Why is the Speed of Light Squared in Special Relativity?

[Swampeast Mike]

Special relativity says that the speed of light is a constant regardless of the speed of the observer.

I've always been troubled with the idea of squaring that speed (C ^ 2) unless speed of light to the zero power (C ^ 0) also has meaning at the same time and in the same space; the net effect to an observer fixed in time being 1 = 1 where 0 = 0.

To an observer fixed in space, 0 would appear equal to 1 at the same time that 1 appeared equal to 0.

To a real observer, part of one appears to be in zero and part of zero appears to be in one.

Please destroy this view of special relativity.

 

[da_willem]

What's wrong with squaring a number? If this is an assault on SR you might want to post this in the theory development section.

 

[pervect]

The speed of light squared is not a velocity. It appears in some expressions where velocity^2 appears, like E=mc^2 (compare to E=(1/2)mv^2).

The speed of light is constant for all observers regardless of their state of motion, this is however totally unrelated to the properties of c^2.

 

[Swampeast Mike]

Nothing wrong with squaring a number.

What I have a problem understanding is squaring a constant that cannot change for an observer who is fixed in both time and space.

 

[Swampeast Mike]

The speed of light squared is not a velocity

My exact problem. Speed of light squared is a concept that depends on your time/space perspective.

 

[Doc Al]

What I have a problem understanding is squaring a constant that cannot change for an observer who is fixed in both time and space.

Why would you have a problem squaring a constant? (I have no idea what you mean by an observer "fixed in both time and space". The speed of light is an invariant: it has the same value as measured by anyone regardless of their motion.)

Do you have a problem with the formula for (non-relativistic) kinetic energy: {KE} = 1/2 m v^2 ? In that formula speed is not a constant, but is frame dependent. I trust that you are fine with that. So, if anything, squaring c should be even less of an issue.

My exact problem. Speed of light squared is a concept that depends on your time/space perspective.

Actually, the speed of light, being frame independent, is less dependent on "time/space perspective" than ordinary, sub-light speeds. (But, even if something is frame-dependent, what prevents you from squaring it?)

 

[eNathan]

Maybe a calculus class may help. I don't fully understand the logic in squaring c either :pondering:

 

[Mortimer]

Whenever you find a c^2 there's probably a way to write it differently, making the source of the square more obvious.
Look e.g. at E=mc^2 which is usually written more correctly as E/c=\sqrt{(m_0c)^2+p^2}. The c on the left part is a scaling factor while the c in the right part is a velocity.
In t'=\gamma(t-vx/c^2)=\gamma(t-(v/c)(x/c)), v/c is a ratio of velocities while in x/c the c is a correction in the scale of x.
Try it. You'll find for most cases either this separation in a velocity and a scaling factor, while in other cases you may find the square to have its roots in a Minkowski equivalent of a "Pythagorean" operation (A^2=C^2-B^2 for Minkowski space, equivalent to C^2=A^2+B^2 for Euclidean space).

 

[εllipse]

Do you have a problem with square rooting pi? r=\frac{\sqrt{A}}{\sqrt{\pi}} This is simply a rearranged version of A={\pi}r^2, which can be used to find the radius of a circle if you know the area.


What about \sqrt{E}=\sqrt{m}c? You don't have to square c; E=mc^2 is just a little more elegant than the former.

 

[russ_watters]

Maybe a calculus class may help. I don't fully understand the logic in squaring c either :pondering:

Its exctly the same as squaring velocity in the kinetic energy equation - that's what you have to do to turn a velocity into an energy (well, that and multiply by mass and a proportionality constant).

Bernoulli's equation also uses velocity squared -- to calculate pressure. So it isn't an unusual thing to square a velocity..

Nothing wrong with squaring a number.

What I have a problem understanding is squaring a constant that cannot change for an observer who is fixed in both time and space.

The speed of light is a constant, but it is also a real, physical speed.

Lets look at it from the other angle: why would it be a problem to square a constant (others have provided examples...)?

 

[pervect]

c is not a "number" in the standard MKS unit system, it is a velocity. Squaring a velocity makes sense in certain cirumstances, as when one squares the velocity and multiplies by the mass to calculate the energy. The units work out correctly

energy - force * distance = mass * acceleration * distance = kg * (m/s^2) * m =
kg * (m/s)^2

Thus we see that unitwise, energy is naturally the product of a mass multipled by the square of a velocity.

Squaring a constant velocity makes sense under the same circumstances that squaring a non-constant velocity makes sense. The fact that the velocity is constant is really irrelevant to the units.

 

[reilly]

Let's see. How about h**0 = c **0= 1 = e**0, etc, where h is Planck's constant, and e is the electron's charge. I'm 6'2 tall. As long as observers are moving perpendicular to me, I will be 6'2 for all those observers. If my height is H, then all those observers will see the same value for H**1/7.

The usual rules of algebra proclaim that the values of c**N, with N any real number from - infinity to + infinity are the same for all inertial observers. Hint: the rules of algebra are Lorentz invariant.

Regards,
Reilly Atkinson

 

[Swampeast Mike]

No Problem Squaring pi

Speed of light involves time; pi involves space.

We comprehend time as a non-zero because nothing can move at that speed.

Yet something appears to change between objects separated by space.

To both the similarity and change between objects, time is zero.

 

[Janus]

Speed of light involves time; pi involves space.

We comprehend time as a non-zero because nothing can move at that speed.

Yet something appears to change between objects separated by space.

To both the similarity and change between objects, time is zero.

This makes no sense what-so-ever.

You have a problem with squaring properties that involve time? What about acceleration which is distance/time²?

 

[pervect]

I suppose we'd better not square Planck's constant, either.

The Bohr radius

http://musr.physics.ubc.ca/~jess/hr/skept/QM1D/node3.html

however does involve the square of Planck's constant. Since h-bar has units of

kg m^2 / sec

I suppose this must be a bad thing since it's even more complicated in its units than a velocity, and it's a universal constant, to boot.

(Personally, I still don't see the problem).

There are numerous other places in physics where fundamental constants are squared. I would just try and tell Swampbeast to "deal with it"

 

[pallidin]

I believe that the OP has an issue with the invariant C being expressed, mathematically, beyond the constants' immutable value.
In a sense the OP is correct is such concern, as a constant value can not be un-reasonably altered in any equation.

 

[quantumdude]

I believe that the OP has an issue with the invariant C being expressed, mathematically, beyond the constants' immutable value.

Nowhere in any equation of special relativity is 'c' anything other than 'c'. Squaring the speed of light doesn't give you a new value of 'c', it gives you a value of 'c²'

In a sense the OP is correct is such concern, as a constant value can not be un-reasonably altered in any equation.

There is nothing correct about the OP.

(edit: except the first line, that is)

 

[pallidin]

Nowhere in any equation of special relativity is 'c' anything other than 'c'. Squaring the speed of light doesn't give you a new value of 'c', it gives you a value of 'c²

QUOTE]


You are incorrect: the squaring of "c" supposes a mathematical and practical scenario which is not possible, thus it becomes invalid by default.
Doing otherwise would be much like saying 1 squared is greater than 1.
They are both immutable constants.

 

[quantumdude]

You are incorrect: the squaring of "c" supposes a mathematical and practical scenario which is not possible, thus it becomes invalid by default.

No I'm not, and no it doesn't. Squaring the speed of light is not in any way contrary to the speed of light postulate. I have no idea of why anyone would think that it is.

Doing otherwise would be much like saying 1 squared is greater than 1.
They are both immutable constants.

This makes absolutely no sense.

 

[pallidin]

Are you then suggesting that c-squared is a valid condition? If so, it's inclusion in mathematics is valid.
What evidence, then, does anyone have for a c-squared phenomenon?

 

[quantumdude]

Are you then suggesting that c-squared is a valid condition?

I don't even know what you mean by c-squared as a condition. If you are asking me if the equations of relativity that contain c² are validly derived from the postulates, and match experimental predictions, then I say "Yes, of course they are."

If so, it's inclusion in mathematics is valid.

This is ridiculous. Since when does any statement of any physical theory have any bearing on what is included in or excluded from mathematics?

What evidence, then, does anyone have for a c-squared phenomenon?

What is a "c-squared phenomenon"?

 

[pallidin]

I don't even know what you mean by c-squared as a condition. If you are asking me if the equations of relativity that contain c² are validly derived from the postulates, and match experimental predictions, then I say "Yes, of course they are."



This is ridiculous. Since when does any statement of any physical theory have any bearing on what is included in or excluded from mathematics?



What is a "c-squared phenomenon"?

Tom, I want to thank you for granting lattitude towards someone(myself) who is not well versed in mathematics. It is my hope that I can learn from such discourse.

In answer to your question "What is a c-squared phenomenon"?:
My position, in light of your statement "Since when does any statement of any physical theory have any bearing on what is included in or excluded from mathematics?" holds my position somewhat mute.

Thanks again for your instruction.

 

[Swampeast Mike]

Glad You Mentioned Planck's Constant

It leads directly to the duality of energy transfer between separated bodies. After all, he saw it as nothing more than a mathematical convenience.

Particle and/or wave, both and/or neither at the same time.

Go back to the original post. [My apologies for trying to express something that can only be described as nothing in simple words.]

My problem wasn't understanding the "speed of light squared" it was understanding how such works as an individual concept relative to both time and space.

If light can be both a something and a nothing, cannot its' speed be only measurable as a product? a sum? a difference? all? some?

Energy, radiation, light or whatever you want to call it always occurs BOTH WAYS between separated bodies yet we tend to conviently overlook the view of the other observer.

 

[Gokul43201]

Dear Mike : There is no physics in this post, or in the OP. Hence, there can be no physical discussion about it. If you can address the problem in terms of well-defined physical constructs, do so, and a physical discussion may result.

Else, this thread will have to be closed.

 

[ZapperZ]

At the risk of reviving a dead horse (have you tried giving mouth-to-mouth to one?), I would like to point out an almost DEFINITIVE treatment on this particular question:

J.J. Prentis Am. J. Phys. v.73, p.701 (2005).

He describes the derivation and the conceptual foundation behind mv^2/2 in painful detail. Anyone trying to make up their own ideas about this is required to first read this paper and understand it inside-out.

Zz.

 

[CarlB]

I've always been troubled with the idea of squaring that speed (C ^ 2) unless speed of light to the zero power (C ^ 0) also has meaning at the same time and in the same space; the net effect to an observer fixed in time being 1 = 1 where 0 = 0.

I have no idea why this would trouble you, however, it does turn out that there is a branch of mathematics / physics that does something sort of similar to this. In D. Hestenes' "Geometric Algebra", the dot product and the cross product are combined into a single "multi-vector" operation. Hundreds of physicists and mathematicians use his theory. Hestenes' website is here:
http://modelingnts.la.asu.edu/

Geometric algebra is a subdivision of Clifford algebra. Where the standard physics education meets Clifford algebra is in the Pauli or Dirac gamma matrices, which are examples of geometric algebras defined on the manifold of 3-space and 4-dimensional space-time, respectively.

The reason I'm bringing this up is because the Geometric algebra crosses the usual boundaries of scalars, vectors and tensors. In the GA, for example, one can add a vector to a scalar and get what is called a "multivector". The laws of E&M, for example, can be written with very few symbols in this manner.

I disagree with the poster who said that "c" is a velocity. I believe instead that it is a speed. A velocity has a direction, "c" does not. But if you think of "c" as a velocity, then the conversion that takes you from c to c squared is a dot product.

The other half of a dot product, in Geometric algebra, is a cross product. In standard physics, a cross product takes two vectors and turns them into a "psuedovector". Undergraduates used to be taught that the result of a cross product is a "vector", but in grad school they get taught differently.

Now the operation of squaring a vector and getting back a scalar takes anobject of dimension 1 and turns it into an object of dimension 0. This is mighty odd stuff. To put it back into Dirac gamma matrix form, this gets back to the fact that the square of a gamma matrix is unity.

In the context of the gamma matrices, "unity" really means a 4x4 matrix with ones down the diagonal. That makes sense to me. But Clifford / geometric algebra is written without reference to matrices, and to me the implications of scaling laws with them is odd in the same way that the Swampeast Mike's comment is odd. Two things I do not understand that seem to be for the same reason.

Carl

 

[Mortimer]

Carl is of course right with his speed-versus-velocity remark. I made the same error in one of my posts (as a bad excuse I could say that in Dutch the distinction doesn't exist ).

Look e.g. at E=mc^2 which is usually written more correctly as E/c=\sqrt{(m_0c)^2+p^2}. The c on the left part is a scaling factor while the c in the right part is a velocity. (Here it is a velocity if you ask me)
In t'=\gamma(t-vx/c^2)=\gamma(t-(v/c)(x/c)), v/c is a ratio of velocities (here it is a speed) while in x/c the c is a correction in the scale of x.
Try it. You'll find for most cases either this separation in a velocity and a scaling factor, while in other cases you may find the square to have its roots in a Minkowski equivalent of a "Pythagorean" operation (A^2=C^2-B^2 for Minkowski space, equivalent to C^2=A^2+B^2 for Euclidean space).

The point I was trying to make still stands, by the way. If anyone can give examples that do not fit the criteria I described, I would be interested.

 

[Swampeast Mike]

Dear Mike : There is no physics in this post, or in the OP.

I disagree. This is the most fundamental of physical questions when space and time are relative.

The speed of light (C) is also the speed of perfect conduction.

 

[Entropy]

Would you prefer E = \frac{m}{\epsilon_0 \mu_0} instead?

 

[Kino]

I like the idea of a scaling factor: E/c = mc, energy divided by c is equivalent to mass multiplied by c.

But perhaps Swampeast Mike, in his original post, mistakenly assumed that the energy was only released when the mass was actually moving at c-squared? That would perhaps explain his confusion...