To integrate the Lorentz factor (γ) over a path between two points, where the velocity for the Lorentz factor is changing, you need to consider the velocity as a function of time along the path.
The Lorentz factor (γ) is given by the equation:
γ = 1 / sqrt(1 - (v^2 / c^2))
Where:
- v is the velocity of the object
- c is the speed of light
If the velocity is changing along the path, you'll need to express it as a function of time. Let's denote the velocity as v(t). Then, the Lorentz factor becomes a function of time: γ(t).
To integrate the Lorentz factor over the path, you need to determine the time interval over which the velocity is changing. Let's denote the initial time as t1 and the final time as t2.
The integral of the Lorentz factor over the path between the two points can be expressed as:
Integral of γ(t) dt from t1 to t2
Mathematically, you would evaluate this integral by finding the antiderivative of γ(t) and then applying the limits of integration.
However, it's important to note that integrating the Lorentz factor can be a complex task, especially if the velocity is a complicated function of time. In practice, the integration may require numerical methods or approximation techniques depending on the specific scenario.