PDA

View Full Version : Time Delation and Unit Circle

Relative
2012-Apr-13, 02:00 PM
If we look at the factor (y) time is delayed as a function of the factor (x) of speed of light, we find a (unit) circle function, likewise Pythagoras:

y := sqrt(1-x^2).

Example: if we travel at 0.6 c, time is delayed by factor 0.8, since 0.8 = sqrt(1-0.36) or written as Pythagoras 0.6^2+0.8^2 = 1.

I just wonder: If the radius of this circle for any velocity is always 1, speed could be seen as just an „angle“ to either (unitless) time or speed of light.

If the angle is 90 degrees, we do not move, and experience the radius 1 along y-axis as „time passing by“.
If the angle is zero, time stands still and we may experience the radius 1 along x-axis as „location passing by“.

So, the Lorentz factor says that this „radius“ always remains: Are (unitless) time and c representing the same value then?

HenrikOlsen
2012-Apr-13, 03:11 PM
Scaling space-time so space is in units of c*s and time is in s, the distance between events becomes sqrt(x^2+y^2+z^2-t^2) and coordinate transformations due to relative velocity becomes pure rotations.

Relative
2012-Apr-13, 03:32 PM
Thanks,
but have I been scaling anything (in particular to arbitrary (s) units?)

HenrikOlsen
2012-Apr-14, 11:33 AM
You scaled velocity to go from 0-1, so yes, you did. (Ok, you scaled time by an unstated unit (which can be 1s for convenience) and scaled distance by c times that unit. It works regardless of which unit you use for time as it was really space you scaled)

Relative
2012-Apr-14, 02:38 PM
You scaled velocity to go from 0-1, so yes, you did. (Ok, you scaled time by an unstated unit (which can be 1s for convenience) and scaled distance by c times that unit. It works regardless of which unit you use for time as it was really space you scaled)
I guess, I See what you mean.
So if you prefer you can write c instead of 1, where the graph intersects x-axis and T instead of 1 where it intersects y-axis. (The last question of OP remains...)

Rough Edges
2012-Apr-15, 09:40 AM
Scaling space-time so space is in units of c*s and time is in s, the distance between events becomes sqrt(x^2+y^2+z^2-t^2) and coordinate transformations due to relative velocity becomes pure rotations.

Can you elaborate on what you mean by "pure rotations"? This can't be a rotation in the simple Euclidean sense. The x^2+y^2+z^2-t^2 quantity is invariant to change of frame, but setting it equal to a constant describes something which has some cross-sections whcih are hyperbolic, not elliptical. So it is not invariant (for example) a rotation in the x-t plane.

Maybe you are referring to some more general kind of rotation?

Relative
2012-Apr-18, 09:10 PM
So, just before this thread fizzles out:
I wonder why there has been such little reaction. Is this silent acknowneldment or speechless astonishment or – neither of which?
The graph of the Lorentz factor has often been discussed, likewise often sentences as „it is negligible for common speeds, but it increases DRAMATICALLY when one gets close to the speed of light“ have occured, so this function has been judged somewhat STRANGE.
But if you don‘t know what to do with, for example, the graph of a hyperbola, you might find even this one „strange“, although – or because – it is just the reciprocal of a simple straight line.
Same with the graph of the Lorentz factor. Its reciprocal is JUST a CIRCLE, so not "strange" at all and the only function graph you can draw with a compass, by the way!
So, the „span of the compass“ (i.e. the radius) is what makes this circle. It remains constant for the entire graph. The question must be allowed then, what the „meaning“ of this radius is. From the Lorentz factor we learn that this is a.) the speed of light or b.) the physical quantity Time.
So, is there anybody out there to recognize or have an idea of the possibilities (just think about it a little...) of a conclusion that time and speed of light may represent the same thing or, even more, ARE the same?

Relative
2012-Apr-20, 08:32 PM
http://www.phys.unsw.edu.au/einsteinlight/jw/module4_time_dilation.htm#time

Section "Pythagoras meets Einstein".

caveman1917
2012-Apr-20, 09:43 PM
Can you elaborate on what you mean by "pure rotations"? This can't be a rotation in the simple Euclidean sense. The x^2+y^2+z^2-t^2 quantity is invariant to change of frame, but setting it equal to a constant describes something which has some cross-sections whcih are hyperbolic, not elliptical. So it is not invariant (for example) a rotation in the x-t plane.

Maybe you are referring to some more general kind of rotation?

It's a rotation in the complex plane, with time on the imaginary axis, which when you square i gives you the minus sign.

HenrikOlsen
2012-Apr-21, 07:27 AM
By pure I meant no translation, no stretching, just rotation.