at a speed of $u^{\prime}= 0.6c$, an observer in frame $S$ will think that the bullet is moving away from him at a speed of $u = v + u^{\prime} = 0.9c + 0.6c = 1.5c$, which seemingly violates the comic speed limit.

What have we done wrong?

We cannot simply add velocities, as we would do in Newtonian mechanics. In special relativity we have to use the Lorentz transformations to add velocities. How do we do this? Let us remind ourselves that the Lorentz transformations can be written as

The Lorentz transformations to go either from reference frame $S \text{ to } S^{\prime}$, or to go from $S^{\prime} \text{ to } S$.

## Calculating a velocity in two different reference frames

To calculate the velocity $u$ of some object moving with a velocity $u^{\prime}$ in reference frame $S^{\prime}$ we need to use these Lorentz transformations.

We start off by writing

$x = \gamma \left( x^{\prime} + vt^{\prime} \right) \text{ (1) }$

and

$t = \gamma \left( t^{\prime} + \frac{x^{\prime}v}{c^{2} } \right) \text{ (2) }$

We will now take the derivative of each term, so we have

$dx = \gamma \left( dx^{\prime} + vdt^{\prime} \right)$

and

$dt = \gamma \left( dt^{\prime} + \frac{dx^{\prime}v}{c^{2} } \right)$

We can now write $dx/dt = u$ (the velocity of the object as seen in frame $S$) as

$\frac{dx}{dt} = \frac{ \gamma \left( dx^{\prime} + vdt^{\prime} \right) }{ \gamma \left( dt^{\prime} + \frac{dx^{\prime}v}{c^{2} } \right) }$

The $\gamma$ terms cancel, and dividing each term on the right hand side by $dt^{\prime}$ gives

$\frac{dx}{dt} = \frac{ \left( dx^{\prime}/dt^{\prime} + vdt^{\prime}/dt^{\prime} \right) }{ \left( dt^{\prime}/dt^{\prime} + \frac{dx^{\prime}v}{c^{2} dt^{\prime} } \right) } = \frac{ \left( u^{\prime} + v \right) }{ \left( 1 + \frac{ u^{\prime} v}{c^{2} } \right) }$

$\boxed{ u = \frac{ \left( u^{\prime} + v \right) }{ \left( 1 + \frac{ u^{\prime} v}{c^{2} } \right) } }$

where $u^{\prime}$ was the velocity of the object in reference frame $S^{\prime}$.

Going back to our example of $v = 0.9c$ and $u^{\prime} = 0.6c$, we can see that the velocity $u$ as measured by an observer in reference frame $S$ will be

$u = \frac{ 0.6c + 0.9c }{ \left( 1 + \frac{ (0.6c \times 0.9c) }{ c^{2} } \right) } = \frac{ 1.5c }{ 1 + 0.54 } = \frac{ 1.5c }{ 1.54} = \boxed {0.974c}$, not $1.5c$ as we naively calculated.

## The constancy of the speed of light

What happens if a person in reference frame $S^{\prime}$ shines a light in the same direction as $S^{\prime}$ is moving away from $S$? In this case, $u^{\prime}=1.0c$. Putting this into our equation for $u$ we get

$u = \frac{ 0.6c + 1.0c }{ \left( 1 + \frac{ (0.6c \times 1.0c) }{ c^{2} } \right) } = \frac{ 1.6c }{ 1 + 0.6 } = \frac{ 1.6c }{ 1.6} = 1.0c$

So they both agree that the light is moving away from them with the same speed $c$!