Lorentz Transformation: How to Calculate Event Coordinates in Different Inertial Frames

Lorentz Transformation is a crucial concept in relativity. It is used to relate the coordinates of an event in one inertial frame to the coordinates in another inertial frame that is moving at a constant velocity relative to the first. This article will guide you through the process of using Lorentz transformation to find the coordinates of a particular event in one frame (Frame S) in another frame (Frame S').

Understanding Lorentz Transformation Equations

The Lorentz transformation equations are essential for this calculation. These equations connect the coordinates (x, t) in Frame S to the coordinates (x', t') in Frame S' with the following formulas:

x' γ(x - vt)

t' γ(t - (vx/c2))

In these equations, the symbol γ (gamma) represents the Lorentz factor, which is a function of the relative velocity between the two frames. The Lorentz factor is given by:

γ 1 / √(1 - (v2/c2))

Where:

c is the speed of light

v is the relative velocity between the two frames

γ is the Lorentz factor

Steps to Transform Coordinates Using Lorentz Transformation

Step 1: Identify the Event Coordinates

Begin with the coordinates (x, t) of the event in Frame S.

Step 2: Determine the Relative Velocity

Identify the velocity v of Frame S relative to Frame S'.

Step 3: Calculate the Lorentz Factor

Compute the Lorentz factor γ using the relative velocity v.

Step 4: Apply the Lorentz Transformation Equations

Substitute the values of x, t, and v into the Lorentz transformation equations for x' and t'. This will give you the transformed coordinates (x', t') in Frame S'.

Step 5: Obtain the Transformed Coordinates

The resulting (x', t') will be the coordinates of the event in Frame S'.

Example Calculation

Suppose an event occurs at coordinates (x, t) (100 m, 2 s) in Frame S, and Frame S is moving at v 0.8c.

Calculate the Lorentz Factor

First, we need to compute the Lorentz factor γ:

γ 1 / √(1 - (0.82/c2)) 1 / √(1 - 0.64) 1 / √(0.36) 5/3

Apply the Lorentz Transformations

Now, plug in the values into the Lorentz transformation equations:

x' γ(x - vt) (5/3)(100 m - 0.8c × 2 s)

Assuming c ≈ 3 × 108 m/s, the calculation proceeds as follows:

x' (5/3)(100 m - 0.8 × 3 × 108 m/s × 2 s) (5/3)(100 m - 480 × 106 m) (5/3)(-479,999,900 m)

t' γ(t - (vx/c2)) (5/3)(2 s - (0.8c × 100 m)/(3 × 108 m/s2))

Calculating these will give you the transformed coordinates (x', t') in Frame S'. You can plug in the values for c and perform the arithmetic to find the exact numerical results.

Conclusion

Lorentz transformation is a powerful tool in relativity that helps us understand and calculate the coordinates of an event in different inertial frames. By following the steps outlined in this article and understanding the Lorentz transformation equations, you can accurately determine the transformed coordinates of events in various reference frames.