## Neutrinos faster than light.

Recently experiment has shown that neutrinos move faster then light. Some claim that “neutrinos might break one of the most fundamental laws of physics”. The question would be “what law of physics would be broken?” In general this all is associated with relativity. But there is no law within relativity that tell us that nothing can move faster then light. What relativity tells us is that the speed of light is a special velocity.

Let us consider a body. We can change the velocity of the body by using of a force. The change in energy is by definition given by $d E = \vec{F} d\vec{r}$. Newton tells us that $\displaystyle \vec{F} = \frac{d\vec{p}}{dt}$. Newton tells us also that $\vec{p} = m \vec{v}$, while Einstein tells us that $\vec{p} = m_o \vec{v} / \sqrt{1 - v^2 / c^2}$. So when we follow Newton for the force $\vec{F}$ and Einstein for the momentum $\vec{p}$, we get

(1) $\displaystyle dE = m_o \vec{v} d \left( \frac{\vec{v}}{\sqrt{1 - \vec{v} \cdot \vec{v} / c^2}} \right) = d\left( \frac{m_o c^2}{\sqrt{1 - \vec{v} \cdot \vec{v}/c^2}} \right)$

So we clearly obtain $\displaystyle E(v) = E_o + \frac{m_o c^2}{\sqrt{1 - v^2/c^2}}$, where we have used $\vec{v}\cdot\vec{v} = v^2$. Since this equation depends on speed, direction is not of importance. Within the Special Theory of Relativity the convention is used that $E_o = 0$, thus we obtain

(2) $\displaystyle E(v) = \frac{m_o c^2}{\sqrt{1 - v^2/c^2}}$

and also the famous equation $E(0) = m_o c^2$. Let $E(v)$ be the energy of a body moving at the speed $v$. Let $\Delta E(v_1,v_2)$ be the energy required to change the velocity of a body from $v_1$ to $v_2$. It is clear that $\Delta E(v_1,v_2) + \Delta E(v_2,v_3) = \Delta E(v_1,v_3)$ as $\Delta E(v_1,v_2) = E(v_2) - E(v_1)$. Let $v_1 < c$ and $v_2 > c$. It is clear that $\displaystyle E(v_1) = \frac{m_o c^2}{\sqrt{1-v_1^2/c^2}}$ and $\displaystyle E(v_2) = \frac{- i m_o c^2}{\sqrt{v_2^2/c^2-1}}$. Therefore

(3) $\displaystyle E(v_1,v_2) = - m_o c^2 \left( \frac{1}{\sqrt{1 - v_1^2/c^2}} + \frac{i}{\sqrt{v_2^2/c^2 - 1}} \right)$.

The amount of energy required to change the speed of a body from a speed slower than light into a speed faster than light is complex. This is not allowed and therefore we cannot change the speed of a body from a speed slower than light into a speed faster than light is complex – but the reverse is also true due to the symmetry; the amount of energy required to change the speed of a body from a speed faster than light into a speed slower than light is complex and is not allowed and therefore we cannot change the speed of a body from a speed faster than light into a speed slower than light is complex. In other words: either bodies always move with a speed slower than the speed of light or bodies always move with a speed faster then the speed of light. A body that always moves faster than the speed of light does not have an observable rest-mass $m_o$ and such bodies a theoretically a complex rest-mass.

When neutrinos move faster than the speed of light then neutrinos have a complex rest mass and that does not “break one of the most fundamental laws of physics”.

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### 3 Responses to Neutrinos faster than light.

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