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bernhard.rothenstein
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Please tell me if it is possible to derive the formula which accounts for the Lorentz contraction from the invariance of the space-time interval.
Thanks
Thanks
Thanks for your answer. That is the derivation of the time dilation formula.morrobay said:Is this the algebra your looking for ?
Let " equal squared
Let 0" -(cT)" equal spacetime interval of stationary observer
Let (vt)" - (ct)" equal spacetime interval of moving observer
-(cT)" = (vt)" -(ct)" invariance of spacetime interval
c"t" = v"t" + c"T" divide by t"
c"=v"+ c"T"/t" divide by c"
1-v"/c" =T"/t" take sq root both sides
(sq rt) 1-v"/c" = T/t
t= T/ (sq rt) 1-v"/c"
kev said:Try this:
[tex] -(c^2T^2) +X^2 = -c^2t^2 + L_0^2[/tex]
Say a clock is transported from A to B (distance [tex]L_0[/tex]) in time t at velocity v relative to your frame.
T is the proper time of the transported clock as measured by itself (by definition).
X is the proper distance the clock moves in its own frame which is zero .
[tex] -(c^2T^2) = -c^2t^2 + L_0^2[/tex]
[tex] T^2 = t^2 - L_0^2/c^2[/tex]
[tex] T^2 = L_0^2/v^2 - L_0^2/c^2[/tex]
[tex] T^2 v^2 = L_0^2 -L_0^2v^2/c^2[/tex]
[tex] T^2 v^2 = L_0^2 (1-v^2/c^2)[/tex]
[tex] vT = L_0\sqrt{1-v^2/c^2} = L[/tex]
L is the distance A to B as measured by an observer comoving with the clock.
bernhard.rothenstein said:Please tell me if it is possible to derive the formula which accounts for the Lorentz contraction from the invariance of the space-time interval.
Thanks
bernhard.rothenstein said:Thanks. Do you know a derivation of the invariance of the space-time interval
bernhard.rothenstein said:Thanks. Do you know a derivation of the invariance of the space-time interval
bernhard.rothenstein said:Imagine systems S and S'. Person A stands at the origin in S and person B stands in the origin of S', which moves with velocity v relative to S. The origins coincide at t=t'=0. Imagine A lights a spark at t=0. A spherical light wave spreads out in all directions. At any point in time A stands in the center of the sphere which has radius ct. So the wavefront obeys the equation:
x^2+y^2+z^2=(ct)^2
Same thing holds in S'. B also stands in the center of the sphere at all times (2nd postulate):
x'^2+y'^2+z'^2=(ct')^2
so whatever happens to the coordinates, it should always be the case that:
x^2+y^2+z^2-(ct)^2=x'^2+y'^2+z'^2-(ct')^2
kev said:I notice in another thread https://www.physicsforums.com/archive/index.php/t-115451 you posted this:
..which seems perfectly valid to me, but you were bothered that the final equation is obtained by subtracting (ct)^2 from both sides of the first two equations so that the first two equations equal zero and then saying if two quantities are equal to zero then they are equal to each other.
As far as know using the technique of setting two equations to zero and saying they equal to each other is a perfectly normal mathematical technique.
bernhard.rothenstein said:Thanks. Do you consider that starting with
c*t*-x*=d* (1) *2
c"t'*-x'*=d* (2)
a consequence of the invariance of distances measured perpendicular to the direction of relative motion d then
c*t*-x*=c*t'*-x'*
is more convincing?
Regards
country boy said:By setting two zero-interval expressions equal to each other you are not demonstrating the in variance for non-zero values. The argument could be completed by deriving the Lorentz transformations from your two light spheres and then showing that all spacetime intervals are invariant.
bernhard.rothenstein said:Thanks kev. It is of big help to me.
Please have a critical look at the following:
Consider the well known light clock experiment performed in its rest frame I(0). Let d be the distance between the two mirrors. Being measured along a direction which is perpendicular to the direction of relative motion between the involved inertial reference frames it has the same magnitude in all of them. Consider the same experiment from I relative which moves relative to I(0) in the standard way. Let x be the horizontal displacement of the upper mirror and ct the distance traveled by the light signal between the two mirrors. Pythagoras' theorem requires
d^2=c^2t^2-x^2 (1)
Invariance of d requires the invariance of the right side of (1) and so in a I' reference frame we should have
d^2=c^2t'2-x'^2. (2)
Equating two different from zero relativistic expressions we have
c^2t^2-x^2=c^2t'^2-x'^2 (3)
Do you consider that (3) is a prove for the invariance of the relativistic space-time interval?
bernhard.rothenstein said:Do you consider that (3) is a prove for the invariance of the relativistic space-time interval?
1effect said:The invariance of the space-time interval is a postulate. You don't prove postulates since they are ...postulates.
When I started to learn English helped by the British Broadcasting Company (BBC) professor Grammar told me that "English is a very flexible language". I think that special relativity is a very flexible theory and you can derive almost every thing starting with something that is in accordance with the two postulates or is a consequence of it. I have seen papers in which the Lorentz transformations are derived from time dilation, from length contraction, from both of them, from the addition law of parallel speeds, from the Doppler Effect, from the invariance of the space-time interval. I have not the competence to decide which is the best way to follow. I think that the invariance of the Minkowski distance is a good way for doing it, but only after having proved its invariance and that is in my opinion the weak point of Taylor's approach to SR.1effect said:The invariance of the space-time interval is a postulate. You don't prove postulates since they are ...postulates.
kev said:Is it a postulate? Minkowski came up with the invariant interval and his spacetime after Einstein formulated Special Relativity which has the invariance of the speed of light and invariance of the laws of physics in any inertial reference frame. Eistien did not build special relativity on the postulate of the invariant interval. (Well as far as I know)
thanks for the derivation. please tell me if it involves clock synchronization the times involved being displayed by Einstein synchronized clocks.dx said:Let an observer (2) be positioned on one side of a rod of length [tex]l_0[/tex] in his rest frame. Let (1) be another observer relative to whom (2) is moving to the left with velocity [tex]v[/tex]. Their initial configuration is shown below in (1)'s frame.
(3)...[tex]l[/tex]...(2)
(1)
the rod is moving leftwards.
Let the length of the rod be [tex]l[/tex] in (1)'s frame.
[tex]E_1[/tex] : (3) and (1) coincide.
[tex]E_2[/tex] : (2) and (1) coincide.
According to (1), the time between the two events is [tex]\frac{l}{v}[/tex], and the distance is zero. Therefore,
[tex]t_1^2 - x_1^2 = \frac{l^2}{v^2}[/tex]
According to (2), the time between the two events is [tex]\frac{l_0}{v}[/tex], and the distance is [tex]l_0[/tex]. Therefore,
[tex] t_2^2 - x_2 ^2 = \frac{l_0^2}{v^2} - l_0^2 [/tex]
Equating the two, we get
[tex] l = l_0\sqrt{1-v^2} [/tex]
Mentz114 said:Dx,
nice derivation. I take it all times are measured in 1's frame.
bernhard.rothenstein said:thanks for the derivation. please tell me if it involves clock synchronization the times involved being displayed by Einstein synchronized clocks.
o.k. the interval contains the times t and t' and works even for the time intervals dt and dt'. It would not be necessary to mention the clocks that display those times, how are they synchronized...? I think that before to postulate Einstein's clock synchronization should be mentioned.1effect said:That is a postulate, so there is no derivation for it.
Lorentz contraction is a phenomenon that occurs in special relativity, where objects appear to be shorter in the direction of motion when measured by an observer in a different frame of reference.
Lorentz contraction is a consequence of the principle of space-time interval invariance, which states that the distance between two events in space-time is the same for all observers, regardless of their relative motion.
The formula for Lorentz contraction is L = L0 / γ, where L is the contracted length, L0 is the rest length, and γ is the Lorentz factor. This formula takes into account the time dilation effect and the relative velocity between the two frames of reference.
Yes, Lorentz contraction only becomes significant at speeds close to the speed of light. At lower speeds, the contraction is too small to be measured.
Lorentz contraction is a real physical phenomenon that has been demonstrated through various experiments and is an essential aspect of special relativity. It has practical implications, such as in the design of particle accelerators and GPS systems.