How do you tell if you are in an inertial frame or not? According to Newton’s model of a fixed space, an inertial frame is one which is at rest or moving with constant velocity through that fixed space. We saw that determining this required us to observe the state of motion through that space of an object Q that was known to be not under the influence of any forces. If you observe Q to be at rest or moving with constant velocity, then you are in an inertial frame. If however, even in the absence of forces, the freely floating object Q appears to be accelerating in some direction with respect to you, you could conclude that this must be because you are in a frame that is an accelerating in a straight line in the opposite direction and hence you are not in an inertial frame.
But the interesting thing that Newton noted is that one could detect if one was in a non-inertial frame even without observing that external object Q if your frame was rotating, because then things in your own frame would start behaving oddly, with cups of tea sloshing around for no apparent reason. In fact, one could even measure the rate of one’s rotation by doing experiments purely within one’s own frame, even if one was in a windowless room so that one could not see outside.
Newton did just such an experiment to arrive at his conclusion that empty space seems to have this property that we cannot tell when we at rest or moving with constant velocity through it but we can tell when we are rotating. What he did was to hang a bucket of water from the ceiling and then twist the rope and let go. When the rope began to unwind causing the bucket to spin, he observed that the surface of the water was no longer flat but had a meniscus, with the water seeming to climb up the sides of the bucket so that the center of the surface was lower than the sides. Thus we can tell if the room we are in is rotating or not by hanging a bucket of water from the ceiling. Even if we do not cause the bucket to spin by twisting the rope, by observing the surface of the water, we can tell if the room is spinning by observing the shape of the surface of the water. We could do this in a closed room and do not need an external backdrop like the stars to know this.
The effects of rotational motion are well known. Rotation, when perceived by an observer who is also rotating, introduces fictitious forces known as centrifugal and Coriolis forces. The former gives the appearance of wanting to make things move away from the center in a radial direction. An ice skater spinning on their axis with their arms hanging down will feel a tug trying to take them away from the body. The bulge at the Earth’s equator is seen by us who are at rest with respect to the spinning planet as being due to the matter near the equator being ‘pulled away’ from the center. Someone who is hanging on the rope in Newton’s bucket experiment and rotating with it would see the water not as rotating but being pulled away from the center and towards the sides. An example of Coriolis forces is when air currents tend to veer away from their expected trajectories. The number of familiar everyday occurrences can be easily expanded. All these are examples of the fictitious centrifugal and Coriolis forces that come into play to explain the deviations from Newton’s laws of motion that we see when we are in a rotating frame.
Newton concluded that space has this property that we cannot tell if we are at rest or moving with constant velocity through it but that we can tell if we are spinning. We can even calculate how fast we are spinning, by doing experiments purely within the rotating frame. The curvature of the water in the bucket increases with how fast it is rotating.
But not all of Newton’s contemporaries were persuaded by his idea of a fixed space having these properties.The philosopher Bishop George Berkeley (1685-1753) for one rejected the idea of absolute space for the modern-sounding idea that it was unobservable. He felt that ascribing properties to space to explain physical effects made no sense and that all physical effects that were felt by any object were due to other physical bodies. If that were the case, then how does one explain the fact of the water surface becoming curved in the rotating bucket and other centrifugal force effects that can be seen even when done in a closed room far away from any other physical body? Berkeley said that it was the distant stars that caused those effects. The fact that we could not see the stars in a closed room did not matter since gravitational forces cannot be shielded. The fact that the stars were very far away could not be used to dismiss their influence since the stars were massive and there were so many of them that we could not conclude that their distance meant that their effects were negligible enough to be ignored. Far from being a passive backdrop like wallpaper, the distant stars exerted dynamical effects.
The difference between Newton’s and Berkeley’s views can be posed more starkly by this thought experiment. If there were no stars, indeed nothing at all in the universe other than Newton and his bucket, and he did the experiment again, would the water surface be curved? According to Newton, the answer is yes, since it is only the bucket’s rotation with respect to space that matters and space remains even after everything else is removed. So for Newton, as far as the bucket is concerned, nothing of significance has changed. But according to Berkeley, the water surface would remain flat because in the absence of the stars we could never tell if we were rotating or not. Similarly, according to Berkeley’s model, in the absence of the distant stars, an ice skater would not feel a tug on their arms to enable them to tell if they were spinning or not and the Earth would have no bulge at the equator.
While one cannot will away the stars to test whether Newton’s or Berkeley’s view would prevail, there is some evidentiary support for Berkeley’s position that it is only rotation with respect to the distant stars, not space, that is relevant. Recall that we can measure the centrifugal and Coriolis forces within a given frame without reference to anything external to it by (say) measuring the curvature of the water surface in a spinning bucket. From that we can calculate the spinning rate even in a closed windowless room. If we do this for the observable phenomena caused by the Earth’s rotation about its axis, we obtain a value for the spinning rate that is very close to the value we obtain using the length of a day. But the length of a day is obtained using the rate of rotation of the Earth with respect to the stars. There is no a priori reason why these two numbers (the first measuring rotation with respect to space, the second with respect to the stars) should come out to be the same. Could this be just a coincidence? Berkeley thought not.
In general, scientists tend to be wary of ascribing events to coincidences. It is not that coincidences cannot happen. For example, the fact that the distance of the Moon from the Earth is such that during a solar eclipse the Moon exactly blocks out the Sun, despite them being of very different sizes, is believed to be a coincidence. There is no deep underlying reason why it should be so and indeed we expect that with time, since the distances will change, the nature of eclipses will change with it.
In the case of values of the two rotational rates being the same, one explanation might be that the stars are fixed in space, so that the rotation rate of the Earth with respect to space also happens to be the same as the rotation rate with respect to the stars. But there are no obvious reasons as to why all the distant stars should be fixed in space. Berkeley proposed that those two rotation rates should be the same because in both cases we are measuring rotation with respect to the stars. Space has nothing to do with it. (We will see later that looking for deeper reasons as to why certain seeming coincidences happen also leads to the important Principle of Equivalence.)
Berkeley’s ideas are now seen as being remarkably modern but in his time, they were seen as a little strange and did not gain much traction. The idea of stars, known even then to be extremely far away, having an effect on the motion of buckets and other objects on Earth seemed almost mystical and while astrologers might like the idea, his contemporary scientists were skeptical. The mathematician Leonhard Euler (1707-1783) said that the alleged influence of stars was “very strange and contrary to the dogmas of metaphysics”.
The search for some frame in which we can measure absolute velocity continues to exert a strong hold. Newton’s idea of an absolute fixed space initially won out and had a powerful effect on scientific thinking for a long time. The idea of an aether that is co-existent with this space and permeates all of space and acts as the medium for the transmission of electromagnetic waves remained the dominant paradigm until the arrival of special and general relativity, when (as we will see later) Berkeley’s views about the distant stars being the reference frame experienced a resurgence due to the work of Ernst Mach and with support from Einstein. Now we also have the Cosmic Microwave Background that permeates all of space and we can measure our velocity with respect to that.
The anisotropy of the cosmic microwave background (CMB) consists of the small temperature fluctuations in the blackbody radiation left over from the Big Bang. The average temperature of this radiation is 2.725 K as measured by the FIRAS instrument on the COBE satellite. Without any contrast enhancement the CMB sky looks like the upper left panel of the figure below. But there are small temperature fluctuations superimposed on this average. One pattern is a plus or minus 0.00335 K variation with one hot pole and one cold pole: a dipole pattern. A velocity of the observer with respect to the Universe produces a dipole pattern with dT/T = v/c by the Doppler shift. The observed dipole indicates that the Solar System is moving at 368+/-2 km/sec relative to the observable Universe in the direction galactic longitude l=263.85o and latitude b=48.25o with an uncertainty slightly smaller than 0.1o.
While we can choose any of these frames as our reference, the choice is arbitrary in the sense that there is no reason to think that any of them are privileged over the others with respect to the laws of physics.
(To be continued.)