1. Orthogonal Vectors
2. Transformation of four forces

Orthogonal Relations in Special Relativity

1. Definition of Orthogonality in Minkowski Spacetime:

   • Two four-vectors $A_\mu$ and $B_\mu$ are said to be orthogonal if their Minkowski dot product (scalar product) is zero:

     $$ A^\mu B_\mu = 0 $$

   • This dot product in Minkowski spacetime is defined as:

     $$ A^\mu B_\mu = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 $$

   Where $A^0$ and $B^0$ are time components, and $A^1$, $A^2$, $A^3$ and $B^1$, $B^2$, $B^3$ are spatial components.

Physical Interpretations of Orthogonality

• Four-Momentum and Four-Force:
  • The four-force $F^μ$ acting on a particle is orthogonal to the four-momentum $p^μ$ of the particle

$$ F^\mu p^\mu = 0 $$

This arises due to the invariance of the particle’s rest mass (which is constant in time).

Four-Velocity and Four-Acceleration

:

• The four-velocity $u^μ$ and four-acceleration $a^μ$ of a particle are orthogonal in Minkowski spacetime

$$ u^\mu a_\mu = 0 $$

This is a direct consequence of the normalization of the four-velocity

$$ u^\mu u_\mu = C^2 $$

where $C$ is the speed of light.