Classical physics (Newtonian) accurately describes the world only in the limiting case of low velocities ($v \ll c$), but fails at high velocities where relativistic effects dominate. The fundamental problem of classical physics lies in its inability to accurately describe phenomena at the microscopic level (atomic level). Towards the end of the 19th century, classical physics reached its limits because experimental observations did not agree with the established theories (Newtonian mechanics, Maxwell's electrodynamics). Classical physics is deterministic and continuous whereas nature at the atomic level is fundamentally probabilistic (dependent on chance) and quantized. The incompatibility of these two worldviews led to the development of Quantum mechanics.

Classical physics assumes absolute quantities:

- Absolute time (the same for all observers)
- Absolute simultaneity
- Absolute lengths

In classical physics, velocities add up by definition:

- Velocities add simply: $v_{total}=v_1+v_2$
- If one particle flies to the right at 0.9c and another to the left at 0.9c, their relative velocity would be 1.8c. This would violate the speed of light as a cosmic speed limit.

The Special Theory of Relativity recognizes:

- Time is relative (time dilation)
- Lengths are relative (length contraction)
- Simultaneity is relative

$\Large{v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}}$

This formula demonstrates that even at extremely high velocities, the resulting relative velocity always remains below the speed of light.

${\Large{v_{\text{rel}} = \frac{0.999999991c + 0.999999991c}{1 + \frac{(0.999999991c)^2}{c^2}}}} \approx 0.99999999999999999996c$
(LHC example)

Einstein's relativity is the key to understanding why particles never reach $c$ and why their energy lies in the TeV range.

Physical quantity

Velocity

Momentum / Relativistic Momentum

Energy / Relativistic Energy

Center-of-mass energy

Mass

Classical physics

Unlimited, since c can be exceeded: 1.999999982c (impossible)

$p = m v$

$E_{\text{kin}} = \frac{1}{2} m v^2$

$E_{\text{CM}} = \frac{1}{2} \mu \, v_{\text{rel}}^2$ // $E_{\text{CM}} = \sum \frac{1}{2} m_i \, v_i'^{\,2}$

constant

Relativistic solution

c as the cosmic limit: ≈c

$p = \gamma m_0 v$

$E_{\text{ges}} = \gamma m_0 c^2$ or $E_{\text{total}} = \gamma m_0 c^2$

$\sqrt{s}$ = 13.6 TeV

Apparent increase with velocity

$m_0$: Rest mass

$m_0$ is called rest mass

Definition: The mass of a particle in its rest frame, i.e when it is stationary relative to the observer (v=0).
Important property: Rest mass is an invariant quantity. It is the same for all observers regardless of their state of motion.
Example proton: $m_0 \approx 1.673 \times 10^{-27}$ kg or equivalently 938 MeV/c²

Rest mass is clearly distinct from relativistic mass $m(v) = \gamma m_0$, which depends on velocity.

$E_0$: Rest energy

$E_0$ is called rest energy
$E_0 = m c^2$

Meaning: The energy contained in the rest mass $m_0$ of a particle (even when it is motionless).
Example: A proton has $E_0$ = 938 MeV

- To be continued -

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