Deriving the speed of light from the Pythagorean Theorem

[frankinstein]

Has anyone heard of the speed of light being derived from the Pythagorean Theorem? Obviously I'm referring to using the time dilation effect of motion.

Frank

 

[JesseM]

Has anyone heard of the speed of light being derived from the Pythagorean Theorem? Obviously I'm referring to using the time dilation effect of motion.

Frank

You may be thinking of the light clock, where in the clock's rest frame the light travels vertically up and down, but in another frame where the clock is moving, it travels on a slanted path that's the hypotenuse of a right triangle whose vertical side is the same height H as in the previous frame, and whose horizontal side has a length equal to the clock's velocity v times the time T it takes for the light to get from the bottom to top in this frame. Since the light must travel at c in all frames, the length of the hypotenuse must be equal to c*T, so by the Pythagorean theorem we have H² + (vT)² = (cT)², from this you can find the time T for the light to get from in this frame, which will be greater than in the clock's rest frame, where it the light just has to travel vertically a distance H so the time is H/c.

 

[frankinstein]

You may be thinking of the light clock, where in the clock's rest frame the light travels vertically up and down, but in another frame where the clock is moving, it travels on a slanted path that's the hypotenuse of a right triangle whose vertical side is the same height H as in the previous frame, and whose horizontal side has a length equal to the clock's velocity v times the time T it takes for the light to get from the bottom to top in this frame. Since the light must travel at c in all frames, the length of the hypotenuse must be equal to c*T, so by the Pythagorean theorem we have H² + (vT)² = (cT)², from this you can find the time T for the light to get from in this frame, which will be greater than in the clock's rest frame, where it the light just has to travel vertically a distance H so the time is H/c.

Yes I was thinking of the light clock, but wouldn't the path of anything in motion be lengthen, its doesn't have to be a photon, it could be a ping pong ball, or a bullet in a vaccum, the path distortion is dependent on the motion of the frame. If that's the case then to reach zero time we should derive the speed of light or something very close?

Frank

 

[JesseM]

Yes I was thinking of the light clock, but wouldn't the path of anything in motion be lengthen, its doesn't have to be a photon, it could be a ping pong ball, or a bullet in a vaccum, the path distortion is dependent on the motion of the frame. If that's the case then to reach zero time we should derive the speed of light or something very close?

The derivation depends on the assumption that light has a speed of c in both frames, which is one of the basic postulates of relativity--for slower-than-light objects like a ping pong ball there's no postulate demanding that the speed be the same in both frames. I don't understand what you mean by the phrase "to reach zero time"...

 

[frankinstein]

The derivation depends on the assumption that light has a speed of c in both frames, which is one of the basic postulates of relativity--for slower-than-light objects like a ping pong ball there's no postulate demanding that the speed be the same in both frames. I don't understand what you mean by the phrase "to reach zero time"...

Understood but if the speed is constant in both frames the time dilation remains because of the longer path formed by moving the frame.

What I mean't by "to reach zero time" is the lengthening of the path that ultimately stops the tick of the clock.

 

[GrayGhost]

Has anyone heard of the speed of light being derived from the Pythagorean Theorem? Obviously I'm referring to using the time dilation effect of motion.

By Pathagorus' theorem ...

c = vt/(t²-tau²)^1/2​

where vt = x

GrayGhost

 

[JesseM]

Understood but if the speed is constant in both frames the time dilation remains because of the longer path formed by moving the frame.

Right, that's why the argument concludes that time dilation must exist in relativity, but my point is you need light for the argument to work, not ping pong balls or other slower-than-light objects.

What I mean't by "to reach zero time" is the lengthening of the path that ultimately stops the tick of the clock.

It can't really stop the clock because it's impossible to accelerate an object with mass to exactly the speed of light, although you can get it moving at a speed arbitrarily close to c in your frame.

 

[JesseM]

By Pathagorus' theorem ...

c = vt/(t²-tau²)^1/2​

where vt = x

Did you get that just by solving the time dilation equation t = tau/sqrt(1 - v²/c²) for c, or did you use some other argument like the light clock? Just solving for c in the time dilation equation wouldn't require use of the Pythagorean theorem...

 

[frankinstein]

Right, that's why the argument concludes that time dilation must exist in relativity, but my point is you need light for the argument to work, not ping pong balls or other slower-than-light objects.

But the effect still happens without having a light clock. So could a constant c actually be caused by the geometry change or manifold expansion due to motion? When objects are moving much slower to c then the effects to the geometry are negligible but at c we reach the maximum distortion?

 

[GrayGhost]

Did you get that just by solving the time dilation equation t = tau/sqrt(1 - v²/c²) for c, or did you use some other argument like the light clock? Just solving for c in the time dilation equation wouldn't require use of the Pythagorean theorem...

JesseM,

By Pathagus' theorem ...

It's just from Einstein's kinematic model ... in a 2-space diagram of system K (ie. x,y,z) with time (t) implied, imagine the system k (ie X,Y,Z, time Tau implied) moving at inertial v along +x with x & X colinear and y & Y always parallel. The origin is marked by the momentary colocation of both system origins at t=Tau=0. Imagine a spherical EM pulse emitted from said origin, and a subsequent consideration of the intersection of said expanding EM sphere with the moving Y-axis in quadrant 1 (ie +x,+y). This interval represents the path of a single photon from origin to the moving Y-axis at some arbitrary time t. This lightpath is defined in system K by Pathagorus' theorem ...

(ct)² = (vt)²+y²​

where x=vt is the location of the system k origin in system K at time t. Since no length contractions exist wrt axes orthogonal to the direction of motion, then y=Y. Add, per system k, Y=cTau, so y=Y=cTau. So ...

(ct)² = (vt)²+y²
(ct)² = (vt)²+(cTau)²​

then solving for c, this becomes ...

c= vt/(t²-tau²)^1/2​

I just figured I'd present it in this way since the poster asked how light's speed could be derived using Pathagorus' theorm as a source of the derivation. There may be other ways, but this was the first one to come to mind :)

GrayGhost

 

[JesseM]

Add, per system k, Y=cTau, so y=Y=cTau.

Ah, gotcha, that was the step I didn't think of, that since the heights are equal you can substitute an expression involving proper time into the equation for the three sides of the triangle in the frame where the clock is moving. My method was just to solve for T in the equation H² + (vT)² = (cT)² and compare it to the proper time of H/c in the clock's rest frame, but yours works equally well.

By the way, the fellow's name was spelled "Pythagoras" ;)

 

[GrayGhost]

By the way, the fellow's name was spelled "Pythagoras" ;)

Yeh, I suppsoe I should honor fellows who have earned it by at least getting their names spelled right. The 2 that given me the most trouble are Pythagoras and Galileo. Galileo I usually put 2 l's, sometimes Gallileo and other times Galilleo. I occasionally look them up first to get the spelling right, but I always seem to forget in short order :( On the other hand, Einstein never gives me any trouble unless I'm typing too fast in 2 finger mode :)

GrayGhost

 

[GrayGhost]

JesseM,

By Pythagoras' theorem ...

It's just from Einstein's kinematic model ... in a 2-space diagram of system K (ie. x,y,z) with time (t) implied, imagine the system k (ie X,Y,Z, time Tau implied) moving at inertial v along +x with x & X colinear and y & Y always parallel. The origin is marked by the momentary colocation of both system origins at t=Tau=0. Imagine a spherical EM pulse emitted from said origin, and a subsequent consideration of the intersection of said expanding EM sphere with the moving Y-axis in quadrant 1 (ie +x,+y). This interval represents the path of a single photon from origin to the moving Y-axis at some arbitrary time t. This lightpath is defined in system K by Pathagorus' theorem ...

(ct)² = (vt)²+y²​

where x=vt is the location of the system k origin in system K at time t. Since no length contractions exist wrt axes orthogonal to the direction of motion, then y=Y. Add, per system k, Y=cTau, so y=Y=cTau. So ...

(ct)² = (vt)²+y²
(ct)² = (vt)²+(cTau)²​

then solving for c, this becomes ...

c= vt/(t²-tau²)^1/2​

This equation has other interesting inherent meanings when presented differently ...

c = v/(1-(tau/t)²)^1/2​

and so the ratio of velocities in terms of the ratio of times ...

v/c = (1-(tau/t)²)^1/2

(v/c)² = 1-(tau/t)²​

and if viewed in terms of the Pythagorean theorem ...

1² = (v/c)² + (tau/t)²

c² = (v/c)² + (tau/t)²​

which may be interpreted that velocity thru 4-space is c, and material motion the result of unparallel speed c vectors.

GrayGhost

 

[bcrowell]

But the effect still happens without having a light clock. So could a constant c actually be caused by the geometry change or manifold expansion due to motion?

The geometry of flat spacetime is the same geometry regardless of what frame of reference you use. In fact, you can even use arbitrary coordinates, and the geometry is still the same. GR defines curvature in a manner that is coordinate-independent.

When objects are moving much slower to c then the effects to the geometry are negligible but at c we reach the maximum distortion?

The distortion grows without bound as the velocity approaches c, so there is no maximum. Material objects can't move at c.

 

[frankinstein]

The distortion grows without bound as the velocity approaches c, so there is no maximum. Material objects can't move at c.

I get that, rest mass cannot reach the speed of light only get very close to it. But as I think about how time dilation happens it would seem the distortions caused by motion could be calculated as the limit of the distortions approaches infinity and the result of the frame's speed should be c?

Also the answers given by GrayGhost don't derive c from the Pythagoras theorem. He actually introduces c, which is not the same...

My real question however is; Has anyone derived c from the limit of the function for distortions caused by motion of a frame without introducing c?

Frank

 

[frankinstein]

For some reason Latex is not working on my browser so please bare with the awkward notation, thanks.

From Pythagoras:

A frame in motion has the following distortion, irrelevant of relativistic terms:

D = 2*sqrt(L^2 + (1/2*vt)^2)

v = sqrt(D^2 - 2*L^2)/t

So evaluating the limit of sqrt(D^2 - 2*L^2)/t, as D and t approach infinity the function should limit toward c?

If this wrong please explain, and if not how do I find the solution of the limits?

Thx in advance.

Frank

 

[frankinstein]

For some reason Latex is not working on my browser so please bare with the awkward notation, thanks.

From Pythagoras:

A frame in motion has the following distortion, irrelevant of relativistic terms:

D = 2*sqrt(L^2 + (1/2*vt)^2)

v = sqrt(D^2 - 2*L^2)/t

So evaluating the limit of sqrt(D^2 - 2*L^2)/t, as D and t approach infinity the function should limit toward c?

If this wrong please explain, and if not how do I find the solution of the limits?

Thx in advance.

Frank

Ultimately L is insignificant and its really the ratio of D/t, where
the rate at which D approaches infinity is different than the rate at which t approaches infinity and those rates remain relative to the velocity of the frame. So again; the function should limit towards c.

Is my conclusion wrong?

 

[GrayGhost]

My real question however is; Has anyone derived c from the limit of the function for distortions caused by motion of a frame without introducing c?

Here's the problem ... The LTs themselves define the distortions, and the LTs are derived using an invariant light speed of c as an apriori postulate. By obtaining c from the LTs, we'd just find ourselves back at the 2nd postulate.

GrayGhost

 

[bcrowell]

For some reason Latex is not working on my browser so please bare with the awkward notation, thanks.

Latex doesn't work properly in preview mode. You have to submit your post, and then the math will show up correctly. If you see a mistake, you can correct it then.

 

[JesseM]

From Pythagoras:

A frame in motion has the following distortion, irrelevant of relativistic terms:

D = 2*sqrt(L^2 + (1/2*vt)^2)

v = sqrt(D^2 - 2*L^2)/t

Can you explain what the terms are supposed to represent physically? Are you still dealing with a light-clock-like scenario where a signal travels vertically in the clock's rest frame, but along a diagonal in the frame where the clock is moving? If so, is v the velocity of the clock, rather than the velocity of the signal, while D/2 is the length of the diagonal path (just from bottom to top, while D is the length of two diagonals, from bottom to top and back to the bottom) and L is the vertical height? And t is the time for the signal to go from bottom to top and back to the bottom, so t/2 is the time for it to just go from bottom to top? If so that would make sense of D = 2*sqrt(L^2 + (1/2*vt)^2). Then for a single diagonal from bottom to top we'd have the horizontal side as v*(t/2), the vertical side as L, and the diagonal side as D/2. But that doesn't seem equivalent to your second equation, since that gives us:

v^2*t^2/4 + L^2 = D^2/4

which simplifies to

v^2 = 4/t^2 * (D^2/4 - L^2)
v = sqrt(D^2 - 4*L^2)/t

...which isn't quite what you had.

Anyway, with the modified equation for v above, both equations would be as true in relativity as they are in Newtonian mechanics. But I don't see why you think taking the limit as D approaches infinity should limit towards c, that just seems like a complete non sequitur since D does not approach infinity in relativity, and you can't derive the precise value of c from any classical limit, that's an empirical matter (after all we could imagine a variety of theories that had the same equations as relativity but different values of c).

 

[GrayGhost]

that gives us:

v^2*t^2/4 + L^2 = D^2/4

which simplifies to

v^2 = 4/t^2 * (D^2/4 - L^2)
v = sqrt(D^2 - 4*L^2)/t

...which isn't quite what you had.

Anyway, with the modified equation for v above, both equations would be as true in relativity as they are in Newtonian mechanics. But I don't see why you think taking the limit as D approaches infinity should limit towards c, that just seems like a complete non sequitur since D does not approach infinity in relativity, ...

Ahhh, I see.

However, I'm trying to figure out what you mean by the highlight here. I'm supposing that you'll say D = ct, because the photon travels with the emitter never departing it, yes? So D depends on the duration t considered, and the specific value of invariant c assumed. Yet, if D/2 is defined by the reflection event, then it must take an infinite amount of time t, and so when v=c then vt = D = infinity (given L is held constant), yes?

GrayGhost

 

[JesseM]

Ahhh, I see.

However, I'm trying to figure out what you mean by the highlight here. I'm supposing that you'll say D = ct, because the photon travels with the emitter never departing it, yes?

Yes, I would say D=ct...I assume by "never departing it" you mean that the photon always remains directly above the current position of the emitter, even as the emitter moves horizontally with velocity v, right?

So D depends on the duration t considered, and the specific value of invariant c assumed. Yet, if D/2 is defined by the reflection event, then it must take an infinite amount of time t

Why would it take an infinite amount of time? Perhaps you think that because of the next part...

and so when v=c

v is the horizontal velocity of the clock, not the velocity of the signal, so it wouldn't make sense to me to set v=c in relativity.

 

[GrayGhost]

Yes, I would say D=ct...I assume by "never departing it" you mean that the photon always remains directly above the current position of the emitter, even as the emitter moves horizontally with velocity v, right?

yes.

Why would it take an infinite amount of time? Perhaps you think that because of the next part...

v is the horizontal velocity of the clock, not the velocity of the signal, so it wouldn't make sense to me to set v=c in relativity.

Indeed, v cannot reach c. Yet, we are imagining what the math requires if v did magically reach c. When frankinstein says an infinite length is required for v=c, he's talking the limit as v approaches c. Assuming the reflection event must always bisect the roundtrip interval, L being held constant, then D/2 = (vt)/2 = infinity ... because the pathlengths have to be parallel (even though they cannot be). So it's the usual case whereby the model blows up at v=c. Thus, we have to assume an infinite duration t since the photon never moves away from the emitter over time, yet has to complete the photon's roundtrip interval. We make the assumption the photon does make the roundtrip per one at rest with the emitter/reflector, and consider the POV of a magic speed c observer.

I don't see any way of geometrically obtaining the value c from infinity. It's like an operation that results in an indefinite, the soln can be any value. The value of c that must arise from the geometry (and the related infinity) can only be what it was postulated at in the first place, so I don't imagine it could be derived geometrically (w/o any other considerations) in the way that Frankinstein would hope ... because it's an anomaly.

GrayGhost

 

[JesseM]

Indeed, v cannot reach c. Yet, we are imagining what the math requires if v did magically reach c.

OK, I thought that by considering the limit as D and t approached infinity, frankinstein wanted to find a value for some quantity that would match what relativity says when you consider the case where the signal's velocity is c, not the clock's. The equations frankinstein wrote (with the small correction I noted) are very general, and would apply equally well to either a light clock or a clock where the signal bouncing between top and bottom had some velocity other than c (and they'd work in either Newtonian physics or relativity). And c does have a role in relativity similar to infinite velocity in Newtonian physics, which was why I was thinking along these lines.

When frankinstein says an infinite length is required for v=c, he's talking the limit as v approaches c.

Yes, your interpretation does seem to make more sense of frankinstein's other comments, I had mainly been looking at the post where he actually wrote the equations, but elsewhere I see now he did talk about the idea of a frame having a speed of c. In any case I agree with you that there's not going to be any way of deriving c from this, since the signal will just move along with the emitter forever and not rise at all in this limit. And the second part of my original comment about this still applies:

you can't derive the precise value of c from any classical limit, that's an empirical matter (after all we could imagine a variety of theories that had the same equations as relativity but different values of c).

 

[frankinstein]

Yes, your interpretation does seem to make more sense of frankinstein's other comments, I had mainly been looking at the post where he actually wrote the equations, but elsewhere I see now he did talk about the idea of a frame having a speed of c. In any case I agree with you that there's not going to be any way of deriving c from this, since the signal will just move along with the emitter forever and not rise at all in this limit. And the second part of my original comment about this still applies:

Here's were I'm ultimately heading at:

a: acceleration
v: frame final velocity
v': frame initial velocity
D: frame distortion
t: time
fd: frame's traveled distance

a = (v-v')/t
v = v' + at

a = ((fd + D)/t - (fd' + D')/t')/t

v = v' + ((fd + D)/t - (fd' + D')/t')

Where the final velocity of the frame due to acceleration is affected by the frame's distortion since the acceleration rate is a type of clock. Ultimately the increases to final velocity due to acceleration tampers even in classical mechanics when the effects of time dilation are taken into account.

 

[frankinstein]

Here's were I'm ultimately heading at:

a: acceleration
v: frame final velocity
v': frame initial velocity
D: frame distortion
t: time
fd: frame's traveled distance

a = (v-v')/t
v = v' + at

a = ((fd + D)/t - (fd' + D')/t')/t

v = v' + ((fd + D)/t - (fd' + D')/t')

D = 2*sqrt(L^2 + (1/2*vt)^2)

a = ((fd - D)/t - v')/t

fd = vt

a = ((vt-D)/t - v')/t

v = v' + at-D

I listed the equation for distortion as a quick reference and simplified the other equations, also made a sign change to the distortion. And now its pretty obvious the distortion counters the velocity as an object accelerates.

 

[frankinstein]

D = 2*sqrt(L^2 + (1/2*vt)^2)

a = ((fd - D)/t - v')/t

fd = vt

a = ((vt-D)/t - v')/t

v = v' + at-D

I listed the equation for distortion as a quick reference and simplified the other equations, also made a sign change to the distortion. And now its pretty obvious the distortion counters the velocity as an object accelerates.

The first test if an answer is right is whether all the terms are in the same type of units, D is not in the correct unit type. After looking at the light clock example using acceleration as a clock changes a few things, where L is really the distance that (at) travels. The distortion is just the right triangle and not the two right triangles as in the light clock. I'll post my all the changes after I'm done.