Changes

(This is expressed in a simplified non-vector form, assuming that the two velocities are in a single dimension. For vectors, the calculation <math>v_1 v_2</math> needs to be performed as a [[dot product]].)

(This is expressed in a simplified non-vector form, assuming that the two velocities are in a single dimension. For vectors, the calculation <math>v_1 v_2</math> needs to be performed as a [[dot product]].)

Note that at low velocities (where <math>v_1 v_2</math> is small) then <math>v_1 v_2 / c^2</math> is close to zero and so the equation gives the same result as the Galilean formulation. Since <math>c</math> is such a large number (<math>c^2</math> being <math>9 \times 10^16 \; \text{m}^2 \, \text{s}^{-2}</math>), the Galilean transformation is sufficiently accurate for the everyday situations which humans encounter, and thus the transformation is sometimes regarded as intuitive. Even for the speeds involved in modern space exploration, Lorentzian adjustments are small. It is only with extremely lightweight bodies (i.e. [[subatomic particle]]s) that high enough speeds can be achieved, and thus devices such as [[particle accelerator]]s need to take account of the differences.

Note that at low velocities (where <math>v_1 v_2</math> is small) then <math>v_1 v_2 / c^2</math> is close to zero and so the equation gives the same result as the Galilean formulation. Since <math>c</math> is such a large number (<math>c^2</math> being <math>9 \times 10^{16} \; \text{m}^2 \, \text{s}^{-2}</math>), the Galilean transformation is sufficiently accurate for the everyday situations which humans encounter, and thus the transformation is sometimes regarded as intuitive. Even for the speeds involved in modern space exploration, Lorentzian adjustments are small. It is only with extremely lightweight bodies (i.e. [[subatomic particle]]s) that high enough speeds can be achieved, and thus devices such as [[particle accelerator]]s need to take account of the differences.

Thus it is debatable whether the Galilean transformation is actually ‘wrong’ since it is still of practical use in many situations – special relativity is a refinement under extreme conditions. Similarly, [[Gravitation#Newtonian Gravitation|Newtonian Gravitation]] is perfectly adequate in many situations – [[general relativity]] is a broadly equivalent refinement. It is often considered that [[Newtonian mechanics]] is an approximation to relativity at low speeds and small gravitational fields.

Thus it is debatable whether the Galilean transformation is actually ‘wrong’ since it is still of practical use in many situations – special relativity is a refinement under extreme conditions. Similarly, [[Gravitation#Newtonian Gravitation|Newtonian Gravitation]] is perfectly adequate in many situations – [[general relativity]] is a broadly equivalent refinement. It is often considered that [[Newtonian mechanics]] is an approximation to relativity at low speeds and small gravitational fields.