C^2 in Lorentz transformations and special relativity.

[accidentprone]

c^2 occurs frequently in special relativity: in the Lorentz transformations, in forumlas for the interval, relativistic energy, and others too. Is there an intuitive reason for the high occurence of c^2?

 

[Pengwuino]

Well, I can't really say this means much, but you're looking at velocity squares, $$ds^2$$, etc etc, so the $$c^2$$ makes sense I suppose.

 

[JesseM]

Yeah, I think most or all of those equations involving c^2 can be derived from the equation for the invariant spacetime interval which can be written as $$d\tau^2 = \sqrt{dt^2 - (1/c^2)dx^2} = dt \sqrt{1 - v^2/c^2}$$, and which plays a role in spacetime analogous to the role of the invariant spatial distance in 2D space given by the Pythagorean theorem as $$\sqrt{dx^2 + dy^2}$$ (see [post=2972720]this post[/post] for more on the geometric analogy)

 

[robphy]

The short answer is "c^2 appears because of the choice of units (that distances and times are measured in different units)". JesseM's post gives more details.

 

[jambaugh]

If you were to measure heights in inches and lateral distances in feet, then work out the height and width of a rectangle containing a rotated yard stick in terms of its slope (in units of feet per inch) you'd get a bunch of c=12 in/ft factors in your formulas:

$$z = z_0 \cos(\theta) + c\cdot x_0\sin(\theta)$$

$$x = -c^{-1}\cdot z_0 \sin(\theta) +x_0\cos(\theta)$$

$$m = \frac{z}{x}= c\cdot \tan(\theta)$$

so
Letting
$$\gamma = \cos(\theta) = \frac{1}{\sqrt{1+\tan^2(\theta)}} =\frac{1}{\sqrt{1+(m/c)^2}}$$
and
$$\sin(\theta) = \cos(\theta)\tan(\theta)=\gamma m/c$$
you have:
$$z = \gamma (z_0 + m x_0)$$
$$x = \gamma (x_0 - (m/c^2)z_0)$$

Now replace (z,x) with (x,t) and the circle trig with hyperbolic trig...
[edit: and replace m with v ]
(above was typed in a rush, check for errors.)

 

[GrayGhost]

c^2 occurs frequently in special relativity: in the Lorentz transformations, in forumlas for the interval, relativistic energy, and others too. Is there an intuitive reason for the high occurence of c^2?

Good question. As JesseM pointed out already, it's due to the Pythagorean theorem. ...

See this post, and in particularly note the 1st equation which defines lengths in one system using the Pythagorean theorem. It also explains Einstein's kinematic model setup, which should help make all this clearer to you ...

https://www.physicsforums.com/showpost.php?p=3254542&postcount=8"​

Pythagoras' theorem is used because the lightpath (ct) is related to an observer moving along +x at v (so vt), and the systems are related by the vertical path y=Y which forms a right triangle. Since Pythagorus' theorem applies, c² arises.

HYP² = ADJ² + OPP²​

That 1st Eqn of the referenced link above, is this ...

(ct)² = (vt)²+y² <- 1st EQN​

so ...

y² = (ct)²-(vt)²
y² = t²(c²-v²)
y² = (ct)²(1-v²/c²)
y = ct(1-v²/c²)^1/2
y/c = t(1-v²/c²)^1/2​

However, since no length contractions exist wrt axes orthogonal to the direction of motion (motion is along x, x being colinear with X), then y=Y, Y being an axis of the moving system that appears always parallel to y, so ...

Y/c = t(1-v²/c²)^1/2​

But in the moving system, from its own POV as stationary, Y = cTau (since light is isotropic), so ...

cTau/c = t(1-v²/c²)^1/2
Tau = t(1-v²/c²)^1/2​

This equation relates the time readouts of 2 clocks that were once co-located prior at their origins (of systems x,y,z,t and X,Y,Z,Tau).

That said, the c² that pops up often in SR is the direct result of Pythagoras' theorem relating lengths of the 2 systems using a right traingle as seen in EQN 1 above. We often see (v/c)² as well, mainly because we like to reduce equations to their simplest form, or a form that presents the most inherent meaning at a glance ... eg Tau = t(1-(v/c)²)^1/2

GrayGhost

 

[robphy]

Good question. As JesseM pointed out already, it's due to the Pythagorean theorem. ...

...snip...

That said, the c² that pops up often in SR is the direct result of Pythagoras' theorem relating lengths of the 2 systems using a right traingle as seen in EQN 1 above. We often see (v/c)² as well, mainly because we like to reduce equations to their simplest form, or a form that presents the most inherent meaning at a glance ... eg Tau = t(1-(v/c)²)^1/2

GrayGhost

c^2 already pops up in (1+1)-Minowski spacetime... due the different choice of units (as mentioned earlier) and the square-interval posted by JesseM. Since the [Euclidean Space] Pythagorean Theorem doesn't play any role in (1+1)-Minkowski spacetime, it can't be the ultimate source of the c^2 in Special Relativity.

If there's a constant that arises from the [Euclidean] Pythagorean Theorem, it's pi.

 

[GrayGhost]

c^2 already pops up in (1+1)-Minowski spacetime... due the different choice of units (as mentioned earlier) and the square-interval posted by JesseM. Since the [Euclidean Space] Pythagorean Theorem doesn't play any role in (1+1)-Minkowski spacetime, it can't be the ultimate source of the c^2 in Special Relativity.

If there's a constant that arises from the [Euclidean] Pythagorean Theorem, it's pi.

The reason c² pops up in Minkowski spacetime, is for the reason I stated. It's all the same.

GrayGhost

 

[Mentz114]

c^2 occurs frequently in special relativity: in the Lorentz transformations, in forumlas for the interval, relativistic energy, and others too. Is there an intuitive reason for the high occurence of c^2?

Minkowski spacetime is defined by the line element

ds² = -c² dt² + dx² + dy² + dz²

c is converting units of time into units of distance.

 

[Naty1]

Maybe it will become more "intuitive" when we have relativity and quantum mechanics fully unified.

That differing velocities may alter our measure of distances, elapsed times, and even different orderings of events is one CRAZY notion...hardly "intuitive" yet so far inescapable.

Seems like we have moved from the Galilean transformation of Newtonian physics, to the Lorentz transformations which Einstein incorporated into relativity... then the more general Poincaré group...(was that later?? I'm not sure) ...In fact it's not even "intuitive" that our man made math should even describe so much of what we observe. Talk about "coincidence".

 

[vilas]

c^2 occurs frequently in special relativity: in the Lorentz transformations, in forumlas for the interval, relativistic energy, and others too. Is there an intuitive reason for the high occurence of c^2?

More important is v and not c. Without v^2 frame with +v wouldn't be equivalent to that with -v