Does the Earth Move?

by George L. Murphy

Einstein’s theory of relativity means, among other things, that a modified version of Tycho Brahe’s earth-centered model of the planetary system is, in principle, as good as Copernicus’ sun-centered model. The question of whether the earth or the sun “really” moves is meaningless in this theory. After dealing with challenges to this claim, implications of relativity for understanding biblical texts that were involved in historical debates about the planetary system, as well as some further theological issues, are considered. An appendix provides some mathematical details.

PSCF 63, no. 2 (2011):

7 comments to Does the Earth Move?

  • Randall Isaac

    George,
      Thanks for a very clear explanation of what it means to view the universe from different reference frames. I’m still puzzled about one aspect, however. When an observer is in a particular reference frame, is it possible to determine whether that frame is “at rest” and what would “at rest” really mean?
      For instance, you state: “Accelerated reference frames can be used in Newtonian mechanics at the cost of introducing “fictitious forces.” These are simply the negative of “mass times acceleration” terms in Newton’s second law moved to the other side of the equation and called forces. Centrifugal and Coriolis forces are examples.”
      Does the need to introduce “fictitious forces” indicate that such a frame is not “at rest?” In other words, does the existence of the Coriolis force differentiate a rotating earth from a “stationary” earth?

      Randy

  • George Murphy

    Randy –
    The critical question you pose is “What would ‘at rest’ really mean?”  What general relativity says – perhaps I should say “asserts” – is that the laws of physics must be the same in all systems of coordinates.   If this is the case – and so far there is no observational evidence to the contrary – then there is no preferred reference frame in the sense of one in which the laws of physics have their “true” form.  
    This contrasts with the pre-special relativity situation in which the aether was suppoed to provide a preferred reference frame and a unique state of rest.  in other frames – i.e., those moving with respect to the aether, the equations for electromagnetic phenomena would be different so that, e.g., waves would propagate at speeds other than c.  In pre-general relativity mechanics, OTOH, there was no preferred state of rest because if Newton’s laws hold in one frame, they hold in any other frame moving with respect to that one with constant velocity.  But these “inertial frames” are preferred because in an accelerated frame Newton II, in the form ma = sum of applied forces doesn’t hold.  You have to include the “fictitious forces” which can be traced to the acceleration of the reference frame.
    However, neither Einstein nor anyone else can keep a person from defining “preferred frames” in some other way.  In some ways any particle’s proper frame – i.e., the one in which it’s at rest – can be seen as preferred.  This is clearly the case when the “particle” is an observer who often finds it convenient to consider her/himself at rest, but it’s also true in other cases.  & as Polkinghorne & others point out, a frame in which the microwave background radiation is most nearly uniform is often the most convenient one for cosmology.  But in neither of these cases is the reference frame in question preferred in the sense that the laws of physics have a form in it different from one moving in some arbitrary way with respect to it.
    So a person claiming that he or she is really at rest and some other people are really moving would have to state clearly what that means.  But if the criterion is the fundamental form of the laws of physics then there is no preferred state of rest.

  • Randall Isaac

    Thanks, George. That helps put to rest the “at rest” perspective. But I’m still wondering whether there is any significance to the “fictitious forces.” If I apply Newton’s laws in the earth’s frame of reference and discover that I need to add the fictitious force of the Coriolis effect in order to explain observed motion, what, if anything, can I deduce from the need to add it? Does that effect alone permit me to conclude anything about the relationship of the earth’s frame of reference to the rest of the universe? If so, what would it be?
    Randy

  • George Murphy

    Formally, what the “need” for fictitious forces tells you is that you’re using a reference frame in which free particles (i.e., one’s on which no applied force acts) don’t move in straight lines with constant speed.  But then there’s the interesting question of why we feel a “need” to have free particles move in that way.  There are probably some interesting psychological issues involved but let me note just one thing that often misleads us.
    As I noted with my quote from Schroedinger in the article, using a particular reference frame doesn’t mean we have to live in it.  (I.e., we can use frames other than our proper one.)  But many of the cases of accelerated frames we think of are in fact ones we inhabit – a car speeding up or slowing down or going around a curve, amusement park rides &c.  & in those cases we may think we actually feel these fictitious forces.  If you’re in a car that’s rapidly speeding up, you think you feel a force pushing you into the back of your seat.  But there is no such applied force that’s applied to you.  The seat is pushing you forward (hence your acceleration) & you are pushing in the opposite direction on the seat (Newton III). 
    The apparent need for a ”centrifugal” force in a rotating frame is more deceiving.  Consider that amusement park ride that consists of a rotating cylinder with the riders standing on the base.  When the cylinder gets up to speed the base drops away & the briders are apparently stuck nto the vertical walls with nothing to stand on.  The common – & wrong – explanation is that centrifugal force is presssing them against the walls, resulting in a frictional force that balances gravity.  Close but not quite because there is no “centrifugal” – literally “fleeing the center” – force acting on the riders.  There is instead a centripetal – acting toward the center – force on them exerted by the walls that keeps them moving in a circle.  The outward force is exerted by the riders on the wall.
    That may all seem like just a review of general physics that doesn’t really get at the heart of your question.  What about the Coriolis force that causes the deflection of Foucault’s pendulum?  Isn’t that – as people thought in the 19th century – proof that the earth really is rotating?
    1st let me reiterate the point I amde about the Copernican & Tychonic system:  General relativity doesn’t say the earth really isn’t rotating.  It just says that a frame in which it is isn’t any more preferred or absolute than one that isn’t.
    But here’s an interesting point that Eddington in The Mathematical Theory of Relativity.  (I can find the page numbers if you’re interested.)  The sun’s gravitational field causes a tiny ”dragging” of the earth’s intertial frame (about 2″/century) – it’s the same effect as that of the earth’s field on an orbiting gyroscope that’s recently been confirmed.  The result is that the rate of rotation of the earth with respect to the “fixed stars” differs very slightly from that given by the Foucault pendulum.  So the old claim that the pendulum proves that it’s the earth rather than the heavens that rotate is problematic.
    Eddington says it bhetter than I have – I’ll try to look up his argument.  For now I hope I’ve at least partially answered your question.

  • Randy Isaac

    Partially, but not completely. True, the dragging effect might alter the Foucault pendulum result a bit, but a rather small bit it is. I guess I can rephrase my question this way. Given all this discussion of reference frames, what does the Coriolis force as demonstrated by the Foucault pendulum really tell us? What can we conclude? Perhaps not with its magnitude or its quantitative precision but by its very existence?

  • George Murphy

    What the Coriolis force tells us is that we’re using a non-inertial reference frame.   Consider instead of the pendulum a particle in a ballistic trajectory.  (It’s a little simpler because then there’s nothing but gravity acting on it.)  So suppose a sub at the north pole launches a missile on a ballistic trajectory at target X.  (There doesn’t have to be a warhead so we can be peaceful!)  It won’t hit X but some point Y to the east of it.  An observer in an inertial frame (e.g., in a spacecraft in free fall toward the pole) will see the missile moving on a portion of an elliptical orbit in a single plane.  The earth’s gravity is the only force acting on it. 
    But if we’re on earth & want to say that we’re stationary then we’ll say that the missile is deflected to the east by the Coriolis force.  Clearly the only thing that “causes” that force is our choice of reference frame.
    Further elaboration:  In Einstein’s theory gravitation is also in a sense a fictitious force due to pretending that we’re in flat spacetime when it’s really curved.  In the proper frame of an observer in free fall (like the spacecraft above) the effects of gravitation can be transformed away locally – i.e., close to the observer’s worldline.  But you can’t transform away real gravitational fields over all spacetime.  Fictititious forces like the Coriolis can be eliminated everywhere by an appropriate choice of reference frame.
    You’re right that the frame dragging effect for the earth is tiny (1.”94″/century)  because the sun’s field is weak & the earth is moving slowly.  But to do justice to Eddington, let me quote him.  (This is from the end of section 44 of MTR.)  He envisions a relativist and a believer in absolute rotation on a cloud covered planet so that the stars aren’t visible and then says:
    “Suppose, however, that the earth’s rotation were much slower than it is now, and that Foucault’s pendulum had indicated a rotation of only -1.”94 per century.  The two disputants on the cloud-bound planet would no doubt carry on a long argument as to whether this was essentially an absolute rotation of the earth in space, the irony of the situation being that the earth all the while was non-rotating in the anti-relativist’s sense … ”
    There’s more but those interested can look it up.
    Let me reiterate that the question isn’t whether or not the earth is rotating but whether or not it’s rotating in an absolute sense – i.e., is it wrong to say it’s not rotating?  Clearly it does rotate with respect to the average distribution of mass in the universe but one can also say that the rest of the universe rotates with respect to it.  Whether or not we could say it was rotating if there were  no other matter in the universe is another question.  I think the jury is still out on Mach’s principle (which answers that question “No”).  Particle physicists think the origin of inertia can be explained by higgs field but I’m not old fashioned relatvist & am not so sure about that. 

  • Randy Isaac

    Many thanks for the helpful insights, George. Great article and explanation!

 

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