What is the significance of the Gamma-Relativity connection in modern physics?

[Wizardsblade]

Hey, I was playing around with Gamma and plotted it as shown in the attachments. I noticed that the area swept out was similar to phi (in spherical coordinates.) So I started wondering if this could have any really relevance. So I looked at it more closely and was able to find length contraction and time dilation. I also noticed that if I were to move the axis to the second frame I would again get the correct dilation/contractions, but this did not sit well with me because each was plotted on different lines, what should be a D space(t,r,theta,phi). So after thinking about this a bit more I came to the conclusion that space must be D (t,r,theta,phi,gamma) but we can only perceive D (t,r,theta,phi), much like watching TV, 4 dimensions(t,x,y,z) are captured but only 3(t,x,y) are shown on the TV.

So my question is this.. Knowing that I am surly not the first to notice this, what is the name for this theory, and does it have any "problems?" I hope not because looking at relativity this way is finally making relativity click for me.

 

[Wizardsblade]

I have done a few google searches and have not found anything that talks about this. But I have been playing around with it some more and it looks like when you integrate the "line" over a plane you get something very close to space-time fabric with craters for gravity.

 

[yuiop]

Hi,

The geometrical property you describe has been used for a long time to draw Minkowski spacetime diagrams showing the rotated lines of simultaneity etc using a compass a ruler.

For example look at step 4 of this tutorial to construct a Lorentz transformation diagram.

http://casa.colorado.edu/~ajsh/sr/construction.html

I guess some people like to view relativity as a projection from a higher dimensional space that we cannot see directly to the 3 spatial and one time dimension we are used to. From this point of view, a length contracted rod is a rod rotated into a higher dimension and we only see the contracted "shadow" of the rod projected into the dimensions we directly percieve. The results are mathematically the same and viewing things that way helps some people to visualise relativity. As you can see in the link I posted the time axis and length axis of the moving frame are rotated relative to the stationary frame so that the time and space axes are no longer at the normal right angles to each other.

 

[shalayka]

Hi Wizardsblade, the relationship you've found is also related to the Pythagorean theorem.

To visualize, start with an isosceles triangle that has a base of practically 0 length, and sides of length 0.5c. As you increase the base from 0 to c in length, the height of the triangle reduces from 0.5c to 0. The Pythagorean theorem relates these by the equation: $$height^2 = base^2 + side^2$$.

I'm sure you're familiar with Einstein's light clock thought experiment. Effectively, the light signal's path is equivalent to the lines traced out by the sides of the isosceles triangle. Where the mirrors and the observer are at relative velocity v = 0, the path is straight "up and down", the base is 0 in length, and each round-trip "tick" of the clock takes 1 second. At relative velocity v = c, the base is c in length, the height is 0, and each "tick" takes an infinite amount of time. This is basically kinematic time dilation in its simplest form: When the light clock and the photon are moving "sideways" at v = c, the photon has no "up and down" movement. To contrast: In Newton's world, the photon would still move "up and down" at one tick per second, regardless of the "sideways" velocity.

If you are more trigonometry minded, it may also help to note that:

$$\sqrt{1 - \frac{v^2}{c^2}} = \cos[\arcsin\left(\frac{v}{c}\right)]$$

 

[A.T.]

So after thinking about this a bit more I came to the conclusion that space must be D (t,r,theta,phi,gamma) but we can only perceive D (t,r,theta,phi), much like watching TV, 4 dimensions(t,x,y,z) are captured but only 3(t,x,y) are shown on the TV.

Length contraction can be interpreted as projection of the proper length onto the space dimensions. An time dilation comes from projecting coordinate time onto the proper time dimension:
http://www.adamtoons.de/physics/relativity.swf

But I have been playing around with it some more and it looks like when you integrate the "line" over a plane you get something very close to space-time fabric with craters for gravity.

With gravitation the above diagram looks like this:
http://www.adamtoons.de/physics/gravitation.swf

 

[peter0302]

Indeed. The major problem with this model is that it makes different predictions than normal SR for addition of velocities.

 

[yuiop]

Indeed. The major problem with this model is that it makes different predictions than normal SR for addition of velocities.

Could you elaborate on that to assist my understanding?

 

[peter0302]

If you model the world in 5 orthogonal, Euclidean dimensions, with the fifth being "proper time" and the fourth being "observed time", indeed you get the Lorentz transformations. But, normal trigonometry (rather than hyperbolic trigonometry) applies in Euclidean space. So velocities add like normal tangents instead of tanh functions, and what happens to the tangent after you pass 90 degrees (the speed of light)? You start decreasing.

 

[yuiop]

If you model the world in 5 orthogonal, Euclidean dimensions, with the fifth being "proper time" and the fourth being "observed time", indeed you get the Lorentz transformations. But, normal trigonometry (rather than hyperbolic trigonometry) applies in Euclidean space. So velocities add like normal tangents instead of tanh functions, and what happens to the tangent after you pass 90 degrees (the speed of light)? You start decreasing.

I see..

 

[Wizardsblade]

If you model the world in 5 orthogonal, Euclidean dimensions, with the fifth being "proper time" and the fourth being "observed time", indeed you get the Lorentz transformations. But, normal trigonometry (rather than hyperbolic trigonometry) applies in Euclidean space. So velocities add like normal tangents instead of tanh functions, and what happens to the tangent after you pass 90 degrees (the speed of light)? You start decreasing.

I do not see how proper and observed time can be orthognal since proper time can also be observed time.

 

[A.T.]

If you model the world in 5 orthogonal, Euclidean dimensions, with the fifth being "proper time" and the fourth being "observed time", indeed you get the Lorentz transformations.

With "observed time" you mean "observer's time" or "coordinate time"? In the model above this is not a 5th dimension orthogonal to proper time and space. It is the length every observed world line.