Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Todd on 2 Sep 2005 08:16 "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... >.... > for example have a look at Einsteins own derivation (from his book > 'Relativity: The Special and General Theory') given at > http://www.bartleby.com/173/a1.html which seems to be a very elegant > way of deriving the Lorentz transformation. > > It is only necessary here to examine the initial equations for this, > which describe the 'equations of motion of a light signal' in the > unprimed and primed reference frames, i.e. > > (1) x-ct=0 > (2) x'-ct'=0 > where c is the speed of light (which obviously has to be a constant >0) > > In the same way, the propagation of a signal in the opposite direction > yields > (3) x+ct=0 > (4) x'+ct'=0 > (note that these equations are not written explicitly in Einstein's > derivation). > ................ > But are the above equations mathematically consistent at all? Let's > subtract equation (1) from (3), which yields > (5) 2ct=0 > which means that for any time t>0 > (6) c=0, > in contradiction to the requirement that c>0. >............. As Bill Hobba says, you must think about what the symbols denote. It might help to write (1) and (3) as (1) x1 - ct = 0 (3) x2 + ct = 0 where x1 is the position of the light pulse that's traveling in the positive x direction and x2 is the position of the other pulse traveling in the negative x direction. Note that x1 never equals x2 except at time t = 0. When you subtract them you get an equation that may be written as x1 - x2 = 2ct This just says that the distance between the pulses is increasing at the rate of 2c, which makes sense. Todd
From: Dirk Van de moortel on 2 Sep 2005 08:39 "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... > Many people maintain that the Lorentz transformation is derived > mathematically consistently and that there is therefore no way to > challenge SR on internal consistency issues. Is this really so? Let's > for example have a look at Einsteins own derivation (from his book > 'Relativity: The Special and General Theory') given at > http://www.bartleby.com/173/a1.html which seems to be a very elegant > way of deriving the Lorentz transformation. > > It is only necessary here to examine the initial equations for this, > which describe the 'equations of motion of a light signal' in the > unprimed and primed reference frames, i.e. > > (1) x-ct=0 > (2) x'-ct'=0 > where c is the speed of light (which obviously has to be a constant >0) > > In the same way, the propagation of a signal in the opposite direction > yields > (3) x+ct=0 > (4) x'+ct'=0 > (note that these equations are not written explicitly in Einstein's > derivation). > > >From equations (1)-(4), the Lorentz transformation is then derived by > some algebraic manipulations. > > But are the above equations mathematically consistent at all? Let's > subtract equation (1) from (3), which yields > (5) 2ct=0 > which means that for any time t>0 > (6) c=0, > in contradiction to the requirement that c>0. Brilliant :-)) Fourth entry already: http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Inconsistent.html This is an excellent follow-up for: http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Cringe.html http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/NotQuiteWithYou.html http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Wrong.html Please don't stop amazing us with your profound insights in the usage of equations. Perhaps you could try to explain how you do it to Androcles. He is a good pupil and I'm sure that your teaching skills at least match your listening skills. Dirk Vdm
From: Dirk Van de moortel on 2 Sep 2005 08:57 "Curt" <curt2(a)ntlworld.com> wrote in message news:jcVRe.5432$w4.3618(a)newsfe5-win.ntli.net... > > "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message > news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... > ....... > > which describe the 'equations of motion of a light signal' in the > > unprimed and primed reference frames, i.e. > > > > (1) x-ct=0 > > (2) x'-ct'=0 > > where c is the speed of light (which obviously has to be a constant >0) > > > > In the same way, the propagation of a signal in the opposite direction > > yields > > (3) x+ct=0 > > (4) x'+ct'=0 > .......... > > > > But are the above equations mathematically consistent at all? Let's > > subtract equation (1) from (3), which yields > > (5) 2ct=0 > > which means that for any time t>0 > > (6) c=0, > > in contradiction to the requirement that c>0. > > > I am no expert in relativity, but I am aware that x, displacement, is a > vector. But that has nothing to do with it. > In the following argument I take c>0, x>0. If I fire a photon in the You better talk about a light signal here. Photons behave rather strange. > positive x direction, from the origin of my frame of reference, then, after > t, the photon has travelled x. Thus, x=ct, in accordance with (1).But how > did you derive (3)? The equation of the light signal path x - c t = 0 describes the coordinates of a signal going in the positive x-direction: at time t the signal is at distance x = c t. The equation of the light signal path x + c t = 0 describes the coordinates of a signal going in the negative x-direction: at time t the signal is at distance x = - c t. The equations talk about different things. Combining the equations like he did (algebraicly "solving a system of two equations with two unknowns") is the analytic geometry equivalent of finding the interesection between the light paths: { x - c t = 0 { x + c t = 0 ==> { 2 c t = 0 { x - c t = 0 ==> { t = 0 { x = 0 So the intersection of the signals happens at time t = 0 at distance x = 0. One can also do the following: The equation of the light signal path xP - c t = 0 describes the coordinates of a signal going in the positive x-direction: at time t the signal is at distance xP = c t. The equation of the light signal path xN + c t = 0 describes the coordinates of a signal going in the negative x-direction: at time t the signal is at distance xN = - c t. When you now subtract the equations, you find xP - xN = 2 c t which gives you an expression for the distance between the two signals at time t as seen by the person who uses these equations to describe the light paths. I have been trying to explain this to Thomas, but I think he is allergic to it, because he didn't even allow me to reach this point :-) Dirk Vdm
From: Daryl McCullough on 2 Sep 2005 09:00 Thomas Smid says... >It is only necessary here to examine the initial equations for this, >which describe the 'equations of motion of a light signal' in the >unprimed and primed reference frames, i.e. > >(1) x-ct=0 >(2) x'-ct'=0 >where c is the speed of light (which obviously has to be a constant >0) > >In the same way, the propagation of a signal in the opposite direction >yields >(3) x+ct=0 >(4) x'+ct'=0 >(note that these equations are not written explicitly in Einstein's >derivation). > >>From equations (1)-(4), the Lorentz transformation is then derived by >some algebraic manipulations. > >But are the above equations mathematically consistent at all? Let's >subtract equation (1) from (3), You can't subtract (1) from (3), since x in (1) refers to a *different* event than the x in (3). It's not the same value of x, and it's not the same value of t. Think about it in terms of a *car* driving down a road that runs East-West at 10 meters/second. Suppose I have a long road running East-West, and I paint a "0" on the road at some spot. Then 1 meter farther down the road to the West, I paint a "1", and then 1 meter farther I paint a "2", etc. 1 meter East of "0", I paint "-1", and then another mile East, I paint a "-2", etc. Next I take a huge number of identical clocks to the point marked "0", and set them all to the same time. Then one by one I slowly carry one clock to each mark on the road and drop it off. So there is a clock at the "0", a clock at the "1", etc. For any event taking place on the road, the "x" for that event is the closest mark. The "t" for that event is the time on the closest clock. Now, if I have a car that is going West at 10 meters/second, starting at mark "0" When it passes mark "1", the time on the closest clock will be 0.1 seconds. When it passes mark "2", the time on the closest clock will be 0.2 seconds. etc. I can summarize the path of the car as follows: x(car) = 10 * t(car) At any moment, x(car) is closest mark to the car, and t(car) is the time showing on the closest clock. Now, if the car had instead been travelling *East", then we would have x(car) = -10 * t(car) So when the car passes mark "-1", the time on the closest clock will be 0.1 seconds, and when it passes mark "-2", the time on the closest clock will be 0.2 seconds, etc. So the statement "the car travels in either direction at speed 10 meters/second" translates into two different equations, depending on whether the car is travelling East or West: car travelling West: x = 10*t car travelling East: x = -10*t Now, do you really think it makes sense to *subtract* those two equations, to get the following? 20*t = 0 -- Daryl McCullough Ithaca, NY
From: Thomas Smid on 2 Sep 2005 09:27
Todd wrote: > As Bill Hobba says, you must think about what the symbols denote. It might > help to write (1) and (3) as > > (1) x1 - ct = 0 > > (3) x2 + ct = 0 > > where x1 is the position of the light pulse that's traveling in the positive > x direction and x2 is the position of the other pulse traveling in the > negative x direction. Note that x1 never equals x2 except at time t = 0. > > When you subtract them you get an equation that may be written as > > x1 - x2 = 2ct > > This just says that the distance between the pulses is increasing at the > rate of 2c, which makes sense. Yes, it would make sense if x2=-x1 i.e. 2x1=2ct, but evidently Einstein's derivation would then not 'work' anymore as it relies on x1=x2=x i.e. 2ct=0. Thomas |