I Linear Accelerator Length Contraction

Orthoceras
Messages
125
Reaction score
48
TL;DR Summary
Effect of relativistic length contraction on the electron bunches in a linear accelerator?
I am trying to understand the effect of relativistic length contraction on the electron bunches in a linear accelerator. Figure B is for nonrelativistic speeds, successive cylinder lengths are progressively longer. However, wikipedia says "At speeds near the speed of light, the incremental velocity increase will be small, with the energy appearing as an increase in the mass of the particles. In portions of the accelerator where this occurs, the tubular electrode lengths will be almost constant", so it should figure D or E. I expect length contraction to occur, therefore D. However, I don't see why the the gap between bunches does not contract.

Which option is right?

linac5.png

Red: electron bunches; grey: cylinders
 
Last edited:
Physics news on Phys.org
Orthoceras said:
Summary:: Effect of relativistic length contraction on the electron bunches in a linear accelerator?

However, I don't see why the the gap between bunches does not contract.
This has to do with how the acceleration is performed. The gap is not a rigid object (not that rigid objects exist in relativity) that maintains the same rest length. The setup is such that the distance between bunches in the instantaneous rest frame of a bunch increases during the process.
 
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...

Similar threads

Back
Top