Thought experiment: Origin of inertia is gravitational?

[johne1618]

Hi,

What do you think of this thought experiment that demonstates a situation where the origin of inertia must be gravitational (Mach's Principle)?

Imagine an empty box with perfect mirrors on its walls.

Now let us pump in a large amount of photons into the box.

Let us assume that the combined mass/energy of the photons is much greater that the mass of the box itself so we can neglect the mass of the box.

Do the confined photons have an inertial mass?

If we apply a force to the box does the box apply an inertial reaction force back proportional to the mass/energy of the photons?

I believe it does.

So where does this inertial force come from?

Apart from elastic scattering from electrons in the walls the only other interaction the photons engage in is gravitational. In particular the photons do not interact with a Higgs field or a zero-point electromagnetic field.

Thus the inertial reaction force can only be due to gravitational interaction with the rest of the Universe.

Now the question is: what type of gravitational interaction?

If we assume an analogy with Electromagnetism then there are two types of gravitational interaction: a static attractive force or acceleration-induced radiation. It seems the latter fits the bill nicely especially if one allows advanced waves so that the resulting inertial reaction force appears instantaneously as it should.

John

 

[enroger0]

Actually I've done a similar thought experiment and found out the box of photons does have an inertia exactly equal to the mass of the photons combined (ignoring the mass of box). But it is not due to gravity, it is due to radiation pressure of photons.

Suppose you push the box, result in some acceleration in say x direction, photons that travel along -x direction would have a blue shift relative to the box, photons travel on x direction would have red shift relative to box.

Thus blue shifted photons will impart more momentum to the box when reflected then before, red shifted photons less.

This change of imparted momentum would behave like reaction force to the outside world. I've calculated the force using only doppler effect and relation between wavelength and momentum and found them to be exact.

The thought experiment I used is also about whether photons have gravitational mass. As photons clearly have inertial mass.

 

[johne1618]

Actually I've done a similar thought experiment and found out the box of photons does have an inertia exactly equal to the mass of the photons combined (ignoring the mass of box). But it is not due to gravity, it is due to radiation pressure of photons.

Interesting - I'd like to see your calculations.

John

 

[mrspeedybob]

The thought experiment I used is also about whether photons have gravitational mass. As photons clearly have inertial mass.

Inertial mass implies gravitational mass. If a photon passes a massive body its path will be deflected. Conservation of momentum means that the massive body must also be accelerated, responding to the gravitational field of the photon.

 

[johne1618]

Maybe enroger0's argument goes something like this:

Imagine a photon in a box with energy

$\Large E = h \nu$

Let us assume the box accelerates with acceleration $a$ for a small time $\Delta t$. Then its velocity $v$ is given by

$\Large v = a \Delta t$

Then, due to the Doppler shift, the energy of the photon as it is reflected off the back wall of the box is given by

$\Large h\nu_d = h\nu (1 + \frac{v}{c})$

$\Large h\nu_d = h\nu(1 + \frac{a \Delta t}{c})$

Therefore in terms of the energy change of the photon

$\Large \Delta E = h\nu_d - h\nu$

we have

$\Large\Delta E = h \nu \frac{a \Delta t}{c}$

Using the relationship

$\Large E = p c$

we have

$\Large \Delta E = \Delta p \ c$

and thus

$\Large\frac{\Delta p}{\Delta t} = \frac{h \nu}{c^2} a$

Now in order to produce this rate of change of momentum we need to apply a force to the box

$\Large F = \frac{\Delta p}{\Delta t}$

Therefore we have

$\Large F = \frac{h \nu}{c^2} a$

Therefore the inertial mass of the photon is $h \nu/c^2$.

This seems to be an argument for the inertial mass of a confined photon using only the concept of Doppler shift and the energy/momentum relations for a photon.

This argument does seem to require that the photon box is moving.

What supplies the inertial reaction force *before* the box moves?

 

[minio]

Well, I might be wrong, but I would say, that there is no inertial reaction force unless the box is changing speed.

 

[johne1618]

Well, I might be wrong, but I would say, that there is no inertial reaction force unless the box is changing speed.

I think just as a particle has a rest mass/energy it also has a corresponding rest inertia defined by F = m a at zero velocity.

 

[minio]

I think just as a particle has a rest mass/energy it also has a corresponding rest inertia defined by F = m a at zero velocity.

But "a" is only about change of speed. At rest or at constant speed it would be 0, so no force...

 

[enroger0]

This argument does seem to require that the photon box is moving.

What supplies the inertial reaction force *before* the box moves?

That's exactly the calculation I did. As to what supplies the inertial force "before", you can think of this box simply as a box holding a bunch of bouncing balls inside. Before any of the balls hit the wall you won't have any reaction force too.

 

[minio]

Maybe stupid question, but if there is red/blue shift, wouldn't it be preserved after you stop applying force and thus stoping the box again?

 

[johne1618]

That's exactly the calculation I did. As to what supplies the inertial force "before", you can think of this box simply as a box holding a bunch of bouncing balls inside. Before any of the balls hit the wall you won't have any reaction force too.

I think the equation F = m a where m is a constant is only exactly true when the box is instantaneously at rest with zero velocity but with non-zero acceleration. I think the Doppler shift argument only works after the box has attained a non-zero velocity. I think we need to explain where the inertial reaction force comes from before the box moves. If you say before the box moves its inertial mass is zero then you are implying that as soon as the force is applied to the box it responds with an infinite acceleration.

 

[johne1618]

Maybe stupid question, but if there is red/blue shift, wouldn't it be preserved after you stop applying force and thus stoping the box again?

You only get a Doppler shift while the box is accelerating and thus providing a difference in velocity between the box and the confined photon. As soon as the box stops accelerating the Doppler shift disappears.

 

[enroger0]

I think we need to explain where the inertial reaction force comes from before the box moves. If you say before the box moves its inertial mass is zero then you are implying that as soon as the force is applied to the box it responds with an infinite acceleration.

Again, the analogy of a box with bouncing balls holds very well. If the mass of the box is very small, then indeed when force is applied you get a very large acceleration (lets not deal with infinities). Until one of the ball hits the box and transferred momentum and bounced the box back and forth.

This is a simple case of delayed reaction/inertia, the whole of the system has a total inertia of an certain amount, but since the members of the system don't interact instantaneously (photons still traveling, balls still moving between the walls...), inertial reaction don't come immediately too but at a delayed time. You can't just apply F=ma in this case since m isn't a rigid body.

 

[enroger0]

Maybe stupid question, but if there is red/blue shift, wouldn't it be preserved after you stop applying force and thus stoping the box again?

Lets consider this in detail:

consider a photon moving back and forth, it will impart equal (but opposite sign) momentum when reflected by both sides.

consider the case when a photon is moving towards -x direction after the box is accelerated, photon is blue shifted, thus when it hit the wall it will impart more momentum then before. But it is reflected as blue shifted photon, so when it hit the other side of the wall it will have the same blue shifted momentum and impart same momentum, reflected back and so forth as before and everything in balance again.

So Doppler shift only cause reaction force (change of momentum) when there is a change of velocity.

 

[bholt]

People talk about photons as if they are true particles with mass. A photon is a unit of light and is a wave that acts like a particle. Light is an electro-magnetic energy transfer from one object to another. Inertia is caused by an objects gravity field. Inertia and mutual attraction are the two functions of the gravity field. Inertia is the primary function and mutual attraction is the secondary function. Electro-magnetism is the inertia and gravity for charges. Inductance does the same thing with current as mass does with velocity. Mass and charges act the same way but operate at different scales.