I’ve been uncom­fort­able with the con­clu­sion that spe­cial rel­a­tiv­ity clearly asserts that it is impos­si­ble to “go faster than the speed of light”. Actu­ally, of course, it doesn’t. Most would allow for tachy­onic solu­tions in which there exist states which cor­re­spond to par­ti­cles trav­el­ling at super-luminal speeds. The prob­lem is actu­ally the 1/x, x->0 in the maths, which we try and explain as a prob­lem of nature instead of a prob­lem with our math­e­mat­i­cal rea­son­ing, and leads us to unre­solv­able con­sid­er­a­tions involv­ing infinity.

From first prin­ci­ples if one starts with the notion that every­one should see light as trav­el­ling at the same speed, then a sim­ple deriva­tion nat­u­rally leads to the Lorentz trans­for­ma­tion from which all of spe­cial rel­a­tiv­ity is con­structed. Then we go on to see that that length con­trac­tion and time dila­tion must occur; a nat­ural con­se­quence of the trans­for­ma­tion equa­tion. And, we have to ask the ques­tion “how can a body have infi­nite mass”? Nat­u­rally we give up and say that it can’t and so it can’t happen.

How­ever, isn’t our inter­pre­ta­tion of the Lorentz trans­for­ma­tion just a descrip­tion of what it is that a sta­tion­ary observer would observe? That is, as a body approaches the speed of light rel­a­tive to the observer, that observer would inter­pret what they see as the body also car­ry­ing an increas­ing mass that rapidly asymp­totes to infin­ity. But wait a minute, I can­not see any­thing in that that says that the body actu­ally has infi­nite mass, it’s just how we inter­pret what we see. The impli­ca­tion of the fact that the speed of light is seen to travel at a fixed speed in all frames, is that there are obser­va­tional impli­ca­tions due to us using light to observe things.

This prob­lem with infin­ity is all an impli­ca­tion of the “approach­ing 1/0″ in the maths; a limit for which we have no mean­ing­ful way of manip­u­lat­ing within the alge­bra. We take the view that the frame is Carte­sian (i.e. flat), and then pos­tu­late that because no par­ti­cle can actu­ally attain infi­nite energy/infinite rest mass, that Lorentz must imply that no par­ti­cle can travel faster than the speed of light.

I’m not so sure.

Imag­ine two observers, and set up a con­stant accel­er­a­tion between the two. Pick one, that’s you. You watch as the other one’s veloc­ity grad­u­ally rises, and approaches the speed of light. Because of the flat-space-time axiom you’ll always be able to see them, if your tele­scope is pow­er­ful enough that is, and you’ll also observe that their mass increases also, so that they appear to be asymp­tot­i­cally approach­ing the the speed of light but never able to get there. You’ll observe them in con­junc­tion with the sin­gu­lar­ity in the maths.

Now imag­ine the other one. Don’t they just define them­selves as being sta­tion­ary? And, it’s you thats get­ting heav­ier and so on. It’s all an obser­va­tional effect. They’re just mosey­ing along in their (very slightly accel­er­ated) rest frame, and all the action is hap­pen­ing to you, not them. It’s an obser­va­tional effect.

Ok, so one argu­ment to that pic­ture is “Ahha! We’ve got an accel­er­at­ing frame, and so spe­cial rel­a­tiv­ity doesn’t count any more”. Fine then, let’s evoke gen­eral rel­a­tiv­ity to explain what’s going on. Now we have the prob­lem that it isn’t mean­ing­ful to make com­par­isons between the the two dif­fer­ent ref­er­ence frames any more, as we have the prob­lem of how to “par­al­lel trans­port” the veloc­ity vec­tors from one frame to the other in order to make mean­ing­ful state­ments about rel­a­tive veloc­ity. We can pos­tu­late that the space/time is flat, and argue that we don’t have to take that into account — don’t we then just recover the spe­cial rel­a­tiv­ity sit­u­a­tion again? Or oth­er­wise we have to let go of the notion that it is pos­si­ble to even mea­sure rel­a­tive veloc­i­ties any more.

I tend to imag­ine that the uni­verse is larger than what we can observe elec­tro­mag­net­i­cally, and that there are E/M dis­con­nected regions. (If you believe the pic­ture of cos­mic infla­tion­ary the­ory you’ll not be able to get away from that fact). I play another thought experiment:

I’m accel­er­at­ing away from you at a nice com­fort­able 1g, I have a nice LENR pow­ered motor, so I don’t need to worry about tak­ing any fuel with me, I just extract what I need from the void as I need it. I can keep this up for ever if I need to; I also have choco­late bars, and a few good books! . Oh, go on then, come along on the trip too!

So, we’re track­ing our veloc­ity with respect to the home planet, and I expect that at some point we will begin to approach the speed of light rel­a­tive to it. How­ever, we’re also track­ing our veloc­ity rel­a­tive to every­thing else that we can see. There’s noth­ing spe­cial about the Earth as a nav­i­ga­tion point with which to take bear­ings. I have a whole gamut of dif­fer­ent veloc­i­ties to observe, arguably with ranges between zero (for my books and choco­late, and of course you) and the speed of light (for all the pho­tons that we are observ­ing from all direc­tions). Our view is lit­tered with stars and galax­ies, gas clouds and other dust.

Now, imag­ine that we found our­selves sud­denly trav­el­ling at faster than the speed of light rel­a­tive to the Earth. Let’s tread care­fully, and ignore any ques­tions of how we came to find our­selves in this sit­u­a­tion, or how we would be able ver­ify that fact; for now let’s just con­sider this a thought exper­i­ment. We went to sleep, and when we woke up we we found our­selves in this state.

How to analyse this sit­u­a­tion? A good ini­tial ques­tion to ask might be, can we still see Earth? I strongly sus­pect not. Stan­dard light cone con­sid­er­a­tions would tell us imme­di­ately that we would be unable to exchange any infor­ma­tion electro-magnetically; we can­not do any analy­sis of veloc­i­ties by using Lorentz trans­for­ma­tion, and so spe­cially rel­a­tiv­ity is out. How­ever, clearly we are still in the same phys­i­cal Uni­verse, but we have become what I’m going to call “Lorentz-disconnected”. We are not longer E/M con­nected with the Earth and in some senses we could be con­sid­ered to be in a totally sep­a­rate E/M Universe[1].

Ok, but what about our rela­tions to every­thing else that we were observ­ing? We’re not going to be trav­el­ling faster than the speed of light rel­a­tive to every­thing we were track­ing before, are we? And what about that asymp­tote or sin­gu­lar­ity, that we passed through in the Lorentz trans­for­ma­tion? We clearly had to do that smoothly, right?

Here’s what I think hap­pened. As we approached the speed of light, the Earth­lings mea­sured us as becom­ing infi­nite in mass, and length con­tracted. As we passed the (rel­a­tive) speed of light we did lit­er­ally dis­ap­pear to them, along with all the obser­va­tional effects that accom­pa­nied that. On the way we began to see other bod­ies that were out of Earth’s E/M uni­verse, but were com­ing into ours as we trav­elled along our geo­desic path tak­ing our E/M frame with us. Ulti­mately we observed the Earth dis­ap­pear into a sin­gu­lar­ity, and we observed some new bod­ies appears from sin­gu­lar­i­ties, and we found our­selves here. Safe and sound.

By ana­log, imag­ine that you’re a mem­ber of the flat Earth soci­ety. You’re watch­ing a ship sail away from you and over the ocean. How do you inter­pret what you see as it approaches, reaches and then dis­ap­pears through the “event” hori­zon that’s there because there is actu­ally a cur­va­ture at play? Naively you might think that it had dis­ap­peared into a sin­gu­lar­ity and would never come back again.

We’re cur­rently mea­sur­ing every­thing that we know about the uni­verse from a sin­gle van­tage point[2]. We con­struct a 3D slice of what we think it looks like from this place, and then make the mis­take of mis­tak­ing that for the whole thing.

You know what? Once we get our acts together and get some space travel going on, and truly get to see things from an inter­galac­tic per­spec­tive, I bet you we’ll be really sur­prises as to how the Uni­verse really looks.

Joe

[1] At this point I’m prone to drift­ing off and imag­in­ing how the inside and the out­side of a black hole are con­nected through the singularity.

[2] More or less; with respect to the size of the Uni­verse our solar sys­tem is infin­i­tes­i­mally small!