Is Energy Really Equivalent To Mass? (a Paradox)

einstein-eMC22-362x289

I got bored, so I put on my “relativistic glasses.”

Cool!! Now everything looks relativistic to me…

Let’s take a look at a massless capacitor with area A and width d such that \sqrt{A}>>d.

Capacitor

Now I charge the plates with +Q and -Q, as a result there is an electric field E=\frac{Q}{\epsilon_{0} A}  between them. Since the capacitor is extremely thin, it’s very easy to calculate the field energy

U_{E}=\int \frac{\epsilon_{0}}{2} E^2 dV=\frac{\epsilon_{0} E^2A d}{2}=m c^2

The mc^2 at the end says that the energy of this system correspond to the rest mass, in fact this frame is the “CoM frame”. As you might have guessed I’ll move to a new frame after this.

Let’s see if I start running to the left with a relativistic velocity v, woahh now the system looks like this

Capacitor1

The capacitor get contracted by a factor of \gamma. And as we know the electric field transform into itself if it points parallel to the velocity. Therefore the energy density stays unchanged… and since the volume is contracted by a factor of \gamma, so is the total energy

U'_{E}=U_{E}/\gamma

or

m'=m/\gamma

hey wait!! hmm…. something looks wrong here, isn’t it?… waitwaitwaitwaitwaitwait…YES IT IS WRONG.

Shouldn’t the mass transform like m'=\gamma m ?

What’s happening?? If this is true, then sdl3$fh%^k@h, and then Einstein will be angry, and then 1/0 and then 0/0 and then… and then….and  then……

Alright calm down bro… let’s look at this more carefully..

The picture above is bugging me, I imagine that if we left the system that way then the plates will move toward each other and BAM. So let’s put something between the plates to avoid the plates crashing into each others. Okay I need someone to help me. Wait a sec… Hey Spiderman! I kidnapped your girlfriend. If you don’t help me, I’ll make sure you won’t be able to see her again! Ok there you go

spiderman

Let’s assume that the Spiderman is massless. In the CoM frame he must hold the force which is equal to the force between the plates

F=\int (\frac{E}{2}) \sigma dA=\frac{\epsilon_{0} E^2 A}{2}

And it is also the same in the moving frame, since force is invariant under parallel Lorentz transformation.Well I can assure myself that everything is fine in CoM frame, but still stuffs in the moving frame still look quite suspicious and also interesting (I’ll make this more precise in a moment). Who knows, maybe the Spiderman can somehow give contribution to the total mass.

In the moving frame, the Spiderman plays a role as the energy transmitter. I mean, first imagine that there is no Spiderman between the plates. Since the plates are attracting each others, the right plate will slow down and the left plate will speed up. Now if there is a Spiderman between them, their velocities will not change at all. So in other words, the Spiderman is taking energy from the left plate at a rate \vec{F}.\vec{v} and transfer it into the right plate to account for the attraction. But, remember that the energy can’t teleport from one plate to the other instantaneously. Some of it might not reach the right plate yet, still located between the plates. Thus in other words:

The “missing mass” might be hiding inside the Spiderman!!! Yeahhh there you are! Gotcha!

However don’t be too happy, let’s see if we really calculate the energy located between the plates. Let’s assume that the spiderman has maximal rigidity, that is energy is transfered through him at the velocity of light c. Thus the time it takes for the energy to be transferred from the left plate to the right one is

\Delta t=\frac{d'}{(c-v)}=\frac{d}{c} \sqrt{\frac{1+v/c}{1-v/c}}

And the total energy taken from the left plate during this time is

\Delta U= \vec{F}.\vec{v} \Delta t=\frac{\epsilon_{o}E^2 Adv}{2c}\sqrt{\frac{1+v/c}{1-v/c}}

Notice that if we add this term to U_{E}/\gamma, we still won’t be able to get the \gamma U_{E} that we wanted because of the annoying square root factor in \Delta U.

Hold on! Maybe even before the energy from the left plate reached the right plate, the right plate has already started receiving energy from the Spiderman. Perhaps it can “borrow” some energy from the Spiderman’s elastic energy. So how do we know when does the right plate start receiving the energy?? Let’s come back to our home, to the CoM frame, things are clearer there. We know that the event one “force start acting on the left plate” and event two “force start acting on the right plate” must happen simultaneously in this frame due to symmetry. From the loss of simultaneity we obtain that these two events aren’t simultaneous in the moving frame, the first event happens

\Delta t'=\frac {\gamma vL}{c^2}

before the second event (rear clock ahead), Thus during this time the Spiderman has stolen an amount of energy from the left plate

\Delta U=\vec{F}.\vec{v} \Delta t'=\frac {\gamma \epsilon_{0} E^2 Ad}{2} \frac{v^2}{c^2}

What a thief! After the second event occurs energy enters and leaves Spiderman at the same rate. So no energy will be stolen anymore. What else? Of course we just need to stick this \Delta U to the U_{E}/\gamma. Are you ready? 3..2…1……

U_{E}/\gamma+\Delta U=\frac{\epsilon_{0} E^2 A d}{2\gamma}+\frac {\gamma \epsilon_{0} E^2 Ad}{2} \frac{v^2}{c^2}=\frac{\gamma\epsilon_{0} E^2 A d}{2}=The\hspace{2mm}Damn\hspace{2mm}\gamma U_{E}

or

m'=\gamma m

btw, thanks Spiderman.

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