PHGN-471 Spring

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Instructors: Frank Kowalski

Measuring Length Contraction

Ideas on how to measure length contraction

1) Imagine the face of a clock, each tick on the outer rim would stick out slightly and be made up of a conductive material. As is rotates, the ticks will come into contact with a circuit that is attached to the time interval analyzer. When the ticks hit, it will complete the circuit and send in an electrical signal. We can measure the amount of time it takes between the ticks, and then compare it to a calculated time to see if the length has contracted.

2) Have a similar apparatus as 1), but now the ticks will hold onto something loosely, but still snug enough to stay on while it rotates. As it speeds up faster, the length with contract between the two ticks, and the clamp should fall off.

3) Have a laser shot onto the edge of the rotating wheel, and a receiver on the other end. The receiver will allow us to view a waveform of the frequency that was sent in and hopefully we can observe where the ticks are on this waveform. Then we can compare it to a calculated value.

4) Same as 3), but can use types of input signals.

5) Instead of having tick marks on rotating circle, there could slits length wise near the edges.

6) Instead of measuring the distance between the slots, the slots could we wider and the size of the slots themselves could be measured to see if they contracted.

7) These slots should probably be like an element dr, instead of rectangular, so that they stay fluid as the circle rotates.

8.) Two arm interferometer with each beam reflecting from a different radius of a spinning mirror.

9.) Measure the torque on a rotating rigid body as length contraction generates forces.

10.) Cone rotating about the symmetry axis. The cone shape would change as the different circumferences are effected by length contraction.

11.) Measure the volume change of the rotating cone.

12.) A charged rotating object should emit a radiation pattern in which length contraction is manifest.

13.) Generate two points at different radii on the rotating object from which to send a time stamps. There should be different times generated.

14.) Put a mirror on the rotating object from which a beam generated in the lab frame reflects.

combining ideas comments

When imagining the apparatus for this experiment, I picture something similar to the microwave diffraction grating that Randy and I recently made in the shop for your Ad Lab 1 class. It would rotate in the same manner, but have a solid front face with some sort of conducting wire that will complete a circuit. My only issue is how we will be able to have an object on the rotating wheel come into contact with a circuit without changing the angular velocity, or simply slowing it down.

Facts needed about length contraction

Idea # 8 requires an effect from light reflecting from a moving mirror. Here are some references

[1]

[2]

I could not easily find any references on the polarization effects on light reflecting from a moving mirror.

- ‘Born rigidity’ - definition of rigid body in special relativity proposed by M. Born in 1909, and ‘rigid motion’ formulated by Pauli, ‘rigidity’ means that distances between respective points of a body in question remain constant in the co-moving frame. This would refer also to non-relativistic mechanics; yet, as applied to SRT, it means that length contraction must satisfy a condition of Lorentz invariance

- M. Planck postulated to separate the problem of geometry on rotating disc from that what happens to the disc in the spin-up phase. In his opinion, the latter requires employing a relevant theory of elasticity.

- 1910 T. Kaluza suggested that geometry on rotating disc is non-Euclidean.

- Non-Euclidean geometry, in elliptic geometry there are no lines that will not intersect, as all that start separate will converge. In addition, elliptic geometry modifies Euclid's first postulate so that two points determine at least one line

- K frame rotating with respect to K', with rods along the periphery (U) and the diameter (D). When K' rotates the rods along the diameter don't experience the Lorentz contraction like those on the periphery. The laws of configuration of rigid bodies with respect to K′ do not agree with the laws of configuration of rigid bodies that are in accordance with Euclidean geometry

- K' stationary $\frac{U}{D} = \pi$

- K' rotating $\frac{U}{D} > \pi$

- The gravitational field influences and even determines the metrical laws of the space-time continuum. If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean.

- Einstein didn’t care about the spin-up phase. Instead, he concentrated solely on the geometry

- The length of periphery remains unchanged during rotation (this is a consequence of the fact that diameter does not undergo contraction). This can be written as R0 = R where R0 and R denote the radius of a disc at rest and in rotation respectively.

- Each particle laid along periphery is contracted in line with the (instantaneous) linear velocity vector, due to the value of Lorentz factor.

- The number of particles composing the periphery of a disc at rest remains unchanged during the spin-up phase.

- Though the geometry on rotating disc seems to be non-Euclidean, it does not give the sufficient grounds to derive the equivalence principle. The reason is that linear velocity, responsible for relativistic effects, is not directly coupled with centrifugal force.

http://redshift.vif.com/JournalFiles/V14NO4PDF/V14N4RYB.pdf

$L' = L \sqrt{1-\frac{v^{2}}{c^{2}}} = \frac{L}{\gamma(v)}$

$\gamma(v) = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

v = ωr

ω = 2πf

$f = \frac{1}{T}$

Controlling the Stanford time interval analyzer with GPIB

Controlling GPIB Devices