How does modem relativity modify the law of conservation of momentum?

In classical physics, the law of conservation of momentum states that the total momentum of a closed system of particles is conserved over time, provided that no external forces are acting on the system. Mathematically, it can be represented as: \(Σ\vec{p} = Σ\vec{p'}\) Here, Σ represents the summation over all particles in the system, \(\vec{p}\) is the initial momentum, and \(\vec{p'}\) is the final momentum.

According to the special theory of relativity, as a particle's speed approaches the speed of light, its mass effectively increases and thus, momentum is no longer given by the classical formula: \(\vec{p} = m\vec{v}\) Instead, it is given by the relativistic momentum formula: \(\vec{p} = \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}}\) where \(m\) is the rest mass of the particle, \(v\) is the velocity, and \(c\) is the speed of light in a vacuum.

In special relativity, the conservation of momentum is similarly based on the principle that the total momentum of an isolated system remains constant. However, it must now be expressed using the relativistic momentum formula: \(Σ \dfrac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}} = Σ \dfrac{m'\vec{v'}}{\sqrt{1 - \frac{{v'}^2}{c^2}}}\) Here, Σ represents the summation over all particles in the system, \(m\vec{v}\) and \(m'\vec{v'}\) are the initial and final momenta of the particles, and \(c\) is the speed of light. This equation demonstrates that even in the context of special relativity, the law of conservation of momentum still holds true, though it must be expressed using the modified relativistic momentum formula to account for the increasing mass and corresponding momentum of particles as they approach the speed of light.