Show the math for lorentz boost functions when a space ship travels with 0.6 c velocity

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In special relativity, the Lorentz boost functions describe how spacetime coordinates transform between reference frames moving at constant velocity relative to each other. Let's consider a spaceship traveling at 0.6 times the speed of light. First, we need to calculate the Lorentz factor gamma. The formula for gamma is 1 divided by the square root of 1 minus v squared over c squared. Substituting our velocity of 0.6c, we get 1 divided by the square root of 1 minus 0.36, which equals 1 divided by the square root of 0.64, or 1 divided by 0.8. This gives us a Lorentz factor gamma of 1.25. The Lorentz transformation equations show how coordinates transform between frames. Time and position in the moving frame are related to the stationary frame by these equations, where gamma equals 1.25 for our spaceship moving at 0.6c. Now, let's substitute our calculated Lorentz factor gamma equals 1.25 and velocity v equals 0.6c into the Lorentz transformation equations. For the time coordinate, t-prime equals gamma times t minus v times x over c squared. Substituting our values, we get t-prime equals 1.25 times t minus 0.6x over c. For the spatial coordinate, x-prime equals gamma times x minus v times t. Substituting, we get x-prime equals 1.25 times x minus 0.6c times t. The y and z coordinates remain unchanged. These equations show how spacetime coordinates transform between the stationary frame and the spaceship's frame moving at 0.6c. In the spacetime diagram, we can see how the coordinate axes are tilted in the moving frame. The light cone remains invariant, showing that the speed of light is the same in all reference frames, which is a fundamental principle of special relativity. At a velocity of 0.6 times the speed of light, relativistic effects become significant. Let's examine two key effects: time dilation and length contraction. Time dilation means that time passes more slowly for moving objects. With our Lorentz factor gamma equal to 1.25, the formula shows that delta t-prime equals gamma times delta t. This means that 1 hour on Earth would be measured as 1.25 hours on the spaceship. Length contraction is the opposite effect - objects appear shorter in the direction of motion. The formula is L-prime equals L divided by gamma. With gamma equal to 1.25, a 1-meter object on Earth would measure only 0.8 meters when viewed from the spaceship. Another important relativistic effect is velocity addition. In classical physics, velocities simply add, but in special relativity, they combine according to this formula. For example, if an object moves at 0.5c relative to the ship, which is moving at 0.6c relative to Earth, the object's velocity relative to Earth is not 1.1c, but approximately negative 0.143c. This ensures that no object can exceed the speed of light, preserving the fundamental principle of special relativity. The Lorentz transformation can also be expressed in matrix form, which is particularly useful for calculations. For a boost along the x-axis with velocity 0.6c, the Lorentz transformation matrix has gamma in the top-left and bottom-right positions, and negative gamma times beta in the off-diagonal elements. Substituting our values with gamma equals 1.25 and beta equals 0.6, we get this matrix. After simplification, we have the Lorentz boost matrix with 1.25 on the diagonal and negative 0.75 in the off-diagonal elements. To transform a four-vector, we multiply it by this matrix. For example, if we have an event with coordinates t equals 2 and x equals 1, the transformed coordinates are t-prime equals 1.75 and x-prime equals negative 0.25. The Lorentz transformation has several important properties. It preserves the spacetime interval, which is a fundamental invariant in special relativity. For our example, the interval s-squared equals negative 3 in both reference frames. Lorentz transformations also form a mathematical group, preserve causality, and maintain the invariance of the speed of light in all reference frames. Let's summarize what we've learned about Lorentz boost functions for a spaceship traveling at 0.6 times the speed of light. First, we calculated the Lorentz factor gamma to be 1.25 using the formula 1 divided by the square root of 1 minus v-squared over c-squared. The Lorentz transformations mix space and time coordinates, with t-prime equal to gamma times t minus v times x over c-squared, and x-prime equal to gamma times x minus v times t. These transformations lead to several relativistic effects. Time dilation means that moving clocks run slower by a factor of gamma, so 1 hour in the stationary frame becomes 1.25 hours in the spaceship frame. Length contraction means that moving objects appear shorter in the direction of motion by a factor of 1 over gamma, so a 1-meter object appears as 0.8 meters. Relativistic velocity addition ensures that no object can exceed the speed of light, preserving the fundamental principle of special relativity. These effects become increasingly significant as velocity approaches the speed of light, and they're essential for understanding the behavior of particles in accelerators, cosmic rays, and other high-energy phenomena in our universe.