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mahapan
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Hello there, a simple special relativity question, how can the Time Dilation and Length Contraction Formulas be derived from the Lorentz Transformation Equations?
Derive them for yourself. Here are a few hints:mahapan said:Hello there, a simple special relativity question, how can the Time Dilation and Length Contraction Formulas be derived from the Lorentz Transformation Equations?
Actually, given the hints and the fact that we know the answer that we're looking for, it should be relatively easy. After all, you're starting with the Lorentz transformations already known. (And in case you can't find them, here they are: https://www.physicsforums.com/showpost.php?p=905669&postcount=3")Naty1 said:I'm not sure doc al's suggestion is realistic for most. yet not impossible either...
mahapan said:Hello there, a simple special relativity question, how can the Time Dilation and Length Contraction Formulas be derived from the Lorentz Transformation Equations?
I would certainly agree, and that's exactly what's done in many elementary treatments of SR. (After first deriving length contraction, time dilation, and clock desynchronization from the more basic assumptions of SR via simple thought experiments.)bernhard.rothenstein said:I think that it is worth to know that the Lorentz transformations could be derived from length contraction and time dilation.
mahapan said:Hello there, a simple special relativity question, how can the Time Dilation and Length Contraction Formulas be derived from the Lorentz Transformation Equations?
The Lorentz Transformation is a mathematical formula used in special relativity to describe how measurements of time and space are affected by the movement of objects at high speeds. It is derived from Einstein's theory of special relativity and is used to explain the concept of time dilation and length contraction.
Time dilation is the phenomenon in which time appears to pass slower for an object in motion relative to an observer. This occurs because the speed of light is constant and absolute, so time must adjust in order for the laws of physics to remain consistent. Essentially, the faster an object moves, the slower time moves for that object.
Length contraction is the counterpart to time dilation, meaning that as time appears to slow down for an object in motion, its length appears to decrease in the direction of motion. This is due to the fact that an object's velocity through space and time are interconnected, and as one changes, so does the other.
The formula for time dilation is t = t0/√(1 - v2/c2), where t is the time measured by an observer, t0 is the time measured by an object in motion, v is the velocity of the object, and c is the speed of light. The formula for length contraction is l = l0√(1 - v2/c2), where l is the length measured by an observer, l0 is the length measured by an object in motion, v is the velocity of the object, and c is the speed of light.
One famous example of time dilation is the "twin paradox," in which one twin travels into space at high speeds and returns home to find that they have aged much slower than their twin who stayed on Earth. This is due to the difference in their velocities and the resulting time dilation. Another example is the use of atomic clocks on GPS satellites, which must account for time dilation in order to accurately measure distances on Earth. As for length contraction, a common example is the apparent shortening of a train when viewed from a stationary platform as it passes by at high speeds. This is due to the train's velocity causing length contraction in the direction of motion.