A Test of the Copernican Principle

May 22, 2008 By Lisa Zyga, Phys.org feature
A Test of the Copernican Principle
This image shows a cross-section of a void universe with an observer (O) in the center, in violation of the Copernican principle. CMB photons (yellow lines) can scatter off reionized gas, and some may lead to CMB distortions. Credit: Caldwell, R. R. and Stebbins, A. ©2008 APS.

The Copernican principle states that the Earth is not the center of the universe, and that, as observers, we don’t occupy a special place. First stated by Copernicus in the 16th century, today the idea is wholly accepted by scientists, and is an assumed concept in many astronomical theories.

However, as physicists Robert Caldwell of Dartmouth College in Hanover, New Hampshire, and Albert Stebbins of Fermilab in Batavia, Illinois, point out, the Copernican principle has never been confirmed as a whole. In a recent paper published in Physical Review Letters called “A Test of the Copernican Principle,” the two researchers set out to prove the 500-year-old principle using observations of the cosmic microwave background (CMB).

“The Copernican principle is a cornerstone of most of astronomy, it is assumed without question, and plays an important role in many statistical tests for the viability of cosmological models,” Stebbins told PhysOrg.com. “It is also a necessary consequence of the stronger assumption of the Cosmological Principle: namely, that not only do we not live in a special part of the universe, but there are no special parts of the universe – everything is the same everywhere (up to statistical variation).

“It is a very handy principle, since it implies that here and now is the same as there and now, and here and then is the same as there and then. We do not have to look back in time at our current location to see how the universe was in our past – we can just look very far away, and given the large light travel time, we are looking at a distant part of the universe in the distant past. Given the Cosmological Principle, their past is the same as our past.”

Cosmic Distortion

When the universe was just 400,000 years old, matter and radiation decoupled and left a remnant radiation that still pervades the entire universe today. By measuring the tiny temperature fluctuations of this CMB radiation, scientists can learn things about the universe such as its shape, size, and rate of expansion. In the latter case, the observations show that the universe is expanding at an ever-accelerating rate, leading scientists to speculate about the existence of dark energy, new laws of gravity, and other possible – and often exotic – theories.

But what if the universe’s accelerating expansion is just an illusion? As Caldwell and Stebbins explained, this scenario is entirely plausible if the Copernican principle is loosened a bit. If, instead of the universe being homogenous and isotropic as the Cosmological Principle states, there is rather “a peculiar distribution of matter centered upon our location,” then the universe would be centered on a low-density, matter-dominated void. Such a universe would be non-accelerating, and there would be no need for dark energy or other similar theories.

That’s why it’s important to know if the Copernican principle is correct: it will ensure that CMB observations haven’t been misinterpreted to indicate cosmic acceleration when there is none. To test the principle, Caldwell and Stebbins developed a “CMB-distortion test”: they looked for deviations of the CMB spectrum from a perfect blackbody as might have been caused by a large, local void. A void or other “non-Copernican structure” would cause ionized gas to move relative to the CMB, and the Doppler-shifted CMB scattered toward us could contain detectable deviations from a blackbody.

“In essence, we use the reionized Universe as a mirror to look at ourselves in CMB light,” the researchers explained. “If we see ourselves in the mirror, it is because ours is a privileged location. If we see nothing [i.e. no peculiar distortions] in the mirror, then the Copernican principle is upheld.”

The Hubble Bubble

As an initial test, Caldwell and Stebbins focused on a universe model consisting of a simple, spherically symmetric void, which is also known as a “Hubble bubble.” This void universe resembles an open (low-density) universe embedded inside a flat (medium-density) universe. The size of the void depends on how gas is distributed throughout the universe. Basically, gas can exist in three zones – neutral, reflection, and Doppler – depending on its redshift. Depending on how these three zones overlap, the void can come in five sizes, from small to “superhorizon,” where the void encompasses the entire observable universe.

Using their CMB-distortion test, the researchers calculated that only the smaller void models could lead to the type of distortion associated with a violation of the Copernican principle. Then, by analyzing data for the CMB spectrum, they were able to rule out nearly all of these non-Copernican Hubble bubble void universes – meaning the Copernican principle passed this first test. However, Caldwell and Stebbins also noted that other models – such as those with a higher density or smaller radius – may still exist that evade this test.

The researchers added that this is not the first time that bits of the Copernican principle have been tested, but it is one of the first tests of the remaining radial inhomogeneity on very large scales. Caldwell explained that, in 1995, physicist Jeremy Goodman of Princeton proposed a similar test of spectral distortions. Goodman’s implementation resulted in a weaker constraint, or test, of the Copernican principle.

“This [large-scale testing] is not easy to do because, when we look far away, we are looking back in time, and it is difficult to say whether what we see is due to changes with time, which does not violate the Copernican principle, or changes with distance, which does,” Stebbins explained. “Thus, it is a hard question to answer, which is why it has not been done.”

More Tests

In the future, the scientists plan to further pinpoint the CMB distortions that could be caused by a local non-Copernican structure, and also apply the test to other more general universe models. These tests should be useful in potentially ruling out alternative hypotheses for dark energy, as Caldwell explained. More fundamentally, the tests could either verify the foundation of centuries of astronomical work, or – and the chance is slim – suggest that the Copernican principle may not be as certain as we think.

“If our test of the Copernican principle were to fail, it would probably not be believed, and a variety of other observations would be required to test it,” Stebbins said. “If all these further tests confirmed the large void, then we would have to rethink our ideas about dark energy, or, namely, unthink them.

“I think the scientific community would not be too unhappy with the idea of a large under-dense region – it is not hard to think of ideas of how they might come to be, even in the context of a hot big bang model. What is hard to understand is why we would be so close to the center of one. No doubt someone would come up with an ‘anthropic’ argument for it – but I've thought a bit about that, and don't really think there is a salable anthropic explanation. (By the way, I don't think there is a salable intelligent design reason, either.) In the end, we might have to live with the Walter Cronkite explanation ‘... and that's the way it is .... ’”

More information: Caldwell, R. R. and Stebbins, A. “A Test of the Copernican Principle.” Physical Review Letters 100, 191302 (2008).

Copyright 2008 PhysOrg.com.
All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.

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1.6 / 5 (7) May 22, 2008
Ok maybe a dumb question but when considering the amount of dark matter in the universe do we consider the weight of light? Every star or object we see is actually the center of an ever expanding sphere of light millions of light years across. The only part of that we see is the every passing wave of photons at any given second that intersects our own point. While any individual photon is of neglible mass billions of spheres (one for each radiant body that exists or has ever previously existed) each potentially millions of light years across constitute energy pushing us appart as a wave and mass pulling us back together as particles. All this we never see just because its just not moving our way.
3.9 / 5 (10) May 22, 2008
Photons are not of "negligible" mass. Photons by definition have NO mass, only energy, as dictated by the famous equation E = hf (energy equals Planck's constant times the frequency of the light).

Perhaps you are referring to the missing mass that was used to produce the photons? Only 0.7% of the original mass in stellar fusion is lost in the chemical process. I can guarantee you astronomers have factored this in to cosmology theory :)
2.5 / 5 (6) May 22, 2008
Not a dumb question at all Paul. The answer, at least as far as is currently known, is that light has no mass at all, and so is not a candidate for the "missing matter" in the universe. See http://en.wikiped...perties.
1.9 / 5 (14) May 22, 2008
Uh... E=hf, but also E=mc^2. Light has mass.

This is sort of basic...
3.2 / 5 (5) May 22, 2008
Has anyone ever looked into whether or not the universe is rotating itself and this force could explain some of the "Dark Matter" mystery? Complete novice here, so forgive my stupidy if this is completely void of possibility.
3.8 / 5 (10) May 22, 2008
Uh... E=hf, but also E=mc^2. Light has mass.

This is sort of basic...

No particle travelling at the speed of light can have mass as that would take an infinite amount of energy.
2.7 / 5 (7) May 22, 2008
Axemaster, according to Dr. Ken Mellendorf of Illinois Central College:

For most average objects, momentum is truly mass x velocity. When motion gets close to the speed of light, we find that the momentum relation p=mv is only an approximation. It is only correct when speed (v) is much smaller than the speed of light (c). The relation that works for all speeds is E^2 = p^2c^2 m^2c^4. It is much less convenient to use, and doesn't help figure anything out until you reach speeds of perhaps thirty million meters per second. For a particle with no mass, the relation reduces to E=pc. This works for a photon. For very small speeds, the system reduces to E=mc^2 (1/2)mv^2, and p=mv. This leads to relations with kinetic energy and momentum: much more convenient to work with and just as accurate until you reach speeds close to the speed of light.
2.6 / 5 (7) May 22, 2008
This is what happens when you have time at work I was originally looking at this from a slightly different perspective. While not to rain down on currently accepted scientific theory they are just that theories.

The problem we face is that what we observe is not explained under the theories we currently have. I see theories starting with dark (unobservable matter)matter and now dark (unobservable) energy. Our mathmatical system starts suffering irregularities when we hit/approach zero. Is it possible that the photon having freqency and spin maintains some neglible mass in singular but signigicant in vast quantity???

Alternatively suppose in space (excluding the mass from the trace gasses and stray atoms you were to instantainesouly enclose one cubic meter (or kilometer for that matter) with a perfectly absorbtive sphere such that all photons with in the space were perfectly absorbed such that we harvested all the energy contained in that space E=mc^2. Even at a trace amount there is alot of cubic volume to concider so thats alot of matter I assume? Does this mass exert gravitational attraction when converted to energy?

:) I know this is where I should have shut my mouth and only be thought a fool instead of proving it :)
3.5 / 5 (6) May 22, 2008
thales you forgot to mention that while a photon does not have mass, it does have momentum.
1.1 / 5 (7) May 22, 2008
Surely as the images of the stars fly towards our eyes there must be an equal and opposite reaction driving the stars away, unless that is balanced by the energy required to thrust the image towards us. Mind you I did learn my physics from a slightly out of date textbook.
3.3 / 5 (4) May 22, 2008
To allay the confusion, it's good to clearly define the terms (i.e. What is mass?).

The wikipedia article, "Mass in special relativity", is very revealing - http://en.wikiped...lativity

Here's a quote:
"The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass, but it is usually reserved for systems which consist of widely separated particles.

The term relativistic mass is also used, and this is the total quantity of energy in a body (divided by c2). The relativistic mass includes a contribution from the kinetic energy of the body, and is bigger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference.

Because the relativistic mass is just another name for the energy, it has gradually fallen into disuse."
2.4 / 5 (7) May 22, 2008
So in other words, you're not a fool to think that photons have mass. Defining mass in that way is useful, but it's not what played out. They do have inertia and gravitational effects are important (I'm assuming that this is the intuition that sometimes leads people to insist that they have 'mass', since this is what did it for me).. but relativity applies.
2.1 / 5 (8) May 22, 2008
actually, I don't think it's accurate to say they have inertia. I'll go hide now.
2.4 / 5 (5) May 22, 2008
ok, my last comment (sorry to come back, but it's for clarification). I can say that photons have 'relativistic inertia'. Again, it's due to relativistic effects, and I believe that they would have no inertia to an observer in their frame of reference.
3.9 / 5 (7) May 22, 2008
The momentum of the photons pushes a solar sail. An object does not need mass to have momentum (see thales' post, above). The energy of the photon is equivalent to a certain amount of mass (as described by the equation E=mc^2) and can be converted into mass under special circumstances, but any such conversion results in the annihilation of the photon. Photons themselves are indeed massless.
2.3 / 5 (4) May 22, 2008
Light has radiation pressure. The article assumes there is such a thing as a spacetime continuum. Einstein himself doubted this foundation of his work the year before he died. In a discontinuous universe a very different perspective necessarily emerges. See the article Gravity, Quantum Relativity and System 3 at www.cosmic-mindreach.com. There is a different explanation for the red shift of distant galaxies and also several candidtaes for the background radiation. (see Verschuur, G. L., High Galactic Lattitude Interstellar Neutral Hydrogen Structure and Associated (WMAP) High Frequency Continuum Emission, Astrophysical Journal, Dec. 10, 1997.)Light is emitted by atoms and defines external linear space relative to the inner spherical space defined by the ionization limit of the maximum photon energy level in the primary hydrogen atom. That is why E=mc^2. Otherwise there is no universal measuring rod in space. See Unified Theories, Fantasy, and Cosmic Order on the same website.
2 / 5 (5) May 22, 2008
Photons are massless under a certain accepted definition of mass. To avoid causing any laypeople needless confusion that can last for years, I think it's better to say publicly that they have no 'invariant mass'.. because photons do generate a gravitational field -- and this is something that people associate with mass (though it's not the accepted definition in the field). The explanation of how we have come to a definition of mass in which photons do not have it can come later.
2.6 / 5 (5) May 22, 2008
Common sense tells us that photons must have mass. But some quaint equations show that they don't. How can something without mass be affected by gravity? But, Einstein has a convenient explanation for that. Gravity warps the fabric of space and all the photons do is follow that wrinkle and hence appears bent. But, I am not able to get a reconcile this in my mind. Just goes to show that I am nowhere near as smart as the rest :(
2.8 / 5 (4) May 22, 2008
Question: Einstein refers to "mass" distorting space-time, thus providing a geometric model for gravity. Which "mass" does Einstein refer to?

I always understood the M in Einstein's relativity was the same as in E=MC^2 which suggests to me that photons, by virtue of their energy, have relativistic mass, and hence warp space... i.e. interact with gravity.

Does only "rest mass" warp space?

The original question by PaulLove asked if cosmologists have considered if the dark energy/mass that is "missing" is simply that contained in the starlight.

I've re-read above posts and still have not seen the answer. I agree the rest mass of the photon is zero, but none of the sunlight protons are at rest relative to us or the stars behind them.

Please... help us out here.

Ontheinternets quotes the definition of relativistic mass as including the kinetic energy of the object. This suggests that a hot earth distorts space a little more than a cold earth, giving the appearance of attracting a satellite a little stronger... the kinetic energy contributing to the relativistic mass.

Is this correct? Is this something that has been taken into account?

I'll quit with one more thought: Black Holes are suppose to be so gravitationally strong, photons cannot escape. This implies that photons indeed respond to gravity... and thus do have mass in that sense.

Well... maybe one more thought: as the photon escapes from an "almost" black hole, it travels at the speed of light, but as it climbs out of the energy well, it shifts from being an x-ray photon to being an infrared photon. Is this true? ...consider the gedanken experiment: as a massive red giant collapses into a black hole, what happens to the last few photons that escape? I've always imagined they turn deeper and deeper red, they become radio waves, then are snuffed out entirely. Is this correct?

I suspect the cosmologists have taken into account the mass of all the starlight... and the "red shift" due to gravity effects... and still have to to conjecture an expanding universe to explain the magnitude of the red shift and still have to conjecture "dark matter" to explain the changes in the rate of expansion.
2.8 / 5 (4) May 23, 2008
This argument about the Copernican principle reminds me of the luminiferous aether or ether. Too often we look for the mythical missing variable to make our equations add up, and tend to forget to question the assumptions we made along the way.
1.6 / 5 (7) May 23, 2008
In a discontinuous universe primary hyrogen atoms are synchronously projected as independent space-frames everywhere at once. Light is emitted from within atoms so it can only transmit a fixed distance in each primary interval of time defined by each synchronously projected space frame in the cosmic movie. Its speed is thus universal with respect to each independent atom, that is nevertheless universally coherent with all atoms in the integrated fabric of space-time. The latter is disctinct from the assumed spacetime continuum of general relativity. There is no evidence that such a thing as a continuum exists. Space and time are concepts derived from creation and it is hardly reasonable to raise them to a priori status to explain creation.

This means that the relative angular motion of galaxies involves the skipping of space-frames defined by light at their centers with respect to their peripheries, which at the limit results in black holes. There is a hole in the fabric of space-time because there is no light. Light can not bridge the quantum jumps in position associated with relative angular motions so some frames must be skipped at the center. It is this that curves the integrated fabric of space-time. Light travelling through this fabric must necessarily bend with it.

One can see how this works from the historic coordinates on the website mentioned above. There is an obvious derivation of the Lorentz Transformations from these coordinates that shows two complementary aspects to them and how gravity works accordingly.

Current approaches based on the assumed spacetime continuum as an independent "thing" that curves according to mass which is embedded in it are necessarily flawed. This discussion is about these contradictory assumptions in general relativity. If there really was a continuum then Zeno's arrow would never reach the target. Space and time are quantized in discrete increments and they are not infinitely divisible. There is a minimum limit to the differential in the calculus associated with the uncertainty principle. Exact position is known in a single frame. Momentum by its nature requires a succession of frames. Both can't be known accurately at once.

Planck's constant is a universal quantum of action associated with the projection of discrete space frames. Light comes to us as a series of pulses in each discret space frame. It takes a while for this to sink in.
2.2 / 5 (5) May 23, 2008
The question about the luminiferous aether can take on a new complexion because atomic space-frames alternate with timeless quantum frames that are holistically integrated as a boundless quantum energy field called the Void (distinct from the vacuum). Matter is both separate on the space frame side and integrated as one boundless energy field on the quantum frame side. Atomic matter is synchronously projected as series of frames as in a cosmic movie.

The Void by its nature is spatially indeterminate and so defies detection on the space-frame side. The Void is orthogonal to the integrated fabric of space-time. Each particulate atom has a conjugate quantum energy equivalent in the Void, and since the Void is timeless the space frames close ranks to present the appearance of continuity. The Void thus has certain characteristics of the ether.

Nevertheless there are irrational holes in space-time associated with the quantization of discrete frames of space-time. Particulate matter oscillates between space and quantum frames such that each oscillation defines one primary interval of time.

Matter is a wave and a particle at the same time. But its wave motion is only detectable at high relative motions attainable with subatomic particles. The Schroedinger wave equation requires a complex conjugate energy equivalent in order to correctly solve the equation when the wave function is said to collapse. It collapses into one final space-frame state that specifies specific conditions at that point in time. This state must take into account qunatum energy equivalents on the quantum side because they accumulate due to relative space frame skipping that accounts for relativistic efects.

The primary interval of time is confirmed by the fact that the orbital angular momentum of the electron in the first orbit of hydrogen is Zero. This means that the electron synchronously recurrs in the same place relative to the proton in each successive projection of the atomic space-frame. The primary interval is thus defined as 1.519x10^-16 seconds.
1.5 / 5 (4) May 23, 2008
Maybe everywhere is the center of the universe because the universe is a giant singularity.
2 / 5 (3) May 23, 2008
General relativity requires that there is no preferred frame of reference but this is not consistent with Mach's principle. For instance Foucault's pendulum is caused to swing by the earth's gravity and yet the direction of its swings is constant with respect to the fixed stars thousands of light years distant. The earth rotates under it. A gyrocompass works on this fact that there is a universally synchronous frame of reference in the universe as a whole. There must be a preponderance of synchronicity for us to perceive the universe as a whole.

This being the case in a discontinuous universe the Copernican assumption is too simplistic. The gravitational mass of the earth is synchronous with that of stellar population of the galaxy as a whole and yet its rotational inertial velocity is completely independent from its gravitational mass.
2.3 / 5 (6) May 23, 2008
Photons have mass, lol next they'll be saying that radio and sound waves have mass too.
4 / 5 (3) May 23, 2008
(my comment got cut off? I'll try to put it together again in time)

CWFlink - As far as I'm aware, current models (and there is more than just one model out there that reasonably matches observations) hold that gravitational fields are generated proportional to relativistic mass. So yes, a hot body would exert greater gravitational attraction than a similar colder body (however, the difference would likely be very slim -- I hope and trust that many physicists have bothered to calculate this, but it would never hurt for others to do it again).

Back to photons- I'm not a physicist, but I believe that the gravitational effect of a photon has never been measured/verified in a lab because current equipment is insufficiently sensitive. It's hard enough to detect the gravitational effect of a small mass (say, less than 1g), and the relativistic energy possessed by a passing photon is considerably less.

It's a bit embarrassing that most educated people will stubbornly hold that a photon does or does not possess mass - as if trying to prove exposure to the authoritative answer on the matter and/or win a trivia contest - rather than be concerned about what a photon does in the real world under various circumstances.
2 / 5 (4) May 23, 2008
dark matter is unecessary to explain the structure of the universe...
not rated yet May 23, 2008
As quantum mechanics got up and running the practice of physics became divorced from the interpretation of physics, however the Copenhagen interpretation became the default interpretation for experimental physics. Nevertheless the same empirical results became subject to a growing number of philosophical interpretations that attempt to model creation after the facts. But none of these interpretations allow of confirmation in phenomenal experience of any kind, not ever. No one will ever see a Big Bang, or probability waves, or exotic dark matter, or infinitesimal strings, or parallel universes or whatever. These interpretations don't really add anything positive to the practice of physics and they generally wander off into fantasy or mathematical obscurity which is the same thing. Einstein and other important founders of modern physics were disturbed by this divorce between practice and interpretation for this reason.

Paradoxically the interpretations available generally accept either implicitly or explicitly the spacetime continuum as fundamental. There is currently no interpretation generally recognized that explores all the ramifications of a discontinuous universe that at the same stroke explores the structural requirements implicit in the concept of universal wholeness.

Meaning derives from the integration of experience and physics seeks a unified theory. Every culture has had a theory of everything. We need a universal worldview to relate to. So there must be such a thing as universal wholeness, unless we are all hopelessly lost forever. But infinitesimal strings or probability waves don't offer much pragmatic direction to the human spirit in the challenges we face in just living.

The thing about a discontinuous universe is that it remarries the practice and interpretation of physics. It requires direct confirmation in phenomenal experience and it can expand the horizons of physics as well as the biological and social sciences. Unlike interpretations divorced from practice it offers a universal methodology that can complement traditional approaches to science in very practical ways.

So this alternative approach is not taken into consideration with the article's investigation of the Copernican Principle. Whatever these researchers find or conclude will be based upon a spacetime continuum that itself has not been confirmed in phenomenal experience and never can be.

As for the photon there are several aspects to consider in a discontinuous universe. Photons examined in experiments are distinct phenomena from the ubiquitous light that saturates our surroundings even in darkness and that defines space in the vast interstellar reaches as well as in our dark cellars. EM radiation reaches everywhere and it is quantized by the discontinuous but synchronous projection of the cosmic movie.

So specific photons produced by a source for experimental study are superimposed on this background. They are emitted in some way from within atoms, by some manipulation of atomic or sub-atomic processes. Within atoms photons define the spherical inner space of the atom in discrete levels consistent with de Broglies wave equation, so light that is emitted is always quantized accordingly across the whole range of the EM spectrum.

The point here is that it is light that implicitly defines mass within the atom, not vice-versa. It is the intimate internal relationship between photon, electron and proton within the atom as well as in the conjugate quantum frame that specifies what mass, space and time are. This is clear in a discontinuous universe and it is directly confirmed by our most fundamental laws which derive naturally from this interpretation.

This interpretation does allow of confirmation in phenomenal experience and no mass for a photon has ever been detected. Light energy is essentially matter in reflux through the space frame side of the movie, but on the space-frame side it does not manifest as mass, only as energy. Nevertheless in a discontinuous universe the neutrino produced in decay processes is essentially a photon remnant in particle form. There is speculation that it may have a very small mass.

One is entitled to have intuitive hunches about anything of course but they should not become blind beliefs as the current interpretations so often are. They must find confirmation in phenomenal experience in either the public or private domain. The advantage of the approach taken at www.cosmic-mindreach.com is that it exhausts all possible structural varieties to phenomenal experience. In this way it can provide valuable intuitive guidance regardless of the circumstance.
4 / 5 (2) May 24, 2008
On the subject of "what pushes a solar sail in space"?

"Photons" do not exist as particles. Neither the particle theory nor the wave theory of light is accurate.

A "Photon" is actually more an event than a particle, since you can never actually observe a photon "in motion". When a WAVE of light leaves point A strikes a surface at point B, a "photon" is said to strike the surface, but no such particle actually exists between point A and point B.

A solar sail works because, when a wave strikes the surface of the sail and experiences total reflection, Newton's laws state that conservation of momentum must be maintained. The wave of light then imparts its own momentum to the sail, and to balance this, it is reflected in exactly opposite direction with double the electromagnetic frequency. This causes the wave to have momentum double its original momentum and in the opposite direction, and the sail gains momentum equal to the light wave's original momentum, conservation is maintained. Theoretically anyway.
not rated yet May 24, 2008
Are you sure that the reflected photons have double the momentum?
1 / 5 (1) May 25, 2008
I'd just like to make a small correction to thales post about the relativistic relation: E^2 = p^2c^2 m^2c^4

That should be E^2 = c^2p^2 m^2c^4
5 / 5 (2) May 25, 2008

Hmm. actualy, I think I have it backwards. The sail gains twice the the reflected waves original momentum in order to conserve the total momentum of the system (wave sail).

If sail is at rest and wave strikes it, the systems momentum before the collision is "p".

Now p must remain the same at all times.

Before the collision, the wave has all of the momentum, so the momentum of the wave is w = p.

The momentum of the sail is s = 0.

Once again, the system's momentum p must remain the same at all times. that is:
p = s w
p' = s' w'

in this case, "w'" has sign opposite "w".

In the case of total reflection, the wave's entire momentum is transferred to the sail, causing the sail to gain momentum "after" the collision equal to twice that of the original wave.
p = w s
s = 0


p = w (if the sail started at rest)


s' = 2w
w' = -w

p' = s' w'


p' = 2w - w

p' = w = p

Ok, so i had it reversed. Sorry about this. The wave leaves with the same frequency and the sail gains double the momentum of the original wave.

2-1 = 1 so momentum is conserved.

That is total reflection.


Anything less than total reflection from the normal angle becomes inefficient.
not rated yet May 25, 2008
'Plus' signs apparently don't work in this comment system, so that makes it a bit more difficult to be sure what your comment is saying.

I didn't look at the equations too closely, but that seems more plausible (it matches my experience of mirrors - which are an example of something that we are still familiar with through everyday experience :)).

You say that the wave leaves with the same frequency. I think that is nearly correct. However, since the sail moves a bit, I would expect Doppler shift to slightly affect the reflected wave (decreasing frequency slightly). This would be part of ensuring that energy is conserved.

Imagine two mirrors opposite one another, and light bouncing back and forth between them -- As the light pushes the mirrors apart, energy has got to be lost somewhere, or else we would potentially have a perpetual motion device. I believe the loss would be in the energy of the photons via Doppler shift.
not rated yet May 25, 2008
I should clarify that this is from the mirror's perspective. The mirror would observe this shift if the light were to somehow come back, since as the light is reflected, it is moving more quickly away from the light. As for other observers.. I don't want to go there because it's confusing enough as it is.
not rated yet May 27, 2008
The sail gains twice the the reflected waves original momentum in order to conserve the total momentum of the system.

Thanks for the explanation Quantum, I think you hit it on the head. As other commenters have noted, light has momentum but not mass. The equation E^2 = c^2p^2 m^2c^4 implies that for E to equal mc^2, then c^2p^2 must equal 1. Thus p is quite small but not zero.

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