grav
03-September-2006, 01:21 AM
In the Light scattered by electrons (http://www.bautforum.com/showthread.php?t=45974) thread, an interesting discussion was brought up about the relation between the black body curve and the Maxwell-Boltzmann distribution of velocities. It is ironic that that these two are exactly what I have been attempting to relate together for over a week and a half now. I have used up a notebook's worth of paper so far and am still only getting started. But I will post what I have so far and see if those that posted in the mentioned thread or others might have any further observations or criticisms to offer.
I won't go into too much detail here, but I basically began by relating the number of photons per volume for a given bandwidth of wavelengths to the total number per volume over all wavelengths. I then set that equal to the Maxwell-Boltzmann distribution of velocities over a very small range of velocities. By considering light to be made up of particles of equal mass regardless of frequency, I ran into dead end after dead end. Finally, I tried the ratio n=hf/mv2, where m is the mass of a particle in a particle medium through which light is transported as a wave, and n is the number of these particles per wavelength.
If we then square both formulas and set them equal, we get a relation of the distribution of frequency to their energy. If x=hf/kT and y=E/kT, where E=n*mv2, the relation becomes y4/(ey)=x5/(ex-1). For small f and therefore E, we can see that this works out, where y=x, or E=hf. But for large f and E (f>kT/h), I almost gave up since it didn't seem it should work out for even the frequencies we usually observe. But after doing the math, I find that for very large f and E the relationship is also about the same. In fact, the only place it really differs much (by about twenty percent) is at and around f=kT/h.
Since I originally did this to find the dispersion velocities through the neutrino medium, I can now find the velocities of light at any particular frequency this way. Since y=E/kT=n*mv2/kT, then y/x=(E/kT)/(hf/kT)=E/hf=(n*mv2/hf=(n*m)v2/hf=(hf/c2)v2/hf=(v/c)2, whereas the ratio of the velocity to the speed of light will vary at around f=kt/h by as much as ten percent. I find that the velocities start off at the speed of light at low f and slowly but steadily rise higher than c until it suddenly dips below c at around f=kT/h, and then begins to rise again until it is once again equal to c at the high end where f is infinity. The difference is extremely slight except for where f=kT/h, where there is a rise and then a quick fall in velocity, like a tiny singular sine wave pulse on an otherwise steady linear graph. The velocity rises at the low end of the frequency spectrum with about v=c[1+(h/kT)*f/16] and the relationship at the large end is about v=c[1-8(kT/h)/f].
Since this progression is not linear, it makes for a complicated matter when it comes to dispersion, especially if the light is redshifted during transit. I will work on that. However, this also demonstrates the mass of the particles involved in the propogation of light. In this case, the mass would simply be m=kT/c2. I have not worked out the details yet, so this may actually include some small variable, but it should still be around that order of magnitude. As it turns out, this is way off the mark for what I wrote in my paper for the mass of the neutrinos or neutrino-like particles that produce gravity and redshift. The last part of the paper demonstrated that if the redshift is the energy lost by a single neutrino collision per wavelength (since hH would be the energy lost per wavelength for tired light), then the mass of a neutrino is m=hH/c2. I now believe this may be incorrect, and that m=kT/c2 may be the mass of a neutrino in free space, where T is the average temperature of free space. If this is the case, mneu=kT/c2=4.191324046*10-40, for an energy of 2.35*10-4eV.
I have also always wondered why the light intensity is found by multiplying the light pressure by (c/4) instead of just c. I figured it must just have something to do with the ratio of the surface area of a sphere to its cross-sectional area. But I think it might be noteworthy to state that I found that by dividing the pressure of light by the photon number per area, instead of just finding for PA directly, we find a difference of a multiple of 2pi2/5=3.94784176. So if the pressure (or intensity) is really meant to be found with I=P(c/3.94784176), then the temperature would then vary by a factor of (4/3.94784176)1/4=1.00328672.
In addition, I am quickly coming to the conclusion that light is purely a wave. The velocities found were for those particles that actually transmit the light. Light itself, however, demonstrates all of the properties associated with waves. The only property one might say doesn't is the E=hf relationship. It been shown that this can be quantized into "packets" of energy of E=n*hf, where n is a whole number fraction. But waves are also known to demonstrate this property as well, where only a whole number fraction of waves will be found between two oscillation points. Also, what does hf mean if it is not associated with waves, or hc/w for that matter? f and w are both properties of waves. If f is the frequency of successive photons, then how can it be used to find the energy of an indivual photon particle? If, however, we consider a photon to be the oscillation of a single wave, from wavecrest to wavecrest, then hf2 is the energy per unit time carried by the wave. We can see this by dividing by the speed of light, which gives us the energy per distance along the wave as E/d=hf2/c. Then multiplying by a single wavelength, we get an energy per wavelength of E=(hf2)w/c=(hf)(fw)/c=hf.
I will continue to work on this. I know I didn't go into much detail here, so any questions are certainly welcome. Just know that I am only beginning to explore these relationships, and my knowledge about it is far from complete. Any and all help or constructive critism (even negative criticism is constructive at this point) is immensely appreciated.
I won't go into too much detail here, but I basically began by relating the number of photons per volume for a given bandwidth of wavelengths to the total number per volume over all wavelengths. I then set that equal to the Maxwell-Boltzmann distribution of velocities over a very small range of velocities. By considering light to be made up of particles of equal mass regardless of frequency, I ran into dead end after dead end. Finally, I tried the ratio n=hf/mv2, where m is the mass of a particle in a particle medium through which light is transported as a wave, and n is the number of these particles per wavelength.
If we then square both formulas and set them equal, we get a relation of the distribution of frequency to their energy. If x=hf/kT and y=E/kT, where E=n*mv2, the relation becomes y4/(ey)=x5/(ex-1). For small f and therefore E, we can see that this works out, where y=x, or E=hf. But for large f and E (f>kT/h), I almost gave up since it didn't seem it should work out for even the frequencies we usually observe. But after doing the math, I find that for very large f and E the relationship is also about the same. In fact, the only place it really differs much (by about twenty percent) is at and around f=kT/h.
Since I originally did this to find the dispersion velocities through the neutrino medium, I can now find the velocities of light at any particular frequency this way. Since y=E/kT=n*mv2/kT, then y/x=(E/kT)/(hf/kT)=E/hf=(n*mv2/hf=(n*m)v2/hf=(hf/c2)v2/hf=(v/c)2, whereas the ratio of the velocity to the speed of light will vary at around f=kt/h by as much as ten percent. I find that the velocities start off at the speed of light at low f and slowly but steadily rise higher than c until it suddenly dips below c at around f=kT/h, and then begins to rise again until it is once again equal to c at the high end where f is infinity. The difference is extremely slight except for where f=kT/h, where there is a rise and then a quick fall in velocity, like a tiny singular sine wave pulse on an otherwise steady linear graph. The velocity rises at the low end of the frequency spectrum with about v=c[1+(h/kT)*f/16] and the relationship at the large end is about v=c[1-8(kT/h)/f].
Since this progression is not linear, it makes for a complicated matter when it comes to dispersion, especially if the light is redshifted during transit. I will work on that. However, this also demonstrates the mass of the particles involved in the propogation of light. In this case, the mass would simply be m=kT/c2. I have not worked out the details yet, so this may actually include some small variable, but it should still be around that order of magnitude. As it turns out, this is way off the mark for what I wrote in my paper for the mass of the neutrinos or neutrino-like particles that produce gravity and redshift. The last part of the paper demonstrated that if the redshift is the energy lost by a single neutrino collision per wavelength (since hH would be the energy lost per wavelength for tired light), then the mass of a neutrino is m=hH/c2. I now believe this may be incorrect, and that m=kT/c2 may be the mass of a neutrino in free space, where T is the average temperature of free space. If this is the case, mneu=kT/c2=4.191324046*10-40, for an energy of 2.35*10-4eV.
I have also always wondered why the light intensity is found by multiplying the light pressure by (c/4) instead of just c. I figured it must just have something to do with the ratio of the surface area of a sphere to its cross-sectional area. But I think it might be noteworthy to state that I found that by dividing the pressure of light by the photon number per area, instead of just finding for PA directly, we find a difference of a multiple of 2pi2/5=3.94784176. So if the pressure (or intensity) is really meant to be found with I=P(c/3.94784176), then the temperature would then vary by a factor of (4/3.94784176)1/4=1.00328672.
In addition, I am quickly coming to the conclusion that light is purely a wave. The velocities found were for those particles that actually transmit the light. Light itself, however, demonstrates all of the properties associated with waves. The only property one might say doesn't is the E=hf relationship. It been shown that this can be quantized into "packets" of energy of E=n*hf, where n is a whole number fraction. But waves are also known to demonstrate this property as well, where only a whole number fraction of waves will be found between two oscillation points. Also, what does hf mean if it is not associated with waves, or hc/w for that matter? f and w are both properties of waves. If f is the frequency of successive photons, then how can it be used to find the energy of an indivual photon particle? If, however, we consider a photon to be the oscillation of a single wave, from wavecrest to wavecrest, then hf2 is the energy per unit time carried by the wave. We can see this by dividing by the speed of light, which gives us the energy per distance along the wave as E/d=hf2/c. Then multiplying by a single wavelength, we get an energy per wavelength of E=(hf2)w/c=(hf)(fw)/c=hf.
I will continue to work on this. I know I didn't go into much detail here, so any questions are certainly welcome. Just know that I am only beginning to explore these relationships, and my knowledge about it is far from complete. Any and all help or constructive critism (even negative criticism is constructive at this point) is immensely appreciated.