Help with E = mc^2

[MalcolmM]

I have been trying to understand the basics of Special Relativity with my very rusty 35 year old engineering maths.

I can follow the derivations of time dilation and length contraction. I love that such weird concepts can so easily be 'proved' with a bit of trig and Pythagoras.

However! Deriving E = mc^2 always seems to come down to approximating an expression as a series (Taylor?), where the first term is the rest energy, second term is the classical KE and subsequent terms are increasingly infinitesimal.

I loosely understand approximating expressions as a series but I really struggle to understand how a particular term in the series (which is an approximation) can turn out to be so significant!

Is there another way of deriving E = mc^2 using 'graduate engineering maths' that my head might understand and accept?

Or maybe another way of looking at the Series approximation that would allow me to make the mental leap to accepting it?

Thanks,

Malcolm

[andrew s]

This is a simple derivation which makes an approximation I used in my physics exam 50 odd yrs ago.   https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/E=mcsquared/proof.html#:~:text=We want to show that,outcomes returns E%3Dmc2.

Regards Andrew 

[MalcolmM]

47 minutes ago, andrew s said:

This is a simple derivation which makes an approximation I used in my physics exam 50 odd yrs ago.   https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/E=mcsquared/proof.html#:~:text=We want to show that,outcomes returns E%3Dmc2.

Regards Andrew 

Thanks @andrew s,

I had actually googled this quite recently and where I can understand it, I sort of feel the initial assumption takes you half way there.

I will accept that there may simply not be a way of deriving it from first principles with my limited maths in the same way as length contraction and time dilation (though I guess even for these you have to make the assertion that c is constant for different observers!)

Thanks again,

Malcolm

[Gfamily]

The derivation I recall was to consider two plates facing each other.

Given the relationship between photon wavelength and momentum, you can show that if a photon of energy E is emitted from one plate and absorbed at the other, this is equivalent to a mass transfer of E/c^2 from one plate to the other.

 

[andrew s]

Try this then 

 

Regards Andrew 

[AstroKeith]

5 minutes ago, Gfamily said:

The derivation I recall was to consider two plates facing each other.

Given the relationship between photon wavelength and momentum, you can show that if a photon of energy E is emitted from one plate and absorbed at the other, this is equivalent to a mass transfer of E/c^2 from one plate to the other.

 

Yes, imagining a body just below the speed of light and working out energy gained by an applied force and change of momentum through a force is often used as the simplest way to 'prove' E=mc^2.

But, the purest will still need to show that the proof works for other velocities!

[MalcolmM]

1 hour ago, andrew s said:

Try this then 

 

Regards Andrew 

This looks promising thanks, but I'll have to watch it slowed down despite his presentation being near the speed of light

Malcolm

[andrew s]

@MalcolmM this is the derivation I was looking for.   http://www.adamauton.com/warp/emc2.html

Regards Andrew 

[Gfamily]

6 minutes ago, andrew s said:

@MalcolmM this is the derivation I was looking for.   http://www.adamauton.com/warp/emc2.html

Regards Andrew 

Yes, that's the one I was thinking of. Not dependent on any relativistic effects/Lorenz contraction or similar - just on Maxwell's finding that photons have momentum. 

ETA - of course, you can't separate out 'momentum carried by photons' from other aspects of light etc.

Edited June 29, 2022 by Gfamily