How was the Lorentz factor derived?

In summary, the Lorentz factor is derived from the assumption that the speed of light is the same in two frames moving at a constant velocity relative to one another, along with the ratios of length and time measurements between the two frames. This can be found in Appendix A of Einstein's book "Relativity: The Special and General Theory", or through other methods such as the one provided in the link provided.
  • #1
Psyguy22
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So I know about the lorentz factor and how it describes time dialition, mass increasing etc.. but I was wondering how it was derived in the first place?
 
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  • #2
Psyguy22 said:
So I know about the lorentz factor and how it describes time dialition, mass increasing etc.. but I was wondering how it was derived in the first place?

You start with the assumption that the speed of light will be the same in two frames moving at a constant velocity relative to one another (this is a consequence of Einstein's two postulates).

Then if the coordinates in one frame are (x, t) and the coordinates of the other frame are (x', t'), the path of two flashes of light, one moving to the left and the other to the right will be:

x = ct (right-moving)
x = -ct (left-moving)

in the (x ,t) frame and

x' = ct' (right-moving)
x' = -ct' (left-moving)

in the (x', t') frame. We also know that (0,0) in the unprimed frame is (-vt',t') in the primed frame, (0,0) in the primed frame is (vt,t) in the unprimed frame, and that the ratio of the length of an object in the unprimed frame to its measured length in the primed frame must be equal to the ratio of the length of an object in the primed frame to its measured length in the unprimed frame.

From there, it's just algebra to find the relationship between x' and t' as functions of x and t, and vice versa. Time dilation, length contraction, and the Lorentz factor fall out of these relationships when you transform the times and places of two clock ticks at the same place and the two ends of a moving rod at the same time, from one frame to the other.

The complete derivation (and as I said, it is just algebra) is in Appendix A of Enstein's book "Relativity: The Special and General Theory" which is readily available online, for example at http://www.bartleby.com/173/
 
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  • #3
There are many ways of deriving it. Nugatory has given one. Here is another, completely different one: http://arxiv.org/abs/physics/0302045
 

What is the Lorentz factor?

The Lorentz factor is a term used in special relativity to describe the relationship between an object's velocity and its observed mass and length. It is represented by the symbol γ (gamma) and is calculated as 1/√(1-(v/c)²), where v is the object's velocity and c is the speed of light.

How is the Lorentz factor derived?

The Lorentz factor is derived through the equations of special relativity, which describe how the laws of physics change for objects moving at different velocities. By considering the effects of time dilation and length contraction, the Lorentz factor is derived as a way to reconcile these changes with the principle of conservation of mass and energy.

Why is the Lorentz factor important?

The Lorentz factor is important because it allows us to understand how the properties of an object (such as mass and length) change at high velocities. It also plays a crucial role in many equations and theories of special relativity, including the famous equation E=mc².

What is the significance of the Lorentz factor in the theory of relativity?

The Lorentz factor is significant in the theory of relativity because it helps us understand the effects of time and space on objects moving at high velocities. It also provides a mathematical framework for reconciling the seemingly contradictory principles of conservation of mass and energy with the observed changes in an object's properties at high velocities.

How does the Lorentz factor affect our understanding of the universe?

The Lorentz factor has greatly impacted our understanding of the universe by providing a way to explain and predict the behavior of objects at high velocities. It has also led to the development of theories such as special and general relativity, which have revolutionized our understanding of space, time, and gravity.

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