# Mathematical Proof that Photons have No Rest Mass

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1. Feb 14, 2016

### Comeback City

Using these equations I am about to prove that photons have a rest mass of zero (mathematically)
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E=hc/λ Photon Energy Equation

E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation

p=h/λ Momentum of a Photon Equation

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Set the First 2 equations equal to eachother...
E=hc/λ ------------------------------------------------> E2=(pc+mc2)2

(hc/λ)2=(pc+mc2)2

h2c22=p2c2+m2c4

{h2c22}/c2={p2c2+m2c4}/c2

h22= (p2c2/c2)+(m2c4/c2)

h22=(p2)+(m2c2)

(h/λ=p+mc)2

[mc=(h/λ)-p]2

[mc(1/c)=(h/λ)(1/c)-p(1/c)]2

[m=(h/λc)-(p/c)]2

(Enter in the Momentum of a Photon Equation)

{m=(h/λc)-[(h/λ)/c]}2

[m=(h/λc)-(h/λc)]2

(m=0)2

m=0

m(photon) = 0

Is my math correct?
If so, is this legible?
And if so again, has this been proved yet and I am just completely unaware that it has?

Last edited by a moderator: Feb 14, 2016
2. Feb 14, 2016

### mathman

Second line does not follow from first, (right side is wrong).

Last edited by a moderator: Feb 14, 2016
3. Feb 14, 2016

### Comeback City

Thanks. I didn't see that until you pointed it out. So I just did the math again (the right way) and it comes out to be...

0 = m2c2

Could this also prove that the the mass of a photon is zero since we already have a set velocity for the speed of light (2.99x108 m/s)?

4. Feb 14, 2016

### ZapperZ

Staff Emeritus
What you have done is simply proving a circular argument. For example

"p=h/λ Momentum of a Photon Equation"

is ALREADY an assumption that the momentum does NOT depend on ANY mass. In other words, you didn't show that this is true, but rather you adopt this form for the photon.

What you have done is simply doing a mathematical derivation from a known set of criteria, rather than showing that those starting criteria or description is correct. This is not a mathematical proof, but rather a logical consequence.

Zz.

5. Feb 14, 2016

### Comeback City

So in simple terms, are you saying that p=h/λ is an equation DERIVED from the other two equations (and possibly others) ?