Gamma Explained: Reasons for Lorentz Transformation

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In summary, gamma, or the Lorentz factor, is necessary in the Lorentz transformation to account for relativistic effects such as time dilation and length contraction. This is because the constancy of the speed of light in all reference frames requires a different understanding of time and length measurements. The Lorentz transformation helps translate between these different measurements by incorporating the concept of proper time and length, which are relative to an object's rest frame. This is necessary in order to satisfy the conditions of linearity and isotropy in the transformation equations.
  • #1
actionintegral
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Does anyone have a brief explanation of why the gamma is necessary in the Lorentz transformation.
 
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  • #2
gamma?

actionintegral said:
Does anyone have a brief explanation of why the gamma is necessary in the Lorentz transformation.
in order to account for relativistic effects like time dilation, doppler shift...
sine ira et studio
 
  • #3
But the constancy of the speed of light is preserved simply by
x'=x-vt and t'=t-vx/cc. I'm trying to see where the scaling factor comes in.
 
  • #4
actionintegral said:
But the constancy of the speed of light is preserved simply by
x'=x-vt and t'=t-vx/cc. I'm trying to see where the scaling factor comes in.

Comes from preserving dx^2 -( c dt )^2
 
  • #5
actionintegral said:
Does anyone have a brief explanation of why the gamma is necessary in the Lorentz transformation.
Gamma, or sometimes called the Lorentz factor, represents the relationship between relativistic time and proper time or relativistic length and proper length.

Proper time and proper length are resp. the time and the length of an object in its restframe while relativistic time and relativistic length are resp. the time and length of an object that is moving relative to the frame of the observer.

The reason we need gamma to translate lengths and times from one frame to another is that the speed of light is always the same independent of motion. So, distance and duration cannot longer be absolute but instead depend on relative motion.
 
  • #6
nakurusil said:
Comes from preserving dx^2 -( c dt )^2

Fair enough - but why would I want to preserve THAT?
 
  • #7
gamma@

actionintegral said:
But the constancy of the speed of light is preserved simply by
x'=x-vt and t'=t-vx/cc. I'm trying to see where the scaling factor comes in.
you are perfectly right. in the paper

arXiv.org > physics > physics/0409121

Physics, abstract
physics/0409121


Three levels of understanding physical relativity: Galileo's relativity, Up-to-date Galileo's relativity and Einstein's relativity: A historical survey
Authors: Bernhard Rothenstein, Corina Nafornita
Comments: 10 pages, 4 figures
Subj-class: Physics Education

We present a way of teaching Einstein's special relativity. It starts with Galileo's relativity, the learners know from previous lectures. The lecture underlines that we can have three transformation equations for the space-time coordinates of the same event, which lead to absolute clock readings, time intervals and lengths (Galileo's relativity), to absolute clock readings but to relative time intervals and lengths (up-to-date Galileo transformations) and to relative clock readings time intervals and lengths.
we have called the equations you mention "uptodate" Galileo transformations. Please have a critical look at it
 
  • #8
actionintegral said:
Fair enough - but why would I want to preserve THAT?

Because it represents the line element.
 
  • #9
actionintegral said:
nakurusil said:
Comes from preserving dx^2 -( c dt )^2

Fair enough - but why would I want to preserve THAT?

There are experimental results [as well as theoretical/mathematical results] that effectively lead to this requirement.
 
  • #10
actionintegral said:
But the constancy of the speed of light is preserved simply by
x'=x-vt and t'=t-vx/cc. I'm trying to see where the scaling factor comes in.

For special relativity, the set of transformations one considers has to satisfy some conditions, including:
they are linear [lines map to lines] and they form a group [which preserves the metric]. [I don't think your transformations form a group.]

Especially for a theory of relativity, the transformations cannot have a timelike eigenvector [i.e. no distinguished observer].
 
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  • #11
The equations

x1 = x - v*t
t1 = t - (v/c^2) x

do have an inverse, so they will form a group. (A mapping must be associative and have an inverse to be a group, and IIRC associativity is a general property of any mapping so it shouldn't be an issue).

But the form of the inverse reveals why it's not the right choice:

t = (t1 + (v/c^2) x1) / (1 - v^2/c^2)
x = (x1 + v*t) / (1 - v^2/c^2)

We _also_ want the transforms to have the property that the inverse is generated just by changing the sign of v. This is justified by requiring that the transforms be isotropic. Isotropy is the missing element in the proposal.
 
  • #12
pervect said:
We _also_ want the transforms to have the property that the inverse is generated just by changing the sign of v.

Thanks, perv. o:) I will investigate the lack of symmetry in the inverse.
 
  • #13
actionintegral said:
Does anyone have a brief explanation of why the gamma is necessary in the Lorentz transformation.

Because people don't want to keep writing [tex]\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}[/tex]. :-p

Sorry... :biggrin:
 
  • #14
If people wrote [tex]\cosh\theta[/tex] instead of [tex]\gamma[/tex] or [tex] \frac{1}{\sqrt{1-v^2/c^2 } }[/tex], these mysterious quantities might not seem as mysterious.
 

Related to Gamma Explained: Reasons for Lorentz Transformation

1. What is Gamma in the context of Lorentz transformation?

Gamma (γ) is a mathematical factor used in the Lorentz transformation equations to account for the effects of time dilation and length contraction in special relativity. It is often referred to as the "relativistic factor" or "dilation factor."

2. How is Gamma calculated?

Gamma is calculated using the formula γ = 1/√(1 - v^2/c^2), where v is the relative velocity between two reference frames and c is the speed of light. This formula is derived from the principles of special relativity and is used to adjust measurements of time and space between two frames of reference moving at different velocities.

3. Why is Gamma important in Lorentz transformation?

Gamma is important in Lorentz transformation because it allows us to accurately describe the effects of time dilation and length contraction on objects moving at high velocities. It is a crucial component in the equations that enable us to reconcile the differences in measurements between different reference frames in special relativity.

4. What is the relationship between Gamma and the speed of light?

Gamma and the speed of light (c) are directly related through the Lorentz transformation formula. As the relative velocity between two frames approaches the speed of light, the value of Gamma approaches infinity, meaning that time dilation and length contraction become more pronounced. This is a fundamental concept in special relativity.

5. Can Gamma be greater than 1?

Yes, Gamma can be greater than 1. In fact, when an object is moving at a significant fraction of the speed of light (v/c), Gamma will always be greater than 1. This indicates that time will appear to pass slower and lengths will appear shorter in the moving frame of reference compared to a stationary frame, as observed in the famous "twin paradox" thought experiment.

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