Deriving Relativistic Mechanics Equation: \gamma

[amcavoy]

It's probably way beyond me at this point, but how was the following equation derived and where did the need come from?

$$\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$

I was reading a wikipedia article that mentioned it, but that was about it. There wasn't much of an explanation. Does anyone know of a good website that explains this?

Thanks for your help.

 

[εllipse]

In case you don't know what the Lorentz transformation is, it's:

$$x'=\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$t'=\frac{t-\frac{v}{c^2}x}{\sqrt{1-\frac{v^2}{c^2}}}$$

It is from the Lorentz transformation that we get $$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$, which allows the equations to be reduced to:

$$x'=\gamma(x-vt)$$

$$t'=\gamma(t-\frac{v}{c^2}x)$$

So, the value for $\gamma$ itself comes straight from the Lorentz transformation. There are actually quite a few ways to derive the Lorentz transformation. Einstein was the one who found the most simple and elegant way of deriving the Lorentz tranformation, showing that the equations come straight from two simple postulates: (1) All inertial frames are equivalent for the description of the laws of nature (the principle of relativity). (2) The speed of light is a constant in all inertial reference frames. A simple derivation, from Einstein himself, can be found here:

http://www.bartleby.com/173/a1.html

where did the need come from?

Just like there are multiple ways to derive the Lorentz transformation, which were realized about the same time, there were also multiple needs. The original need, that Lorentz came up with the Lorentz transformation for, was the Michelson-Morely experiment, which seemed to show that the speed of light is always the same no matter how fast you are moving. Lorentz suggested that maybe lengths contract in repsonse to motion through the ether, so that could be why the speed of light always appears the same. However, Einstein arrived at the Lorentz transformation from a different angle. He realized that Maxwell's theory of electromagnetism and Galilean mechanics didn't fit well together. Einstein didn't think it made sense for Maxwell's theory to be applied to any particular ether reference frame, so he sought the replacement of Galilean mechanics that would allow it to be applied to all reference frames, and ended up with the same equations Lorentz had found.

 

[amcavoy]

That's really interesting. I guess I will get to study that further in future physics classes. Thanks a lot for the long post and information, it really helped.

Alex

 

[Perspicacious]

How was the following equation derived and where did the need come from?

$$\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$

The simplest derivation is given here:
http://www.pa.msu.edu/courses/2000spring/PHY232/lectures/relativity/dilation.html

The most elegant derivation follows these lines:
http://www.everythingimportant.org/relativity/
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000043000005000434000001
http://arxiv.org/PS_cache/physics/pdf/0302/0302045.pdf

 

[pmb_phy]

It's probably way beyond me at this point, but how was the following equation derived and where did the need come from?

$$\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$

I was reading a wikipedia article that mentioned it, but that was about it. There wasn't much of an explanation. Does anyone know of a good website that explains this?

Thanks for your help.

That is not exactly an equation. That is a definition of the symbol $\gamma$. To see where this comes up you can see the following

Relativistic Kinematics -
http://www.geocities.com/physics_world/sr/light_clock.htm
http://www.geocities.com/physics_world/sr/lorentz_contraction.htm
http://www.geocities.com/physics_world/sr/lorentz_trans.htm

Relativistic Dynamics -
http://www.geocities.com/physics_world/sr/inertial_mass.htm

Pete

 

[HallsofIvy]

This was just covered, a couple of days ago, here:
https://www.physicsforums.com/showthread.php?t=85978