When adjusting the GPS satellites for special relativity, only the first-order approximation of the Lorentz factor is used instead of the full binomial expansion. This simplification is made because the higher-order terms in the expansion are significantly smaller and can be neglected without compromising the accuracy of the calculations.
The Lorentz factor, denoted by γ, is a fundamental concept in special relativity that relates the time dilation and length contraction effects observed when objects move relative to each other at speeds approaching the speed of light. It is given by the equation:
γ = 1 / sqrt(1 - (v^2 / c^2))
In the case of GPS satellites, they are moving at high velocities relative to the Earth's surface (around 14,000 kilometers per hour). Therefore, the time dilation effect due to their motion needs to be taken into account to ensure accurate positioning.
When calculating the time dilation correction for the satellites, the first-order approximation of the Lorentz factor is used, which neglects the higher-order terms. Mathematically, this approximation is expressed as:
γ ≈ 1 + (v^2 / (2c^2))
The reason for using this approximation is that the higher-order terms, such as (v^4 / (8c^4)), are several orders of magnitude smaller and have a negligible impact on the overall calculation. Ignoring these higher-order terms simplifies the calculations while still providing accurate results for the correction needed.
In practical terms, using the full binomial expansion of the Lorentz factor would introduce only a minuscule improvement in accuracy while adding unnecessary complexity to the calculations. Hence, the simplified approximation is sufficient for the purpose of adjusting the GPS satellite system for the effects of special relativity.