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  • Einsteins Most Famous Equation, E = mc2

    Richard Conn Henry 2017 April 7

    1

    Einsteins Theory of Relativity is simply a trivial extension, to 4-dimensions, of the famous Theorem of Pythagoras. The Pythagorean theorem itself can easily be seen to be true, just by glancing at these two (equal-area) big squares:

    Obviously, the square on the hypotenuse equals the sum of the squares on the other two sides of (any) red triangle. The invention of algebra allowed us to write this famous Pythagorean theorem as an equation: ds2 = dx2 + dy2 Relativity asserts that Pythagorass theorembut in our 4-dimensional universeis: ds2 = dx2 + dy2 + dz2 dt2 where ds is the separation in space-and-time of any two events (for example, your snapping your fingers twice, as you wave your arm around)and asserts that ds is invariant: that is, is the same for any observer of the two events. Note the critical minus sign: this is the only way that time differs from being merely one more space dimension! How do we know that relativity is true? Becauseas we will shortly seeit successfully predicts the atomic bomb! Newton taught us that, for a moving mass m, the important quantities are momentum mv and kinetic energy 1/2 mv2 So!we will be keeping a sharp eye out for Newtons mv and 1/2 mv2 as we explore the implications of Einsteins remarkable pseudo-Pythagorean assertions. What Einstein specifically claimed was that ds (the separation of two events) is the same for a stationary observer, and a moving observer. In the following two diagrams, the stationary observer is you, with your x-y-z coordinate system, while the moving observer is your sister (redheart, with her x-y-z moving coordinate system): !

  • Einsteins Most Famous Equation, E = mc2

    Richard Conn Henry 2017 April 7

    2

    We are going to discover that redheart's velocity u (the red arrow) is predicted, from Einstein's Pythagorean claim, to be greater than the velocity v (the black arrow) that you find for her : dx divided by your clock's change, dt. In your stationary coordinate system redheart moves a distance dx, in time dt (by your two synchronized clocks). The two events that we are considering are redhearts snapping her fingers at one moment in her motion (our first panel), and then snapping her fingers again at some later moment in her motion (our second panel): those two events are what is illustrated in the two panels.

    Einstein's claim is that dx2 + dy2 + dz2 dt2 = ds2 = dx 2 + dy 2 + dz 2 dt 2 i.e, ds is invariant. In our example, with the motion being entirely in the x direction, dy= dy = 0 and dz= dz = 0, and so we can write Einsteins claim as that:

    dx2 - (c dt )2 = ds2 = dx 2 - (c dt )2 The cs that I inserted (multiplying the dts) are to convert the units of time from seconds to metersrequired so that the units will be the same for everything in the equation: but the cs have no physics significance at all. The required value of c was long ago determined from experiment to be about 299,792,458 meters per secondvery close to the the speed of lightand this number is now defined to be the exact value (just as there are exactly 3 feet in a yard). Light moves at that speed because light is masslessall massless particles, including for example gluons and gravitons, move at that identical speed, which is set by the remarkable geometry of spacetime. [If we dont use different units for time (compared with space) then c = 1 and can be omitted from all equations entirely.] Snap test: what is the value of dx ? Right! dx = zero! For dx is the spatial separation of those two finger snaps in redhearts own coordinate systemwhich moves right along with her! Since dx is zero only in redheart's coordinate system and in no other, redheart's coordinate system is special indeedso, instead of writing the time separation (in her special coordinate system) as dt we choose to write (from now on) instead of dt, d (the Greek letter tau). So, instead of dx2 - c2 dt2 = ds2 = dx 2 - c2 dt 2 we now have

    c2 d2 = c2 dt2 - dx2 and this equation will now expose us to the shocking true nature of photons of light. For consider the case that redheart is a photon of light created on the sun, and absorbed (in your retina) on Earth. Distance equals velocity times time: dx = v dt, and so for light, dx = (c dt) and so c2 d2 = c2 dt2 - c2 dt2 = 0 and Einstein predicts that d which is the time, by the photon's own watch, of the photon's tripfrom its creation on the sun to its ceasing to exist in your retinais: zero! By its own clock, the photon never exists! So what did Albert Einstein think of this? Einstein (1954): "All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick, and Harry thinks he knows it, but he is mistaken." Ready for one more major consequence of our 'trivial' postulate? Suppose our "photon" from the sun were some kind of "magical" photon that could go at speed Cjust a bit faster than 299,792,458 meters per second. Well, look at our latest equation, which, for this case, would state that c2 d2 = c2 dt2 - C2 dt2 . On the left hand side we have c2 d2 and since any number squared must be positive, the left hand side of our equation is positive, yet the right hand side would be negative for any such magical photons! We not only conclude that there could be no magical photons, we also conclude that no velocities greater than that of light can ever occur: if anyone says that they can, that would be closely analogous to someoneabsurdlyannouncing that they can go north of the north pole. So! Speed limit is onethats why we introduce 299792458: otherwise, your car would go at 0.000000089 (= 60 mph). Now remember that we are seeking the famous equation E = mc2. To get that, we must launch a bunch of algebra. But it is nothing but algebra: no additional physics, of any kind. (But how does Mother Nature know algebra, which we humans (e.g. Omar Khayyam) developed? No one has any idea: but, know it Mother Nature does!) Redheart's rockets speed is = / as measured by us on Earth [and = / as measured by redheart herselfthe very same distance dx of course, but (as we shall bring out) a quite different time interval d]. So, we have:

  • Einsteins Most Famous Equation, E = mc2

    Richard Conn Henry 2017 April 7

    3

    ! !"!"

    != ! ! , so

    =

    =

    =1

    1 !

    !

    = 1 !

    !

    !!!= 1 +

    12!

    !+38!

    !+

    That last step is a binomial expansion (invented by Isaac Newton). Since frequently the velocity v is very much smaller than c, that last term (plus all the higher-order terms: +) can often be ignored. (Be sure to memorize the above definition of ! Also, note that when v = 0, is onebut that as v approaches c, approaches infinity!) Now, return to our equation !!! = !!! !! where I have now multiplied both sides of the equation by m2 (we seek E = m c2, so m, the mass of redheart (or of redheart plus her rocket) must be introduced somehow!) From that equation,

    !! = !! !"!"

    !! !"

    !"

    ! or !! = ! !!

    Notice that u = v [and also that dt = d] if you insert u = v into the last equation, it dissolves. Try it!

    !! = 1 +12!

    !+38!

    !+

    !

    !!

    !! = !!! +

    + !

    ! !

    !

    !!+

    ! ! or, if we now abandon Newton and, with Albert Einstein,

    we redefine momentum to be: p = mu = mv and total energy to be: = ! + !!! + !

    ! !

    !

    !!+ ! +

    (with K being the kinetic energy) then

    ! = ! + ! ! or ! = ! +! You verified all of that? OK! If redheart is not moving at all, her momentum p is zero, and we have, at long last:

    = ! ( or, E = m ) Simply our deriving E = mc2the most famous formula in all of physicsfrom a tiny tweak of the Pythagorean theoremis not enough to have us also understand how the atomic bomb results. So lets go through that historic application of this most famous equation, so as to fully understand what E = mc2 really means in practice.

    If a neutron hits a nucleus of 235U (mass mU) that nucleus decays into a nucleus of 92Kr , plus a nucleus of 141Ba , plus 3 neutronsand those 3 neutrons can then go on to hit 3 more 235Us, producing a chain reactionand thus a bomb. The Uranium, if stationary, had total energy E = mU c2. The resulting Kr and Ba nuclei are observed to each have, in addition to their own internal rest energies mKr c2 and mBa c2 huge kinetic energies K. The total energy after the decay, ignoring the low-mass neutrons, is, according to our above-expounded special relativity,

    EKr + EBa = (mKr c2 + KKr) + ( mBa c2 + KBa)

    Conservation of energy tells us that energy E = mU c2 = EKr + EBa = (mKr + mBa) c2 + (KKr + KBa)

    so the total kinetic energy is K = KKr + KBa = c2 [mU - (mKr + mBa)]

    It is an experimentally observed fact that the sum of the masses of the Krypton and the Barium nuclei, plus those neutrons that are also produced, is significantly LESS than the mass of the original Uranium nucleus: and when this difference is put into Einsteins equation and so is multiplied by the enormous number c2 well:that atomic bomb! And all that, from just these three steps:

    a) the pseudo-Pythagorean Theorem of Albert Einstein: : ds2 = dx2 + dy2 + dz2 dt2 b) a small bunch of algebrain other words, nothing new: no additional facts, or ideasjust algebra! c) one observed fact regarding the masses of the U, Ba, and Kr nuclei

    The most amazing item on the list: our human development of algebra: so, thank you Omar Khayyam (et al!)

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