# Can neutrinos travel Faster Than Light **URL:** \ **Category:** Factual Questions\ **Created:** [September 22, 2011, 10:59pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266 "2011-09-22T22:59:06Z")\ **Posts on this page:** 20\ **Page:** 20
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 18, 2011, 8:56pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/381 "2011-11-18T20:56:25Z")
First of all, the curvature of space due to the intervening rock would be negligible, even with the level of precision they had. Second, even if they did have enough precision to detect it, the effect would have the opposite sign: It should appear to slow down the neutrinos, not speed them up. ---
**Author:** ![Dob](https://avatars.discourse-cdn.com/v4/letter/d/b2d939/32.png) [@Dob](https://boards.straightdope.com/u/Dob)\ **Post date:** [November 18, 2011, 9:38pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/382 "2011-11-18T21:38:30Z")
If we take the results as verified and tested, meaning we have observed FTL travel… what does that mean? How does this shape our understanding of the universe from that moment on? It doesn’t throw out Einstien I wouldn’t think, right?? ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [November 18, 2011, 10:08pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/383 "2011-11-18T22:08:43Z")
> [@Dob](#): > > If we take the results as verified and tested, meaning we have observed FTL travel… what does that mean? How does this shape our understanding of the universe from that moment on? It doesn’t throw out Einstien I wouldn’t think, right?? It throws out Einstein the same way that Einstein threw out Newton – yes, but not really. Any new theory has to explain why the old theory was successful. If relativity is wrong, it is wrong in a subtle way that one could be forgiven for not noticing in a century. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 18, 2011, 10:36pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/384 "2011-11-18T22:36:02Z")
Again, even that not really. There are various “extensions” of standard theory that allow for this and even without such Einstein’s math merely disallows a particle with mass going _at_ the speed of light; it does not disallow particles that are _always_ faster. IF the neutrinos created include two populations popping into existence, one subliminal and one supraliminal of luminal velocity, neither ever crossing the threshold from either direction, that would be kosher. (See “tachyons”.) Question to our resident experts, that I don’t think was already answered - neutrinos are modelled as changing flavors. One hypothesized flavor is the sterile neutrino that does not interact with matter at all (a flavor hypothesized to be relatively quite heavy), could there be a flavor that is _massless_, that goes the speed of light, and which switches to a flavor of mass on the other side? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 19, 2011, 2:38am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/385 "2011-11-19T02:38:41Z")
One (and only one) massless flavor is, so far as I know, consistent with all observations: We have upper bounds on the masses, and we have values for the mass splittings, but we don’t have any minimum mass for the lightest one. I think the hypothetical high-mass sterile neutrino is out of vogue now, though. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 19, 2011, 3:09am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/386 "2011-11-19T03:09:11Z")
Well enough in vogue that it was recently [featured in _Science_](http://www.sciencemag.org/content/334/6054/304.summary)anyway, which how I know about it. > [@](#): > > … Recently, however, the case for sterile neutrinos has grown stronger, bolstered by a new analysis of data from nuclear reactors. So last month 60 physicists from around the world gathered to hash out the arguments for and against the existence of sterile neutrinos and to try to decide whether it’s worth staging a dedicated experiment to settle the matter. From beyond the wall: > [@](#): > > … The strongest experimental evidence for sterile neutrinos comes from the Liquid Scintillator Neutrino Detector (LSND), which ran at Los Alamos National Laboratory in New Mexico from 1993 through 1998. Using a particle accelerator, physicists generated muon antineutrinos that streamed through a detector filled with 167 tons of mineral oil. > > Those low-energy muon antineutrinos should have passed right through. However, an electron antineutrino could interact with the detector by merging with a proton to create a positron and a neutron—essentially, the weak decay of the neutron run backward. Thus, physicists could spot electron antineutrinos appearing in a beam of muon antineutrinos. And they spotted 88 of them, give or take 23. “Lo and behold, we saw an excess of events,” says Los Alamos’s William Louis. > > Reported in 1996 and 2001, the LSND results might seem to show muon antineutrinos mixing into electron antineutrinos. But it couldn’t be that simple, Louis says. Different flavors of neutrinos mix at a rate that depends on the difference in their masses: The bigger the mass difference, the faster the mixing. Studies of atmospheric and solar neutrinos had placed limits on the mass differences among the three neutrino flavors, and the values were too low to explain the lickety-split mixing that LSND saw as the particles flew just 30 meters, Louis says. > > LSND researchers could explain their results, however, if muon antineutrinos oscillated first into sterile antineutrinos and then into electron antineutrinos. The sterile neutrinos would have to be heavier than ordinary neutrinos by 1 electron volt—about 100 times the differences among ordinary neutrinos. > > Hints of extra neutrinos also come from the heavens. For example, cosmologists think the universe burst into existence in the big bang as an ultrahot, ultradense soup of particles. Tiny fluctuations in the density of the soup then stretched to immense proportions during a faster-than-light growth spurt known as inflation and seeded the formation of galaxies. The fluctuations also limit the number of neutrino types, theorist Kevork Abazajian of the University of California, Irvine, said at the meeting. > > The density fluctuations can be thought of as waves of various wavelengths randomly piled on one another. The fluctuations’ intensity and gravitational pull grow stronger as their wavelengths decreases; then the intensity peaks at a certain wavelength and starts to fall again. The position of that peak in a graph of intensity versus wavelength depends on the relative amounts of radiation and matter in the early universe. And because lightweight neutrinos acted like an additional form of radiation, the position of the peak also reveals the number of types of light neutrinos. > > To deduce the distribution and the peak in it, scientists measure tiny variations in the afterglow of the big bang—the cosmic microwave background radiation—across the sky, as NASA’s space-borne Wilkinson Microwave Anisotropy Probe did from 2001 to 2010. Scientists also measure the distribution of the galaxies, as the Sloan Digital Sky Survey has done using a telescope at the Apache Point Observatory in New Mexico. The results suggest a fourth neutrino, Abazajian says. “There’s a 1-in-20 chance that it’s a statistical fluctuation, and those sorts of things go away all the time,” he says. “Still, it’s intriguing that the data’s converging to that value” of four types of neutrinos. > > The newest bit of evidence comes from nuclear reactors. Earlier this year, a team of theorists argued that reactors are putting out more electron antineutrinos than detectors tens of meters away show. The result suggests that some antineutrinos escape detection by morphing into sterile antineutrinos. > > Within a nuclear reactor, nuclei of the isotopes uranium-235, uranium-238, plutonium-239, and plutonium-241 split randomly to make myriad smaller nuclei that release copious antineutrinos. For example, a uranium-235 nucleus can split to make a nucleus of krypton-89. Krypton-89 then changes identity to rubidium-89, strontium-89, and yttrium-89, as one neutron after another in the nucleus spits out an electron and an antineutrino and turns into a proton. Thousands of other chains or “branches” of decays also occur. > > David Lhuillier of France’s Alternative Energies and Atomic Energy Commission in Saclay and colleagues kept track of all those branches in a new calculation. Previous calculations showed that a score of reactor measurements taken over decades observed 97.6%, give or take 2.4%, of the expected antineutrino flux—fine agreement with the prediction. With the new calculation, the measurements average 94.3%, plus or minus 2.3%—a significant difference that suggests neutrinos are disappearing. “Before, all the experiments were in agreement with the prediction,” Lhuillier says. “Now everybody is below the prediction.” Physicists say this single result triggered the workshop. > > The disparate hints are tantalizing, says Joseph Formaggio of the Massachusetts Institute of Technology in Cambridge. “What’s nice is that these anomalies come from different directions,” he says. … Of course elsewhere in the article they pre-emptively dismiss my speculation: a massless neutrino at speed of light, by virtue of traveling at that speed is traveling with time standing still for it, therefore cannot change. (Thus changing flavors proved neutrinos have mass.) Of course you know my obsession - I fixate on the possibility that “changing flavors” is reorientation of the same n-dimensional particle in n-dimensional space with us merely seeing different 3-D slices of the same n-D thing. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 19, 2011, 4:37am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/387 "2011-11-19T04:37:51Z")
> [@](#): > > Of course you know my obsession - I fixate on the possibility that “changing flavors” is reorientation of the same n-dimensional particle in n-dimensional space with us merely seeing different 3-D slices of the same n-D thing. You can describe it mathematically this way, though the dimensions involved are not necessarily spatial dimensions. Basically, there’s a 3-D (or 4-D if you have a sterile one too) neutrino flavor space, which can either be described by the e-mu-tau coordinate axes (if you’re considering neutrinos distinguished by their flavors), or by the 1-2-3 coordinate axes (if you’re considering them distinguished by their masses). These two sets of axes are related by a set of parameters called the “mixing angles”: Based on the atmospheric and solar results (ignoring the LSND and the implied sterile neutrino), it looks like the e and mu axes lie almost exactly in the 1-2 plane, but rotated 45 degrees, while the tau axis is almost coincident with the 3 axis. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [November 19, 2011, 7:29am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/388 "2011-11-19T07:29:44Z")
> [@DSeid](#): > > …Of course elsewhere in the article they pre-emptively dismiss my speculation: a massless neutrino at speed of light, by virtue of traveling at that speed is traveling with time standing still for it, therefore cannot change. > > Of course you know my obsession - [snip] . I am intrigued by “time standing still” for _it_, I think. What other particles are there that are massless and travel at the speed of light? Any? Gravitons? ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 19, 2011, 1:32pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/389 "2011-11-19T13:32:00Z")
Well, photons of course … Thank you **Chronos**. And of course, not necessarily not (curled up) spatial dimensions. In that case the n-D neutrino particle could have some upper bound of mass of the complete particle with the lesser masses reflecting the various 3-D extensions of the various preferred orientations in n-D curled space. Another question (bear with the inarticulate phrasing). IF such was the case (as a thought experiment), if all neutrinos actually had a certain somewhat slightly larger amount of mass in curled up n-D space but were oriented in such ways that only so much interacted with 3-D space in each particular orientation, but whose mass would never the less create gravity that would spread through all n dimensions, how much previously unaccounted mass would be thereby accounted for? (IOW, are n-D extensions of neutrinos of various flavors, with or without sterile flavor, if such existed, viable as a partial source of dark matter, albeit neither WIMP nor MACHO?) ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [November 19, 2011, 5:18pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/390 "2011-11-19T17:18:27Z")
> [@Leo\_Bloom](#): > > I am intrigued by “time standing still” for _it_, I think. What other particles are there that are massless and travel at the speed of light? Any? Gravitons? Photons ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 19, 2011, 6:24pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/391 "2011-11-19T18:24:28Z")
**DSeid** , I think you need to express your question much more rigorously before a meaningful answer is possible. And to the tally of massless particles, you can also add gluons, to the fact that they’re colored and strongly self-interacting makes them behave qualitatively differently from other massless particles. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 19, 2011, 7:10pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/392 "2011-11-19T19:10:04Z")
I will try, but readily acknowledge that I may not have the chops to do so. How many neutrinos are there estimated to be in the universe? What is the upper bound of neutrino mass for any of the standard flavors? If we assumed that all neutrinos actually always have that mass, even if most of it was hidden from our view, extending mostly elsewhere in curled up n-D space when in different orientations, how much mass would that be? How much of a portion of dark matter could that account for? (My suspicion is only a very small fraction, but the principal of matter and, for that “matter” antimatter, having variable amounts hidden from interaction in extended 3-D space (the slice of n-D space that we observe and interact with) and some of it in varying orientations in curled up n-D space being the source of the gravity of dark matter is my admittedly annoyingly persistent theme. :)) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 19, 2011, 11:54pm UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/393 "2011-11-19T23:54:25Z")
Oh, I got that you were aiming for dark matter. What I meant was that you need to be more rigorous about what you meant by “hidden from view, extending in other dimensions”. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [November 20, 2011, 12:14am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/394 "2011-11-20T00:14:57Z")
> [@Terr](#): > > Photons So from the photon’s point of view, so to speak, how does it know where it is? Is the “knowledge” available from outside observation (Heisenberg) the explanation; or, more broadly, if a photon were to care, _that_ self-realization-- would affect it’s base “standing-still,” and Heisenberg would again come into play? This sounds silly. But my intrigue into “no time” – as physical phenomenon and challenge to epistemology -— is quite real. Best, Leo ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 20, 2011, 12:16am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/395 "2011-11-20T00:16:58Z")
Ah. Well think of _Flatland_ for the analogy. Just as Flatland creatures could only “see” a slice of 3-D objects passing through and rotating through its plane, objects that had extensions that went beyond the plane of Flatland, so too would hypothetical n-D objects existing in hypothetical (curled-up) n-D space be “seen” only in various and varying 3-D cross-sections by creatures who exist in the 3 extended dimensional space, with the objects’ observed nature from that 3-D perspective being contingent upon their exact orientation and motions within the limited and curled up constrained other n dimensions. But _gravity_ travels at all times through all the extended and curled up dimensions the same. So, in Flatland, the sphere has the same gravitational effect measurable in Flatland whether its intersection with Flatland is none, a point, or the maximal diameter circle, and that gravitational effect is, in all those cases, greater than that expected from even the circle alone. By analogy a neutrino would be hypothesized to actually be an n-D object of some specific mass that we observe as different flavors contingent upon its various possible orientations in the curled other (n-3?) dimensions. That mass could be greater than that observed from the extended 3-D perspective as most of it is not observable from that perspective anymore than the complete sphere is observable from a Flatlander perspective. But its gravity would be measurable and have an effect. I doubt that this explanation qualifies as rigorous, but I hope it suffices to clarify what I am asking. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [November 20, 2011, 12:35am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/396 "2011-11-20T00:35:25Z")
**Leo** , my stab at it. The photon cannot “know” where it is. The universe may have lived out its complete existence and it may have travelled its complete breadth, but for _it_ no time has passed as no information from the outside reaches it. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [November 20, 2011, 12:49am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/397 "2011-11-20T00:49:47Z")
> [@DSeid](#): > > Ah. > > Well think of _Flatland_ for the analogy. Just as Flatland creatures could only “see” a slice of 3-D objects passing through and rotating through its plane, objects that had extensions that went beyond the plane of Flatland, so too would hypothetical n-D objects existing in hypothetical (curled-up) n-D space be “seen” only in various and varying 3-D cross-sections by creatures who exist in the 3 extended dimensional space, with the objects’ observed nature from that 3-D perspective being contingent upon their exact orientation and motions within the limited and curled up constrained other n dimensions. > > But _gravity_ travels at all times through all the extended and curled up dimensions the same. So, in Flatland, the sphere has the same gravitational effect measurable in Flatland whether its intersection with Flatland is none, a point, or the maximal diameter circle, and that gravitational effect is, in all those cases, greater than that expected from even the circle alone. > > By analogy a neutrino would be hypothesized to actually be an n-D object of some specific mass that we observe as different flavors contingent upon its various possible orientations in the curled other (n-3?) dimensions. That mass could be greater than that observed from the extended 3-D perspective as most of it is not observable from that perspective anymore than the complete sphere is observable from a Flatlander perspective. But its gravity would be measurable and have an effect. > > I doubt that this explanation qualifies as rigorous, but I hope it suffices to clarify what I am asking. I think there are inconsistencies in that. If there is a constant neutrino mass, of which we only observe a slice, how do you satisfy conservation of energy? The lightest neutrino wouldn’t take enough energy away from a reaction if it only took enough energy for it’s slice of rest mass. Also, you wouldn’t satisfy Newton’s third law or equal-and-opposite-reaction for the sphere’s (neutrino’s) gravity. The sphere (neutrino) is pulling with a large 3D (n-D) mass, but is itself only pulled with enough force for its slice of mass. ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [November 20, 2011, 1:35am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/398 "2011-11-20T01:35:47Z")
So…this is GQ. _Can_ neutrinos travel faster than light? (This seems like it may be a tough question, so we may need to pass it on to Cecil :p) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 20, 2011, 6:52am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/399 "2011-11-20T06:52:58Z")
INSUFFICIENT DATA FOR MEANINGFUL ANSWER i ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [November 20, 2011, 10:06am UTC](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266/400 "2011-11-20T10:06:27Z")
> [@Chronos](#): > > INSUFFICIENT DATA FOR MEANINGFUL ANSWER > > i 😃 As for the whole mass and higher dimensions thing, consider the following: mass is given by the magnitude of the [4-momentum](http://en.wikipedia.org/wiki/4-momentum), or, in higher dimensions, of the ‘n-momentum’. The 4-momentum has the components (p[sub]0[/sub], p[sub]1[/sub], p[sub]2[/sub], p[sub]3[/sub]) = (E, p[sub]x[/sub], p[sub]y[/sub], p[sub]z[/sub]) in units where c = 1, with E being the particle’s energy, and **p** = (p[sub]x[/sub], p[sub]y[/sub], p[sub]z[/sub]) being the ordinary three-dimensional momentum. The magnitude, given by the Minkowski norm, evaluates to E[sup]2[/sup] - p[sub]x[/sub][sup]2[/sup] - p[sub]y[/sub][sup]2[/sup] - p[sub]z[/sub][sup]2[/sup] = E[sup]2[/sup] - **p** [sup]2[/sup], which we know from special relativity is equal to m[sup]2[/sup]; the usual way to write this is -p[sup]μ[/sup]p[sub]μ[/sub] = m[sup]2[/sup], where μ runs over the dimensions of the spacetime, conventionally numbered 0 to 3. If we then have a particle in an n-dimensional spacetime (in which (n-1) dimensions are spatial), we can generalize this formula, yielding -p[sup]A[/sup]p[sub]A[/sub] = m[sub]n[/sub][sup]2[/sup], where A now is an index running from 0 to n, and m[sub]n[/sub] is the ‘n-dimensional mass’. If we now get rid of the (n - 4) extra dimensions, say by compactification, we can split up the summation into the 4-dimensional observable and (n - 4)-dimensional hidden part: -p[sup]A[/sup]p[sub]A[/sub] = -p[sup]μ[/sup]p[sub]μ[/sub] + p[sup]a[/sup]p[sub]a[/sub] = m[sub]n[/sub][sup]2[/sup] (a runs over the extra dimensions, i.e. from 4 to n), from which we see that the 4-dimensional mass of the particle must be -p[sup]μ[/sup]p[sub]μ[/sub] = m[sub]n[/sub][sup]2[/sup] - p[sup]a[/sup]p[sub]a[/sub]; but p[sup]a[/sup]p[sub]a[/sub] = -p[sub]4[/sub][sup]2[/sup] - p[sub]5[/sub][sup]2[/sup] - … - p[sub]n[/sub][sup]2[/sup] \< 0, and thus, m[sup]2[/sup] = -p[sup]μ[/sup]p[sub]μ[/sub] = m[sub]n[/sub][sup]2[/sup] - p[sup]a[/sup]p[sub]a[/sub] \> m[sub]n[/sub][sup]2[/sup], i.e. the mass of the particle in 4 dimensions is greater than the mass in n dimensions – in particular, a n-dimensional massless particle looks massive in 4 dimensions. Hmm, not sure that was all that clear. Let’s write it out: -p[sup]A[/sup]p[sub]A[/sub] = p[sub]0[/sub][sup]2[/sup] - p[sub]1[/sub][sup]2[/sup] - … - p[sub]n[/sub][sup]2[/sup] = E[sup]2[/sup] - p[sub]x[/sub][sup]2[/sup] - p[sub]y[/sub][sup]2[/sup] - p[sub]z[/sub][sup]2[/sup] - p[sub]4[/sub][sup]2[/sup] - … - p[sub]n[/sub][sup]2[/sup] = m[sup]2[/sup] - p[sub]4[/sub][sup]2[/sup] - … - p[sub]n[/sub][sup]2[/sup] = m[sub]n[/sub][sup]2[/sup], therefore m[sup]2[/sup] = m[sub]n[/sub][sup]2[/sup] + p[sub]4[/sub][sup]2[/sup] + … + p[sub]n[/sub][sup]2[/sup]. So you can’t hide extra mass in extra dimensions – you’ll end up with even _more_ mass in the usual four, and the gravity exerted by the particle is entirely explicable in terms of this mass. Note, however, that the same higher-dimensional particle can give rise to multiple lower-dimensional ones, which differ only in their mass – the reason is basically that the momentum in the direction of the extra dimensions is quantized, i.e. can be increased only in discrete steps, the step-size being related to the inverse of the radius of the extra dimension. So the same higher dimensional particle with different quanta of momentum in the direction of the extra dimensions shows up as different particles with different masses in our usual 4-D spacetime. This is known as the ‘KK-tower’, KK standing for Theodor Kaluza and Oskar Klein, who proposed a five-dimensional generalization of Einstein’s general relativity to unify it with electromagnetism. I think these KK excitations have indeed been proposed as a candidate for dark matter, but I don’t really know much about that. [Previous page](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266.md?page=19) [Next page](https://boards.straightdope.com/t/can-neutrinos-travel-faster-than-light/597266.md?page=21)