Given 2 clocks with a relative velocity that are spatially separated, we have an ambiguous situation.
If we "assume" the 2 clocks were synchronized when they passed each other, this alone says nothing about the rate of either clock because it is only 1 event.
Work must be done to one of the clocks to bring them back together. It is the clock that had no work done to it, that will have the longest elapsed time. Okay?
...You talk about objects traveling through the earths gravitational field with velocities relative to the center of mass of the earth but in fact we are much deeper into the sun's gravity field than we are the earth's. Shouldn't we use the sun as the center of gravity of of our "region of space"? I think your whole idea of such a "region of space" is incoherent.The whole point of a transform like the Galilean transform or the Lorentz transform is that it allows you to calculate how things look from any frame of reference at all. Until you define a mathematical WarpTech transform that defines your "region of space" and allows us to transform between between any region at all you have nothing. I think it will be more difficult than you think to come up with such a transform. But good luck.
Quote from: WarpTech on 12/16/2017 04:39 am It is the clock that had no work done to it, that will have the longest elapsed time. Okay?That is true of a completed journey where the clocks end up in the same place. But that isn't exactly new. Of course you have to accelerate in order to travel a different path through space/time. It is simple geometry.But the situation is still symmetric. Say you have two spaceships with twin one in ship one and twin two in ship two. Now say both twins crawl into a hibernation unit after which one of the ships accelerate to some velocity away. The twins awaken to see relative velocity between them but don't know which of them accelerated.
It is the clock that had no work done to it, that will have the longest elapsed time. Okay?
So what can they do to determine who has the real velocity and slow clock? After all they see each others clock as slow. If twin one travels back to twin two he will find that the twin one clock is slow. But if twin two travels back to twin one he will find that the twin two clock is slow. Exactly reciprocal.
Under special relativity neither clock can be said to be slower or faster than the other until they end up at the same place. Yes it is the acceleration of one or the other that causes one clock to be slower. But it isn't some physical effect of acceleration on the clock mechanism that causes a clock to slow. Ultimately the speed of the clock depends on the path it takes through space/time and acceleration changes that path. It is simple geometry....
I'm sorry if this was already mentioned somewhere in the previous pages, but I wonder if anyone here reads Sabine Hossenfelder's blog. ...
One reason I suspect there is no actual time travel involved in relativity, only an apparent time travel, is the behavior of current.Lets take a current loop in the lab frame. When a current is applied to a wire the charge picks up a very small velocity. If all the current around the loop actually Lorentz contracts representing some time travel aspect we will get some serious change in charge in the lab frame and a non-conservation of charge. The reason for this is because if a Lorentz contraction happens all around the loop we get a time helix of charge where the perimeter is larger than 2*pi*r.
One bit of advice: it is important to view knowledge as sort of a semantic tree — make sure you understand the fundamental principles, ie the trunk and big branches, before you get into the leaves/details or there is nothing for them to hang on to.
Quote from: dustinthewind on 12/17/2017 05:21 pmOne reason I suspect there is no actual time travel involved in relativity, only an apparent time travel, is the behavior of current.Lets take a current loop in the lab frame. When a current is applied to a wire the charge picks up a very small velocity. If all the current around the loop actually Lorentz contracts representing some time travel aspect we will get some serious change in charge in the lab frame and a non-conservation of charge. The reason for this is because if a Lorentz contraction happens all around the loop we get a time helix of charge where the perimeter is larger than 2*pi*r. This argument makes no sense.Time travel in special relativity doesn't happen because for time travel to happen, there needs to be FTL travel. In ordinary situations, things can't go FTL to begin with, so charges travelling in a loop will obviously say nothing about FTL.Lorentz contraction does not predict any change in the charges in the wire, they will all be uniformly distributed in the lab frame just like you would expect. I am not sure how to clear up whatever misunderstanding led you to your conclusion, because your description includes phrases like "time helix of charge" which simply don't mean anything.It is clear that you don't know what you are talking about here. You should also be able to realize this when you come to conclusions like "special relativity fails in this simple case," since scientists would have noticed that sometime in the last 100 years. Learning is good, so when you get to such a point showing your work and asking where you went wrong is good. Stating your work as if it was a fact is simply spreading ignorance, and has the potential to confuse other people who come by and don't have a strong physics background. It also reflects poorly on you, because instead of demonstrating a curiosity and desire to learn, it comes off as somewhat arrogant (I am having trouble finding the right word, it is basically like some form of self-centeredness where you either think you know a lot more than you do, or want to look like you know more than you do.)Please stop making posts like this, it is doing nobody any good. Sharing ideas you had is fine, but your post is full of incorrect statements where you should be asking questions.There is an interesting phenomenon called "hidden momentum" that happens in a current loop when fields cause the charges to move non-uniformly through the loop. I don't think you should bother reading the link I just posted though, at least not before you have spent some time learning basic relativity and intermediate electrodynamics, it simply wouldn't benefit you and you are likely to come away with more misunderstandings than learning.Quote from: Elon MuskOne bit of advice: it is important to view knowledge as sort of a semantic tree — make sure you understand the fundamental principles, ie the trunk and big branches, before you get into the leaves/details or there is nothing for them to hang on to.
Quote from: dustinthewind on 12/17/2017 07:44 pmI think your projecting your self onto me. This is just a personal attack. My post was explaining that you don't understand what you are talking about, and nicely asking you to respect the fact that scientists have been working with relativity for over 100 years, and it won't be overturned with a back of the envelope calculation.
I think your projecting your self onto me.
Quote from: dustinthewind on 12/17/2017 07:44 pmOne might surmise time travel is possible via traveling into the future using relativity. If they really think that then they also think an instant jump backwards while carrying their forward momentum will jump them backwards in time. The question is if there is any actual travel into some future time or not. Your statements here only serve to show that you do not have any understanding of relativity. It shows that you have not let go of your preconceptions, and therefore you are drawing conclusions that don't actually make sense. I won't explain in detail here in PM, especially because your response to my post indicates that you have no interest in learning.
One might surmise time travel is possible via traveling into the future using relativity. If they really think that then they also think an instant jump backwards while carrying their forward momentum will jump them backwards in time. The question is if there is any actual travel into some future time or not.
Quote from: dustinthewind on 12/17/2017 07:44 pmA time helix is the shortened length (pancaking) of charges because they exist via a gradient in time with in the loop, allowing more charge to exist in the loop. I am saying this seems to not be the case. "Colorless green dreams sleep furiously." Your words form perfectly valid English, but they don't communicate anything. "gradient in time" is another term you seem to have made up. (There are probably contexts where that makes sense, but this is not one.)
A time helix is the shortened length (pancaking) of charges because they exist via a gradient in time with in the loop, allowing more charge to exist in the loop. I am saying this seems to not be the case.
Quote from: dustinthewind on 12/17/2017 07:44 pmI suppose it's possible there is time travel for each individual charge not considering the entire perimeter of the loop. The time gradient exist on the charge with its forward velocity maybe causing more charge to exist perpendicular to its velocity but less forward and behind via some rotation of the charge out of our space/time (slowing its time) So for extra charge perpendicular to vector v we also have subtraction parallel to v resulting in a conserved quantity when integrating its flux. Leaving me unsure about actual time travel in that case. This paragraph is basically gibberish again. You are conflating time dilation (a real, measured effect) with time travel (something predicted to happen if and only if FTL exists)You are also talking about rotations through imaginary directions, rather than understanding that in the lab frame, the loop is some shape with constant density. In another frame, the loop will be distorted and have a different density. In any given frame all of the charge exists all of the time.Lorentz transforms tilt the coordinate axes, but they are not a rotation, because instead of moving in the same direction, the axes squeeze towards each other.Please stop pretending you know what you are talking about.
I suppose it's possible there is time travel for each individual charge not considering the entire perimeter of the loop. The time gradient exist on the charge with its forward velocity maybe causing more charge to exist perpendicular to its velocity but less forward and behind via some rotation of the charge out of our space/time (slowing its time) So for extra charge perpendicular to vector v we also have subtraction parallel to v resulting in a conserved quantity when integrating its flux. Leaving me unsure about actual time travel in that case.
Quote from: ppnl on 12/16/2017 11:22 amQuote from: WarpTech on 12/16/2017 04:39 am It is the clock that had no work done to it, that will have the longest elapsed time. Okay?That is true of a completed journey where the clocks end up in the same place. But that isn't exactly new. Of course you have to accelerate in order to travel a different path through space/time. It is simple geometry.But the situation is still symmetric. Say you have two spaceships with twin one in ship one and twin two in ship two. Now say both twins crawl into a hibernation unit after which one of the ships accelerate to some velocity away. The twins awaken to see relative velocity between them but don't know which of them accelerated.This is what I mean by "ambiguous". Just because they do not know which one accelerated does not mean that it didn't happen.
Quote from: ppnl on 12/16/2017 11:22 amSo what can they do to determine who has the real velocity and slow clock? After all they see each others clock as slow. If twin one travels back to twin two he will find that the twin one clock is slow. But if twin two travels back to twin one he will find that the twin two clock is slow. Exactly reciprocal. Say it was twin one who accelerated and twin two did not. Didn't this acceleration affect his clock? If twin two accelerates to catch up to him, then when they meet and are at rest wrt each other, both of their clocks are running slower than before they went into hibernation. Had twin one returned to twin two, then the rate of their clocks would be the same as it was before hibernation. The situation is not symmetrical, it only "appears" to be because the set-up of the problem is ambiguous.
Quote from: WarpTech on 12/14/2017 06:19 pmAdd a 3rd clock to step 1 whose energy is never changed by any kicks and it remains in the initial rest frame. Then at the end of Scenario 2, the two clocks that are in motion will be running slower than the 3rd clock. Whereas, at the end of Scenario 1, they are all three running at the same rate, but have elapsed different times during the experiment. See the difference?No, because, you can just have the third clock already moving with speed v instead. Your choice of relative speed for the third clock is arbitrary, and has no effect on anything. It is equivalent to picking the frame you are working in, having the clock moving at speed v to start with is what I did when I said "use the frame of clock 1(after the unnecessary step 1)" Everything therefore still remains unchanged.
Add a 3rd clock to step 1 whose energy is never changed by any kicks and it remains in the initial rest frame. Then at the end of Scenario 2, the two clocks that are in motion will be running slower than the 3rd clock. Whereas, at the end of Scenario 1, they are all three running at the same rate, but have elapsed different times during the experiment. See the difference?
Quote from: WarpTech on 12/17/2017 03:28 amQuote from: ppnl on 12/16/2017 11:22 amQuote from: WarpTech on 12/16/2017 04:39 am It is the clock that had no work done to it, that will have the longest elapsed time. Okay?That is true of a completed journey where the clocks end up in the same place. But that isn't exactly new. Of course you have to accelerate in order to travel a different path through space/time. It is simple geometry.But the situation is still symmetric. Say you have two spaceships with twin one in ship one and twin two in ship two. Now say both twins crawl into a hibernation unit after which one of the ships accelerate to some velocity away. The twins awaken to see relative velocity between them but don't know which of them accelerated.This is what I mean by "ambiguous". Just because they do not know which one accelerated does not mean that it didn't happen.It doesn't matter if it happened or not. The only thing that matters is that at the start there is a relative velocity between the 2 clocks. You cannot define an experiment that can tell which one accelerated, because there is literally nothing physically different between them.Quote from: WarpTech on 12/17/2017 03:28 amQuote from: ppnl on 12/16/2017 11:22 amSo what can they do to determine who has the real velocity and slow clock? After all they see each others clock as slow. If twin one travels back to twin two he will find that the twin one clock is slow. But if twin two travels back to twin one he will find that the twin two clock is slow. Exactly reciprocal. Say it was twin one who accelerated and twin two did not. Didn't this acceleration affect his clock? If twin two accelerates to catch up to him, then when they meet and are at rest wrt each other, both of their clocks are running slower than before they went into hibernation. Had twin one returned to twin two, then the rate of their clocks would be the same as it was before hibernation. The situation is not symmetrical, it only "appears" to be because the set-up of the problem is ambiguous.The bolded statements are false. You are stating that in one case their clocks are both ticking "slower" at the end and in the other they are both ticking at the "original rate" but there is actually no way to tell these situations apart, ....
Theoretically, the clock that was accelerated will have a higher temperature relative to the CMB temperature, because when it was accelerated, provided it was accelerated long enough to achieve equilibrium with the vacuum, the Unruh radiation will have elevated its temperature. So there is an experiment that can be done to tell which one was accelerated. It just has to be extremely sensitive.
Quote from: meberbs on 12/17/2017 09:32 pmQuote from: dustinthewind on 12/17/2017 07:44 pmI think your projecting your self onto me. This is just a personal attack. My post was explaining that you don't understand what you are talking about, and nicely asking you to respect the fact that scientists have been working with relativity for over 100 years, and it won't be overturned with a back of the envelope calculation.I wasn't taking your earlier comment about me seriously and was instead batting your comment back at you in jest.
I made no claims to have overturned relativity. The calculation and discussion illustrates that Lorentz contraction doesn't happen for current accelerated in the lab frame which agrees with Purcell in his book.
Quote from: meberbs link=action=profile;u=48207 date=1513549948Quote from: dustinthewind on 12/17/2017 07:44 pmOne might surmise time travel is possible via traveling into the future using relativity. If they really think that then they also think an instant jump backwards while carrying their forward momentum will jump them backwards in time. The question is if there is any actual travel into some future time or not. Your statements here only serve to show that you do not have any understanding of relativity. It shows that you have not let go of your preconceptions, and therefore you are drawing conclusions that don't actually make sense. I won't explain in detail here in PM, especially because your response to my post indicates that you have no interest in learning.It's your prerogative to presume that.
Quote from: meberbs link=action=profile;u=48207 date=1513549948Quote from: dustinthewind on 12/17/2017 07:44 pmA time helix is the shortened length (pancaking) of charges because they exist via a gradient in time with in the loop, allowing more charge to exist in the loop. I am saying this seems to not be the case. "Colorless green dreams sleep furiously." Your words form perfectly valid English, but they don't communicate anything. "gradient in time" is another term you seem to have made up. (There are probably contexts where that makes sense, but this is not one.)Some good questions here. A gradient in time is a slope in time.
A slope in time is what happens when you see an object with relativistic velocity. Such a length contracted object appears to exist in a sloped time.
a form of time travel but it always forces one into the future via the forwards direction. I suppose if the time travel is real then instantly jumping back while retaining ones forward velocity would send one into the past but this seems unlikely to be possible.
Quote from: meberbs link=action=profile;u=48207 date=1513549948Quote from: dustinthewind on 12/17/2017 07:44 pmI suppose it's possible there is time travel for each individual charge not considering the entire perimeter of the loop. The time gradient exist on the charge with its forward velocity maybe causing more charge to exist perpendicular to its velocity but less forward and behind via some rotation of the charge out of our space/time (slowing its time) So for extra charge perpendicular to vector v we also have subtraction parallel to v resulting in a conserved quantity when integrating its flux. Leaving me unsure about actual time travel in that case. This paragraph is basically gibberish again. You are conflating time dilation (a real, measured effect) with time travel (something predicted to happen if and only if FTL exists)You are also talking about rotations through imaginary directions, rather than understanding that in the lab frame, the loop is some shape with constant density. In another frame, the loop will be distorted and have a different density. In any given frame all of the charge exists all of the time.Lorentz transforms tilt the coordinate axes, but they are not a rotation, because instead of moving in the same direction, the axes squeeze towards each other.Please stop pretending you know what you are talking about.I was poking at the reason for electric field pancaking in the lab frame via the charge with velocity still having charge conserved and no Lorentz contraction between the charges. This seems to allow a time gradient to exist on the individual charges, allowing time travel on an individual charge level around the circular path and preserving charge. I was poking if it were possible no actual time travel is involved and instead just the local clock was being modified, while an instant jump would still keep one stuck in the now (non time sloped) universe. This would restrict an instant jump to the proper time plane of the universe such as jump P to Q below. An instant jump where actual time travel is taking place might make it possible to jump forward or backward in time such as jump Q to R. The question being if any such jumps are allowed, can they move backward and forward in time or not.
I resolve resolutely not to let paradox push me around. Instead i suggest showing paradox who's boss.It is entirely possible that the appearance of paradox is itself a indicator that something is not understood enough; that something has been overlooked. And it may well be that that thing is in the picture of how things work in "established science." Then the paradox is a mirage that fades as you approach it. It does not always have to accrue to an experimenter or observer of something weird. I would be surprised if finding something new didn't in someway tilt established science at least to a small degree. new doesn't mean old.
And with FTL even though i knew reference frames were a bear i had no idea i was dealing with a mega-fauna version of a bear.People very very often reverse the reference frames for the effects of time dilation for a traveler traveling to a distal star from earth. Simple things like who ages and who doesn't or thinking that both do or don't. i spend pages and days arguing that over and over with lots of people who should know better. And that is one of the easiest bits WRT relativity to understand.
Quote from: Stormbringer on 12/26/2017 12:48 amI resolve resolutely not to let paradox push me around. Instead i suggest showing paradox who's boss.It is entirely possible that the appearance of paradox is itself a indicator that something is not understood enough; that something has been overlooked. And it may well be that that thing is in the picture of how things work in "established science." Then the paradox is a mirage that fades as you approach it. It does not always have to accrue to an experimenter or observer of something weird. I would be surprised if finding something new didn't in someway tilt established science at least to a small degree. new doesn't mean old.I agree, many paradoxes are not paradoxes the "twin paradox" is a good example, it is only a paradox if you don't think it through completely.Quote from: Stormbringer on 12/26/2017 12:48 amAnd with FTL even though i knew reference frames were a bear i had no idea i was dealing with a mega-fauna version of a bear.People very very often reverse the reference frames for the effects of time dilation for a traveler traveling to a distal star from earth. Simple things like who ages and who doesn't or thinking that both do or don't. i spend pages and days arguing that over and over with lots of people who should know better. And that is one of the easiest bits WRT relativity to understand.I am having trouble telling if you understand it yourself. There simply isn't a meaningful answer unless they come back together. When they are separated neither can be determined as "older or younger," which is the most basic part of special relativity, that no frame is special, and different frames don't agree on what "simultaneous" means for spatially separated events. (This is the most fundamental part of relativity, but not the simplest, since it seems to be the root of the most confusion.)
A third assumption is that there exists a reference frame in which matter is at rest: thisassumption is based on what Einstein terms the “most important fact we draw fromexperience as to the distribution of matter”, the low velocities of the stars:QuoteThe most important fact that we draw from experience as to the distribution of matteris that the relative velocities of the stars are very small as compared with the velocity oflight. So I think that for the present we may base our reasoning upon the followingapproximate assumption. There is a system of reference relatively to which matter maybe looked upon as being permanently at rest.Einstein then embarks on a simple analysis in which he derives values for the componentsof the field equation tensors...Thus Einstein’s assumption that “there is a system of reference relative to which matter may be looked upon as being permanently at rest” seems reasonable. I
The most important fact that we draw from experience as to the distribution of matteris that the relative velocities of the stars are very small as compared with the velocity oflight. So I think that for the present we may base our reasoning upon the followingapproximate assumption. There is a system of reference relatively to which matter maybe looked upon as being permanently at rest.
We analyze the data on the comparison of clock rates between a ∞ying clock and a clock at ground, carried out by Alley and coworkers at the end of '70's .The flt to such data is in favour of an energy-dependent metric for gravitation. We discuss also the results of a recently proposed electromagnetic test of breakdown of local Lorentz invariance - based on the detection of a voltage induced by a stationary magnetic fleld - and show that the obtained positive evidence for such an efiect seems to support the derived form of the energy-dependent gravitational metric.
A test I believe might indicate a relative velocity with respect to such a frame would be to measure the static dipole electric field of a magnetic field. In the lab frame. If there is no difference in clock rates for the individual charges in the lab frame no dipole electric field should form in the lab frame. However if for some reason the clock of a charge sped up moving in some particular direction in the lab, as opposed to another direction, there should be a corresponding dipole electric field to go with it. Such an experiment with sufficient sensitivity, I believe could determine velocity relative to a local field that determines the clock rate. I suspect existing in a gravity field may induce some effect but I would be curious of it's exact orientation of the lab frame dipole electric field. Parallel or perpendicular to Earth's surface.
Quote from: Einstein’s 1917 Static Model of the Universe: A Centennial ReviewQuoteapproximate assumption
Quoteapproximate assumption
approximate assumption
Indeed, many years were to elapse before the discovery of a linear relation between the recession of the distant galaxies and their distance (Hubble 1929), the first evidence for a non-static universe.
There also appears to be an experiment in which a test of the dipole electric field of a solenoid in the lab frame did turn up some interesting results. Also had some unusual readings in the lab frame with a solenoid and very low capacitance capacitor myself but didn't pursue it at the time.
Quote from: dustinthewind on 01/01/2018 05:35 amQuote from: Einstein’s 1917 Static Model of the Universe: A Centennial ReviewQuoteapproximate assumptionAs Einstein stated, it was an approximate assumption made while deriving a specific solution to his equations. It allows for picking a frame that makes the math easier, but it does not make that frame "special" as you are presenting it. Also if you actually bothered to read the paper you referenced, you would see that his assumptions were incorrect for our universe, since the universe is expanding. In fact the last sentence you quoted was immediately preceded by:Quote from: Einstein’s 1917 Static Model of the Universe: A Centennial ReviewIndeed, many years were to elapse before the discovery of a linear relation between the recession of the distant galaxies and their distance (Hubble 1929), the first evidence for a non-static universe.The entire paragraph it was a part of was explaining that our modern knowledge that his assumptions do not describe our universe was not available to Einstein, so his assumptions were reasonable from his perspective. By pulling that sentence out of context you completely changed its meaning.
Einstein soon found that the hypothesis of closed spatial geometry was not sufficient to achieve a successful relativistic model of the universe. A consistent solution could only be achieved with the introduction ofan additional term 𝜆𝑔𝜇𝜈 to the field equations, where 𝜆 represented a constant that later became known as the ‘cosmological constant’. Thus Einstein’s model appears to have evolved according to the following sequence of assumptions: uniform, static distribution of matter → closed spatial geometry → introduction of additional term to the field equations. ...“That term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars”...... Einstein gave his first physical interpretation of the cosmological term, namely that of a negative mass density: “In terms of the Newtonian theory…a modification of the theory is required such that “empty space” takes the role of gravitating negative masses which are distributed all over the interstellar space” (Einstein 1918c)...However, the error may be significant with regard to Einstein’s interpretation of the term. Where he intended to introduce a term to the field equations representing an attenuation of the gravitational interaction at large distances, he in fact introduced a term representing a very different effect. Indeed, the later interpretation of the cosmological term as representing a tendency for empty space to expand would have been deeply problematic for Einstein in 1917, given his understanding of Mach’s Principle at the time....“It appears to me that one can raise a grave argument against the admissibility of this solution…..In my opinion, the general theory of relativity is a satisfying system only if it shows that the physical qualities of space are completely determined by matter alone. Therefore no 𝑔𝜇𝜈- field must exist (that is no space-time continuum is possible) without matter that generates it” (Einstein 1918f).
Quote from: dustinthewind on 01/01/2018 05:35 amThere also appears to be an experiment in which a test of the dipole electric field of a solenoid in the lab frame did turn up some interesting results. Also had some unusual readings in the lab frame with a solenoid and very low capacitance capacitor myself but didn't pursue it at the time. Their paper seems to be discussing a distorted metric in the presence of gravity and reconciling this with certain experiments. We already know about this, it is called general relativity, so it is strange that they repeatedly insist on comparing to Minkowski space-time, when it is known that gravitational potential causes time dilation.As for the circuit, I would find it somewhat surprising, but it is possible they found a case where the difference in gravitational potential energy could affect the circuit. It isn't worth it for me to go through since there are multiple indicators that they are crackpots, and the general idea was already worked out in detail 100 years ago.As for your experiment, it is really, and truly irrelevant. Actually, none of what you just posted is relevant to this thread, especially since it is a combination of misinterpretation and meaningless claims.
However—at the light of the discussion of Sec. 5.2—the parameters (23) and (29) have to be regarded actually as the result of two contributions to the LLI breaking effect, one of gravitational ......Needless to say, the experiment must be independently repeated,before one can claim that a LLI breakdown was actually observed.We attempted an explanation of the results we found in terms of thephenomenological metrics of electromagnetic and gravitational interactionsderived in the framework of DSR. If this interpretation is sound, we canstate that the LLI violation is expected to occur below an energy thresholdE0,e.m. ≈5µeV. In other words, contrarily to what commonly believed, LLIbreakdown (at least for electromagnetic interaction) would occur at low,not at high, energies. This would explain why it was not observed before,and implies that LLI violation would not affect dramatically most of thephenomena in high-energy physics, astrophysics and cosmology.Finally, we expect that the effect of LLI breakdown we seeminglyobserved may affect some photon interference results, in particular thephoton–photon cross section. This seems indeed to be the case.(33)