Question

Asked 11 October 2013

# Why can a Sagnac Interferometer see the rotation of the earth, but a Michaelson Interferometer can't see orbital speed?

Could someone with knowledge of Relativity explain why a Sagnac Interferometer can plainly see the rotation of the earth (or at least a rotating reference frame of 1 day) , yet the Michaelson Interferometer can not detect the earth's translational motion around the sun. There seems to be a paradox here and I can't see a mathematical or theoretical way out of it. If there is no Aether....then how can the Sagnac Interferometer see the earths rotation and at the same time the Michaelson Interferometer can't see translational motion? I have heard that some of Michaelsons later experiments had curious results if placed on a moving platform, but these don't seem well documented and are brushed off because of stresses in the devices. I'm open to observances and theory. This is a fascinating subject for me.

## Most recent answer

The acceleration due to the orbit of the earth around the earth-sun barycenter is much smaller than the acceleration of the earth's surface due to the rotation of the earth around its axis.

Both the Sagnac effect and the Foucault pendulum are sensitive to centripetal acceleration, not linear velocity. Absolute acceleration can be measured according to the theory of special relativity. Vector acceleration can be measured by dynamical means, unlike vector velocity.

The vector acceleration of an object fixed on the surface of the earth can be decomposed into several vector components. Components of vector acceleration include contributions from the earths rotation around its axis, the orbit of the earths center of mass around the sun, and the orbit of the sun's center of mass around the center of the galaxy. However, the acceleration of the earths surface due to rotation is by far the largest component of acceleration.

All circular or elliptical motion is associated with centripetal acceleration. You can figure out the different components of acceleration using the formula for centripetal acceleration, a=v^2/r. Here, a is the centripetal acceleration, v is the speed of the object in question, and r is the distance to whatever axis defines the rotational motion.

So the reason that the orbital motion can't be measured is basically because the orbital acceleration is too small. It can be measured in principle. However, the measurement of small accelerations is very difficult. Further, the acceleration of the earths surface due to rotation is a very large background. It is difficult to measure a small signal on a large background.

You can work backwards from the centripetal acceleration of an object to get the speed in the inertial frame of the center of motion. However, note that the object itself is not in an inertial frame. The center of motion is by definition separate from the object. Thus, special relativity does not forbid this speed from being measured. Also note that this speed is not a vector velocity.

The speed measured by the Sagnac experiment must not be confused with the uniform velocity of an inertial frame. The speed in a Sagnac measurement is always associated with a centripetal acceleration. Thus, the speed is not a 'uniform velocity' as defined in special relativity.

## All Answers (6)

That rotation is detectable wheras translation is not, was known already to Newton (rotating bucket experiment). For the much slower rotation of the Earh's body we had to wait till Foucauld did his famous pendulum experiment. From a non-emotional point of view we should not be surprised to see that the propagation of light is sensitive to circumstences to which the motion of massive bodies is known to be sensitive.

It is a basic assumption of special relativity (and special relativity is the physical theory that best seems to fit measurements made over fairly small regions of space and time) that the speed of light is the same in all inertial frames of reference. But it need not be the same in an accelerating frame of reference as in an inertial one, and rotating frames involve acceleration. So you can't detect absolute velocity (because there's no such thing), but you can detect, and even measure, absolute acceleration.

The wiki page

gives a pretty clear description and explanation, and has good references to follow up.

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Eindhoven University of Technology

With a sensitive laser gyroscopes you can certainly see the difference between the rotation period of 24 hours and 24 (1/365.25) hours. In other words you see the total rotation of the earth (in 24 hours) which comes from the rotation of the earth spinning around its own axis (in 24 1/365.25) hours and the additional rotation around the sun.

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I can give a clear answer in terms of special relativity, not general relativity. However, clearly special relativity is sufficient in situations where the gravitational potential is effectively constant of the dimensions of the measuring apparatus. This is the case on earth for the two measurements that you mentioned.

Both the Sagnac interferometer and the Foucault pendulum does not really detect absolute velocity. They really detect absolute acceleration. Therefore, it does not violate special relativity.

The acceleration of the experimental apparatus in principle impacts these two apparatus in other ways. The acceleration changes the internal forces that keep the framework of these devices rigid. These internal forces have never been measured directly, but their existence has to be hypothesized to analyze the results of the measurement. One has to assume that the mirrors in the Sagnac experiment are a fixed distance apart. One has to assume that the rope that holds the weight in the Foucault pendulum stays at a constant length. The centripetal force caused by the earths rotation has to stress these rigid objects.

These devices can not be hypothesized to be in an inertial frame. If the devices were in an inertial frame, the earth's rotation could not change the internal stress in their material. Since they are subject to a centripetal force, they are not in an inertial frame. Hence, whatever these devices measure has to be related in some way to the dynamical acceleration (a=F/m). In principle, one may be able to get the same information by measuring stress on the frame that holds the mirrors together or the cord that holds the weight. However, this is not really practical.

The postulates of special relativity are consistent with a measurement of absolute acceleration defined in terms of dynamical properties (e.g., force). I am not sure that this hold true in the case of general relativity. I don't know general relativity that well. However, I can provide an educated guess as to why general relativity isn't violated either.

Both experiments directly measure the angular velocity of the experimental apparatus relative to the center of the apparatus. However, the angular velocity is not really 'relative'. The 'speed' often quoted is an extrapolation using both the angular velocity and the radius of the earth. The angular velocity measured by both the Sagnac interferometer and the Foucault pendulum are not sufficient to determine the speed of the earths surface. Measurements that determine the radius of the earth are not local.

General relativity postulates that the speed of light is invariant only for local measurements. In order to calculate the speed of the earths surface, a nonlocal measurement has to be made. Someone has to measure the size of the earth. Maybe that someone actually travels around the world. This is not a local measurement in the sense of general relativity.

The devices are not in free fall because they are attached to the surface of the earth. Hence, you can't really assume that their time line is a geodesic. If you place these devices in orbit, then maybe they would be in free fall. However, they are attached to a 'rigid body' which is the earth. So equations relevant to a body in free fall are not appropriate to analyzing these experiments.

Old Dominion University

Thanks for the inputs. Do you know if anyone has done a Sagnac interferometer experiment to see both the rotation around the sun and the earth's rotation. That is there should be statistical data for the support of both rotations assuming one had a large enough Sagnac Interferometer. I've heard that the GPS satelites could essentially do this because they continually need "Sagnac" corrections.

The acceleration due to the orbit of the earth around the earth-sun barycenter is much smaller than the acceleration of the earth's surface due to the rotation of the earth around its axis.

Both the Sagnac effect and the Foucault pendulum are sensitive to centripetal acceleration, not linear velocity. Absolute acceleration can be measured according to the theory of special relativity. Vector acceleration can be measured by dynamical means, unlike vector velocity.

The vector acceleration of an object fixed on the surface of the earth can be decomposed into several vector components. Components of vector acceleration include contributions from the earths rotation around its axis, the orbit of the earths center of mass around the sun, and the orbit of the sun's center of mass around the center of the galaxy. However, the acceleration of the earths surface due to rotation is by far the largest component of acceleration.

All circular or elliptical motion is associated with centripetal acceleration. You can figure out the different components of acceleration using the formula for centripetal acceleration, a=v^2/r. Here, a is the centripetal acceleration, v is the speed of the object in question, and r is the distance to whatever axis defines the rotational motion.

So the reason that the orbital motion can't be measured is basically because the orbital acceleration is too small. It can be measured in principle. However, the measurement of small accelerations is very difficult. Further, the acceleration of the earths surface due to rotation is a very large background. It is difficult to measure a small signal on a large background.

You can work backwards from the centripetal acceleration of an object to get the speed in the inertial frame of the center of motion. However, note that the object itself is not in an inertial frame. The center of motion is by definition separate from the object. Thus, special relativity does not forbid this speed from being measured. Also note that this speed is not a vector velocity.

The speed measured by the Sagnac experiment must not be confused with the uniform velocity of an inertial frame. The speed in a Sagnac measurement is always associated with a centripetal acceleration. Thus, the speed is not a 'uniform velocity' as defined in special relativity.

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