# What part does 'squaring' the speed of light play in the equation?

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I have never read anywhere, our heard an explanation of, WHY the speed of light figure, @ 300,000,000 meters per sec, has to be squared, light speed times itself, in order for the conversion of energy to mass or vis-versa can be understood. The squaring of the number seems to suggest that an enormous speed of @ 9x10 to the 16th meters per second is needed to convert mass energy to light energy or visa-vie.

Has anyone read anything on this matter?

The equation in question is E=MC2...

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It is not a speed, it is a speed squared, which has different units and certainly not the the same physical meaning...

What is your maths level? I suspect the best way to try and get some physical understanding from this is to understand the derivation of the equation.

It's important to be clear when talking about mass in relativity. Normally on this forum we try to us m to be rest mass, in which case the full equation is actually...

E2=m2c4+p2c2

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c squared is not a speed. meters2 per sec2 is not a velocity. It is a constant.

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c squared is not a speed. meters2 per sec2 is not a velocity. It is a constant.

reply: I do understand that C squared is used as the constant in the equation; but my question is WHY must the speed of light number be squared in order for the formula to reveal that Energy = the static mass of an object?

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reply: I do understand that C squared is used as the constant in the equation; but my question is WHY must the speed of light number be squared in order for the formula to reveal that Energy = the static mass of an object?

You need the units to make sence, so you must have Energy ~ mass x velocity x velocity.

Based on the units it is clear that $E = m c^{2}$ has the right units, i.e. Joules, but that is all we can tell and so does not really explain the equation nor tells us that it has any physical significance.

You need to look into special relativity further to get at this.

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The derivation of the equation is what causes the squaring of the speed of light. What I'm about to say is certainly not the same thing as E=mc², but when you look at the kinetic energy of an object, it's equation is KE=½mv². Momentum, on the other hand, is M=mv. So, you have two equations concerning kinetic energy and momentum, one with velocity squared, and the other not. But it all comes from the derivation, just as with E=mc², and once you read through the derivation, you'll realize that it all makes sense.

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The derivation of the equation is what causes the squaring of the speed of light. What I'm about to say is certainly not the same thing as E=mc², but when you look at the kinetic energy of an object, it's equation is KE=½mv². Momentum, on the other hand, is M=mv. So, you have two equations concerning kinetic energy and momentum, one with velocity squared, and the other not. But it all comes from the derivation, just as with E=mc², and once you read through the derivation, you'll realize that it all makes sense.

Ewmon....Thank you for your reply. The phrase 'derivation of the equation' makes a lot of sense with regard to my question. My real thought is Why is this constent, C squared, the majic bullet that reveals the amount of energy that exists in a static mass? Second, and even possibly more abstract, HOW did Einstein determine that by squaring the value of the fastest known speed inthe universe was the constsant that was needed to unlock the key to the fact that enegry and static mass are eqaul?

Here are the reasons I have such questions. The speed of light is the absolute known speed in our universe. Everything else is relative and subserviant to be. It sets the 'speed limit' for all other things. Therefore it fastinates me that Einstein would conclude that it was necessary to go vastly far beyond that number, @ 186,000 miles per second, and square such an absolute value in order to make his conclusion about the equality of static mass and energy make sense mathematically.

Does anyone know or has anyone read HOW Einstein hit on that value for his constant in this most well known and famous of all physics formulas?

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Ewmon....Thank you for your reply. The phrase 'derivation of the equation' makes a lot of sense with regard to my question. My real thought is Why is this constent, C squared, the majic bullet that reveals the amount of energy that exists in a static mass? Second, and even possibly more abstract, HOW did Einstein determine that by squaring the value of the fastest known speed inthe universe was the constsant that was needed to unlock the key to the fact that enegry and static mass are eqaul?

Here are the reasons I have such questions. The speed of light is the absolute known speed in our universe. Everything else is relative and subserviant to be. It sets the 'speed limit' for all other things. Therefore it fastinates me that Einstein would conclude that it was necessary to go vastly far beyond that number, @ 186,000 miles per second, and square such an absolute value in order to make his conclusion about the equality of static mass and energy make sense mathematically.

Does anyone know or has anyone read HOW Einstein hit on that value for his constant in this most well known and famous of all physics formulas?

Bare bones:

It starts with the postulate of SR that the speed of light is invariant. (this is very important, the fact that there is even such a thing as an invariant speed results in a whole series of consequences.)

This in turns leads to the ideas of time dilation and length contraction ( which both include c² in their formulas)

If you then apply these ideas to a moving object you get a formula which gives the energy of that moving object in Relativity, which turns out to be different than than the one for Newtonian physics (E= mv²/2).

This formula, when solved for v=0 (the object is at rest) leaves E=mc².

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Janus.... Thank you It is very understandable, and helpful, to know that the same content, c2, is used in the associated concepts of time dialation and length contracrion. But...both of these phenomenon are directly related to Einstein's Theory of Relativity that is fundimentally expressed in the profound equation E=MC2. Therefore the use of C2 in the two additional calculations is clearly consistant with the same constant used in Einstein's primary equation E=MC2.

Additionally, the compound use of the same constaint, C2, makes its importance even more pervasive and profoundly significant to understand. This brings us full-circle back to the initial inquiry as to why SQUARING the speed of light is the crucial factor to the equation that unlocks one of the greatest secrets of the universe, that all static mass can be directly expressed and understood as energy. Why then does the squaring of the known speed of light play the crucial role in unlocking this secret? Why wasn' t, for example, just using the speed of light alone sufficent to unlock this profound and critically fundimental secret? What does the squaring of the constant crirtically do to this equation?

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Why then does the squaring of the known speed of light play the crucial role in unlocking this secret?
Because the equations for energy require a velocity squared.

A unit of energy is the joule, and the joule's units are kg * m2/sec2. To make the units come out correctly you must square velocity.

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AGC..... Thank you for sighting this principle. I will definitely need to look into the understanding of how a Joule is expressed and why the squaring of any veliocity is essential. I appreciate this explanation. I read your earlier reply that also implicated the Joule but failed to see the point of its application as clearly as I have in this explanation. Your generalization of the necessity of squaring the velocity when dealing with the Joule unit of energy seems to be the key to my question. Again, thank you very much for readdressing your initial response.

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Have a look here: http://sciencesense-eyesopen.blogspot.gr/2008/02/why-c-squared.html

Quoting myself, I'll re-post here my comment about that (you''ll find it in the link in the discussion part)

"Anonymous said...

Hello Eyesopen, sorry to be late, I like your blog very much, very interesting.

I found it on a general search for information about c squared.

Actually, I would like to have your comment on the following.

I believe the question could go like this: In our equation, we use C2, but in observing the reality, we don't measure C2, but only C. Physics looks like telling us that what we are measuring is only the square root of what is really happening. The true constant is C2 (call it K, as you did), and write down e=mK which is the right equation. The question which follows is thus why the hell are we measuring only the square root of K instead of K, and what is that constant with strange units m2/s2 representing?

Michel, born 1960."

The question still stands.

-------------------

as much as I know, things in physics go like this:

Physicists measure some quantities according to their ability to measure: they measure weight, time, distance, velocity, acceleration, etc.

Then, completely randomly, they try to make mathematical operations with theses measurements.

When they succeed to get a correct measurement from the mathematical operation of 2 others, they (the physicists) encounter orgasmic pleasure.

For example, Newton established the law of universal gravitation on the basis of the square of the distance not because it follows from some logic, but because it gives the correct result. Simple logic would tell that the attraction between 2 bodies would be a function of their mass and of the distance between the 2 bodies. The use of the square distance is alltogether a touch of genius and a very strange feature.

Why the distance squared? Not even twice the distance, but suddenly a measurement in meters that you have to square to get a surface in squared meters: something that you cannot measure. Exactly like we were measuring the square root of "something else".

The same strangeness occurs in e=mc^2, but since the equation gives the correct result, who cares?

Edited by michel123456
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For example, Newton established the law of universal gravitation on the basis of the square of the distance not because it follows from some logic, but because it gives the correct result. Simple logic would tell that the attraction between 2 bodies would be a function of their mass and of the distance between the 2 bodies. The use of the square distance is alltogether a touch of genius and a very strange feature.

Why the distance squared? Not even twice the distance, but suddenly a measurement in meters that you have to square to get a surface in squared meters: something that you cannot measure. Exactly like we were measuring the square root of "something else".

The same strangeness occurs in e=mc^2, but since the equation gives the correct result, who cares?

Distance squared makes a lot of sense if you stop to think about it.

If you have somekind of quantity which is conserved or preserved, and you are spreading it out over a three dimensional volume, and the source is in the middle and it's coming out in a roughly spherical shape, then your quantity will be spread over a spherical area that gets bigger at the same rate a sphere gets bigger. This rate is r^2.

I don't think/know whether Newton used this reasoning (unlikely as he came up with some of the maths which was later used to prove this principle and talk about the idea of conserved fields in general), but it's a very simple argument that can be stated and proved with calculus, providing the premise is assumed (spherically symmetric and conservative field).

There is some level of ad-hoc reasoning and intuition in any physical theory, but this is perfectly acceptable. As long as the theory is internally consistent (mashing symbols together is fine -- if often unproductive -- as long as you follow the rules. Because following the rules of maths _is_ logical reasoning, even if you have no idea what you're doing).

However, the assumptions you are working with are completely up to you. If these assumptions are small in number, differ from existing assumptions minimally, are provably consisting with all known results and -- this is the important bit -- yield novel, specific and preferably precise predictions about things that are presently unknown (often also unexpected) which then prove correct, you have a good physical theory.

Both Newtonian mechanics and relativity did this. And understanding the difference between this and just pulling something from your posterior and kneading it until it fits the data is important.

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I haven't really read through this website, but it seems to show Einstein's derivation step by step:

The Derivation of E=mc2

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Distance squared makes a lot of sense if you stop to think about it.

If you have somekind of quantity which is conserved or preserved, and you are spreading it out over a three dimensional volume, and the source is in the middle and it's coming out in a roughly spherical shape, then your quantity will be spread over a spherical area that gets bigger at the same rate a sphere gets bigger. This rate is r^2.

I don't think/know whether Newton used this reasoning (unlikely as he came up with some of the maths which was later used to prove this principle and talk about the idea of conserved fields in general), but it's a very simple argument that can be stated and proved with calculus, providing the premise is assumed (spherically symmetric and conservative field).

There is some level of ad-hoc reasoning and intuition in any physical theory, but this is perfectly acceptable. As long as the theory is internally consistent (mashing symbols together is fine -- if often unproductive -- as long as you follow the rules. Because following the rules of maths _is_ logical reasoning, even if you have no idea what you're doing).

However, the assumptions you are working with are completely up to you. If these assumptions are small in number, differ from existing assumptions minimally, are provably consisting with all known results and -- this is the important bit -- yield novel, specific and preferably precise predictions about things that are presently unknown (often also unexpected) which then prove correct, you have a good physical theory.

Both Newtonian mechanics and relativity did this. And understanding the difference between this and just pulling something from your posterior and kneading it until it fits the data is important.

Michel, Schrodinger, ewmon....

Thanking each of you for your inputs. Taking time to review and digest your references and ideas...

Also trying to learn the mechanics of pasting partial quotes from prior posts into responses. Haven't found the combination yet. Is there a Help location this type of thing?

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Adam, to understand why $c^2$ even appears in the equations, we will take a look at deriving time dilation. The following image illustrates a light clock at rest:

This clock measures time according to:

$\Delta t = \frac{2 L}{c}$

where $L$ is the distance separating the reflective plates and $c$ is the speed of light. This gives us:

$\frac{\text{distance}}{\text{speed}}=\frac{\text{distance}}{\text{distance}/\text{time}} = \text{time}$

We have twice the distance, $2L$, because the light must travel to the top plate and be reflected back to the bottom plate in order to measure time.

An observer in motion relative to the light clock will not see the path of light as a straight up and down path. Instead, they will see the following:

As we can see from the above image, an observer moving at a speed of $v$ relative to the light clock will see the path of the light elongated to a distance $R$, which is equal to the hypotenuse of the triangle:

Because the speed of light is the same regardless of one's frame of reference, an observer moving at a speed of $v$ relative to the light clock will see that the clock is measuring time according to:

$\Delta \tau = \frac{2 R}{c}$ where $R=\sqrt{\left(\frac{v \Delta \tau}{2}\right)^2+L^2}$ as determined by the distance formula (Pythagorean theorem).

Multiplying both sides of the equation by $c$ and squaring the result, we get:

$c^2 \Delta \tau^2 = 2^2 R^2 = 4 \left( \frac{v^2 \Delta \tau^2}{4} +L^2\right)=v^2 \Delta \tau^2+4L^2$

Subtracting $v^2 \Delta \tau^2$ from both sides yield:

$c^2 \Delta \tau^2 - v^2 \Delta \tau^2 = 4L^2$

Factoring $\Delta \tau^2$ on the left side yields:

$\Delta \tau^2 \left(c^2 - v^2 \right) = 4L^2$

Now we can divide both sides by $\left(c^2 - v^2 \right)$:

$\Delta \tau^2 = \frac{4L^2}{ \left(c^2 - v^2 \right)}$

Factoring $c^2$ from the denominator on the right side and taking the square root of the result yields:

$\Delta \tau = \frac{2L}{ c \sqrt(1 - \frac{v^2}{c^2})}$

Because $\Delta t = 2L/c$ we can substitute $\Delta t$ into the final result to yield the time dilation formula:

$\Delta \tau = \frac{\Delta t}{ \sqrt(1 - \frac{v^2}{c^2})}$

As you can see from the above formula, $\Delta \tau = \gamma \Delta t$ where $\gamma = \frac{1}{ \sqrt(1 - \frac{v^2}{c^2})}$ is the Lorentz factor and it contains $v^2$ and $c^2$ (Gamma, $\gamma$, forms the basis for transformation in special relativity). Applying this to other equations, such as those for energy, yields:

The energy and momentum of an object with invariant mass $m$ (also called rest mass in the case of a single particle), moving with velocity $\text{v}$ with respect to a given frame of reference, are given by

$E=\gamma m c^2$

$\text{p}=\gamma m \text{v}$

Edited by Daedalus

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