When analyzing a moving body with respect to a moving observer we can use the Galilean or Lorentz transformations. As we approach the speed of light we know the Lorentz transformation gives the correct results. At low speeds they are "supposed" to be the same, but they do not converge to exactly the same result. There is a slight discrepancy between the two. The question is: what is the physical intuition behind this difference and what is it telling us?
They are equivalent. And it's known how to show this, cf.
http://www.pnas.org/content/39/6/510.full.pdf
The question doesn't have anything to do with physics, in fact, it's a purely mathematical statement, about how groups are related in a consistent way.
They are equivalent. And it's known how to show this, cf.
http://www.pnas.org/content/39/6/510.full.pdf
The question doesn't have anything to do with physics, in fact, it's a purely mathematical statement, about how groups are related in a consistent way.
They are 'exactly equal' only when the speed is 'exactly zero'. Otherwise, there will be a small correction term in Galilean transformation. Mathematically, Lorentz transformation approaches to Galilean transformation as the speed between the observers approaches to zero.
True, when the speed approaches to zero, but we deal wit finite speeds in physics.
The orbiting speed of the Earth is cca 30000m/s, therefore, for clocks distanced cca 10^13 m (the Solar System diameter) we'll have a measurable synchronization error (vx/c^2), even if we put gamma = 1.
Galilean and Lorentz transformations are equivalent if we consider the limit of low speed and small space region.
Derek,
please explain in which situations something related to your question 'does not converge to exactly the same result'. I don't think that there is something of that kind when we consider limits v --> 0. Of course there are differences in the structural implications of Poincare-invariance and Galilei-invariance, n-body mechanics being a striking example.
In Galilean transformation the time is univesral constant. Time is the same in all inertial coordinates systems. Inertial coordinate systems are systems which are moving with a constant velocity.
Velocity v= v1 in the train and velocity on the ground v2 are related by
v2= v1 in the moving frame+velocity of moving frame. Therefore
x= x'+ v t
This is not true in the Lorentz transformation. When the velocities are comparable to the velocity of light then the coordiante transformations are related by when x is as seen on the groung and x' as seen on the moving frame are related by
x= (x '+ v t')/ sqrt(1- v²/c²)
time are related by
t =(t' + x'v/c²)/sqrt(1-v²/c²)
You can say the Lorentz transformations are universal and under approximation they are equivalent to Galiliean transformations. Please check the Lorentz transformation equations.
@Vaman Kulkarni,
they are not, since in the term x'v/c² (the synchronization error) we can have finite contributions if x' is big and v is small, but finite, as I commented before. Therefore, as Ivanov said, to have equivalence under approximations (not the limit) we have to consider a small space regions. This is not a trivial fact. Consider objects in the Milky Way galaxy, they are rather slow relative to the Earth, however, due to large distances, approximated LT will give results different than GT. The Lorentz transformations imply the time non-simultaneity, which cannot be approximated to zero at large regions of space. In fact, this becomes evident even at dimensions of the Solar system.
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