6 which tells us that c is the limiting speed that a material photons exist.Similarly,tachyons may also exist and. object can attain.(Notice that "material"here means like photons,(a)should be created instantaneously (that "the one with which an inertial frame can be associated". is,can't be created at rest.and then accelerated),and The photons do not fall into this category,as will be (b)should not be allowed to form a "legitimate"inertial discussed shortly!) reference frame. The possibilities therefore are: What about violation of causality? 1.c=+oo (Newtonian mechanics;contradicts (71)); 2.c>0 and finite (Special Relativity); B 3.c=0(Contradics observations.) So we stick with option 2. What if an object were created to have v'>c from the start (a so-called tachyon),like in the recent superluminal <IA neutrino controversy?We'd get some strange results. For example,if we take v=c/2 and v'=2c,we get w'=c/2)+(2g)_5 1+e22=4 (75) so in k the object would move to the right at a slower speed than relative to K',while K'itself is moving to the right relative to K.Bizarre,but OK,the two speeds are measured by different observers,so maybe it's not a big deal.... FIG.4:(Color online.)Violation of causality by a hypothet- ical tachyon. Step 14:Lorentz transformation in standard form Indeed.the Lorentz transformation shows that tachyons violate causality.If we consider two events, However,if we consider the resulting Lorentz transfor- A(tachyon creation)and B(tachyon annihilation)with mation. tB>tA such that tachyon's speed,vpc is greater than c as measured in K(see Fig.4),then in (76) frame K'moving with velocity v <c relative to K we'll have from(77)and (78), or, a-=[te-ta)-2cB-r小 t-x VEB-工A (77) =y1- (tB-tA) V1-器 c2 tB-tA x= x-vt =-)-a, (81) (78) V1-盟 B-TA =Y[(B-A)-v(tB-tA)] = (79) =Y1-v tB-tA EB-EA」 (TB-TA) (80) (82) we notice that in a reference frame K associated with hypothetical tachyons moving with v>c relative to K (imagine a whole fleet of them,forming a grid which where y=1/V1-v2/c2,which shows that it is possible makes up K),the spacetime coordinates of any event to find v <c such that te-t<0;that is,in K'event would be imaginary!In order for the spacetime measure- B happens before event A.This seems to indicate that ments to give real values for (t',',y,2),the reference tachyons are impossible.However,causality is a conse- frame K'made of tachyons must be rejected. quence of the Second Law of Thermodynamics,which What about a reference frame made of photons?In is a statistical law,applicable to macroscopic systems; that case,coordinates would be infinite and should also it does not apply to processes involving individual ele- be rejected.So a fleet of photons cannot form a "le- mentary particles.As a result,the existence of tachyons gitimate"reference frame.Nevertheless,we know that cannot be so easily ruled out.6 which tells us that c is the limiting speed that a material object can attain. (Notice that “material” here means “the one with which an inertial frame can be associated”. The photons do not fall into this category, as will be discussed shortly!) The possibilities therefore are: 1. c = +∞ (Newtonian mechanics; contradicts (71)); 2. c > 0 and finite (Special Relativity); 3. c = 0 (Contradics observations.) So we stick with option 2. What if an object were created to have v 0 > c from the start (a so-called tachyon), like in the recent superluminal neutrino controversy? We’d get some strange results. For example, if we take v = c/2 and v 0 = 2c, we get v 00 = (c/2) + (2c) 1 + (c/2)(2c) c 2 = 5 4 c, (75) so in K the object would move to the right at a slower speed than relative to K0 , while K0 itself is moving to the right relative to K. Bizarre, but OK, the two speeds are measured by different observers, so maybe it’s not a big deal . . . . Step 14: Lorentz transformation in standard form However, if we consider the resulting Lorentz transfor￾mation,  x 0 t 0  = 1 q 1 − v 2 c 2  1 −v − v c 2 1  x t  , (76) or, t 0 = t − v c 2 x q 1 − v 2 c 2 , (77) x 0 = x − vt q 1 − v 2 c 2 , (78) y 0 = y, (79) z 0 = z, (80) we notice that in a reference frame K0 associated with hypothetical tachyons moving with v > c relative to K (imagine a whole fleet of them, forming a grid which makes up K0 ), the spacetime coordinates of any event would be imaginary! In order for the spacetime measure￾ments to give real values for (t 0 , x0 , y0 , z0 ), the reference frame K0 made of tachyons must be rejected. What about a reference frame made of photons? In that case, coordinates would be infinite and should also be rejected. So a fleet of photons cannot form a “le￾gitimate” reference frame. Nevertheless, we know that photons exist. Similarly, tachyons may also exist and, like photons, (a) should be created instantaneously (that is, can’t be created at rest, and then accelerated), and (b) should not be allowed to form a “legitimate” inertial reference frame. What about violation of causality? x ’ t’ t x B A t’ A t’ B < t’ A FIG. 4: (Color online.) Violation of causality by a hypothet￾ical tachyon. Indeed, the Lorentz transformation shows that tachyons violate causality. If we consider two events, A (tachyon creation) and B (tachyon annihilation) with tB > tA such that tachyon’s speed, vp = xB−xA tB−tA > c, is greater than c as measured in K (see Fig. 4), then in frame K0 moving with velocity v < c relative to K we’ll have from (77) and (78), t 0 B − t 0 A = γ h (tB − tA) − v c 2 (xB − xA) i = γ  1 − v c 2 xB − xA tB − tA  (tB − tA) = γ  1 − vvp c 2  (tB − tA), (81) x 0 B − x 0 A = γ [(xB − xA) − v (tB − tA)] = γ  1 − v tB − tA xB − xA  (xB − xA) = γ  1 − v vp  (xB − xA), (82) where γ = 1/ p 1 − v 2/c2, which shows that it is possible to find v < c such that t 0 B − t 0 A < 0; that is, in K0 event B happens before event A. This seems to indicate that tachyons are impossible. However, causality is a conse￾quence of the Second Law of Thermodynamics, which is a statistical law, applicable to macroscopic systems; it does not apply to processes involving individual ele￾mentary particles. As a result, the existence of tachyons cannot be so easily ruled out