By Don Koks, 2017.

# Rotating Coordinates in Relativity

Relativity is built on the existence of inertial frames.  Suppose you hold a mass in front of you and let it go.  If it floats without moving, then you inhabit an inertial frame.  If the mass falls or moves somewhere else, then you don't inhabit an inertial frame.  The first thing relativity does with an inertial frame is define coordinates for it.  These coordinates are meaningful: in particular, the time coordinate is constructed such that two events in the frame that have the same time coordinate will be measured by all inhabitants of the frame to occur at the same time.  Other coordinates can be constructed that don't obey this idea.  These other coordinates have limited value because, in particular, two events that occur simultaneously might be allocated different values of such a time coordinate.  Coordinates that don't respect simultaneity are misleading; after all, if my plane takes off at 7 p.m., then I would like to think that I could aim to be at the airport at 6 p.m. on my watch.  But if my watch's time is not set to agree with the airport's time, then things fall apart very quickly and the practical use of time becomes pointless.

A global time coordinate that is well behaved in the above sense of respecting simultaneity can be constructed for an inertial frame.  It can also be constructed in a slightly limited sense for a "uniformly accelerated frame".  This is the frame of a rocket that accelerates in such a way that its occupants (a) feel a constant acceleration (a constant weight if they are in deep space far from any sources of gravity), and (b) measure each other as stationary with respect to themselves.  (It turns out that when measured in an inertial frame, the tail of this rocket has to accelerate more strongly than the head of the rocket.  As the rocket accelerates to ever-higher speed, this stronger acceleration of the tail compresses the rocket, and in fact the compression factor is precisely the "gamma factor" of special relativity, if we make allowance for the more complicated kinematics of the situation.)  Two observers onboard such a rocket will agree on the simultaneity of all events, and they will also be able to give the same time coordinate to all simultaneous events.

A natural and very old question in relativity is: can such a global and meaningful time coordinate be constructed for the observers sitting on a disk that rotates at, say, constant angular velocity?  It turns out that it cannot.  The observers sitting on such a disk cannot agree with each other on the simultaneity of events.  In a real sense, this means that no matter how perfect the clocks might be that we use on our rotating Earth, they will still disagree on the simultaneity of events.  In fact, this disagreement amounts to tens of nanoseconds, globally.  (But that doesn't mean GPS suffers, as I'll explain later on.)

The question of constructing rotating coordinates is tied historically to the question of the relativistic physics of a rotating disk.  You can find the physical rotating disk discussed in the FAQ entry The Rigid Rotating Disk in Relativity.  Picturing a rotating disk can help us think about rotating coordinates, but it ceases to be helpful if taken too literally.  Discussion of how the disk rotates, how it gets accelerated and so on, are completely irrelevant to the analysis of any sort of rotating frame within the context of special relativity.  That analysis looks at what is simultaneous with what; and for that, we must appeal to the Clock Postulate to construct a plane of simultaneity at each event of interest.  This, by the way, has nothing to do with the generally accepted meaning of general relativity as a theory of gravity.  If one draws planes of simultaneity at various events in a rotating system, then one can form a halfway coherent picture of those events.  It turns out to be impossible to construct a proper set of coordinates, but we can at least analyse the events in some way.  That analysis simply has nothing to do with questions of the disk's structure and how it got spun up.  To keep this distinction clear, we'll avoid the word disk for the rest of this page, and use "platform" instead.

This idea of not worrying about the physics of creating a moving platform is universally taken for granted in the standard Lorentz transform studied throughout special relativity.  There too, just how the "primed frame" was made to move is never discussed, and never has to be discussed if we wish only to discuss how the world appears from the platform's final state of motion.  The platform is simply taken to have always been moving, forever.  Any other treatment of its motion would only mask and complicate the underlying ideas, which concern the Lorentz transform and not questions of how to accelerate physical objects.  We are content to treat a constant-velocity primed frame as having had its state of motion forever, so we do likewise for the rotating platform.  We also don't demand that a constant-velocity frame must be associated with a massive object (such as a train) that might be moving so quickly that its momentum will contribute to the energy–momentum tensor to curve spacetime.  Such a discussion applies to moving masses, but not to frames.  A study of accelerating physical objects certainly has its place in special relativity, and there's nothing wrong with asking how a platform has been accelerated.  But that is an advanced question of dynamics.

One question with rotation involves the idea of the speed of light.  Does rotation give problems with this speed?  If we spin around on the spot, we see the Moon move around us in a huge circle, so isn't it travelling faster than light in our frame?  It is travelling faster than the tabulated value of "c", but it is not travelling faster than light in its locality; after all, we agree that light is still escaping from its surface.  In the language of special relativity, the Moon's world line always remains within its local light cone; it remains "timelike".  In fact, this behaviour is no different to the well-accepted and well-understood behaviour of light in a uniformly accelerated frame, where the measured speed of light depends upon where in that frame it currently is.  This speed can in fact have any value, from zero to infinity.  (See the FAQ entry Do moving clocks always run slowly? for further discussion of this.)  It's only in inertial frames that light's speed is postulated to have the value "c".

A construction of a rotating platform that treats its points the most "democratically" gives the same helical shape in spacetime to the world lines of all points that lie at equal radii from the centre of rotation.  In that case, the circumference of any circle of radius r is then 2πr by construction.  If you draw various planes of simultaneity at a selection of events on the circumference of such a circle, you'll find that these planes are inclined, and are themselves spinning around.  The result is that observers stationed at each of those events—fixed to points on the platform—measure their neighbours to be farther away than would be the case if the platform were not spinning.  Those neighbours are measured as pushed somewhat toward the opposite side of the circle from the viewpoint of each observer.  Also, each observer measures that observers immediately to his "east" (by which I mean those situated in the direction of rotation) are older than himself, neighbours to his immediate "west" are younger than himself, and observers on the opposite side to him from the centre of rotation have the same age as himself.  This all means that the observers at rest on the platform cannot agree on the simultaneity of events.  They can never construct a time coordinate that has the real meaning of time in the way that it does in an inertial frame.  And this means that those observers simply don't constitute a frame.  So the phrase "relativistic rotating frame" is a contradiction in terms; no such frame can exist.

The non-existence of a relativistic rotating frame is recognised in the field of GPS, the Global Positioning System that is used ubiquitously in our modern world to run both our positioning systems and our clocks.  Our modern Greenwich Mean Time is provided by GPS.  This modern version of GMT is defined a little more stringently than GMT was, and so has been given the new name "UTC" (the acronym's meaning isn't relevant here).  UTC is a valid global time for the "Earth-Centred Inertial Frame" (ECI) in which Earth spins, because inertial frames do admit a global time.  UTC is not a global time for the "Earth-Centred Earth-Fixed frame" (ECEF) in which we live our daily lives, because the ECEF is a rotating frame in the non-relativistic sense of the word, and so cannot support a global time coordinate with the same meaning as the time used in an inertial frame.  That is, when two events have the same UTC time, they are simultaneous for all observers at rest in the ECI, but they are not necessarily simultaneous for any observers in the ECEF, and in fact those two events can be off by as many as 30 nanoseconds in the ECEF.  The operation of the GPS location algorithm isn't affected by this, because that algorithm with its global time coordinate is based on the ECI.

You can appreciate that the time shown by a GPS receiver, while being an ECI time, is not a true ECEF time, when you ask what the famous twins of the "Twin Paradox" would see if they each carried such a receiver with them on their separate trips, as well as ideal wrist watches.  Before parting, their receivers and wrist watches would show identical times, as they should.  The stay-at-home twin then sits for a year, while the other twin goes travelling at high speed (either in space or on Earth).  When they re-unite, the stay-at-home twin is older and his wrist watch shows a later time than his sibling, as expected.  But the twins will say that the same amount of UTC elapsed for each, because this time is what is shown on their GPS receivers, which take their time from satellites.  So elapsed UTC time is not a real time for the twins in the way that their wrist watches and ageing bodies show real time (proper time).

The Born metric that you'll find in discussions of rotation in relativity sets the time coordinate t' on the rotating platform to equal the time coordinate t of the surrounding inertial frame.  This is akin to using UTC in the ECEF; it is essentially nothing more than the Galilei transform dressed up in relativity language, and it has no relativistic significance.  Such "rotating coordinates" can be trivially defined for an inertial frame, but they don't create a rotating frame; they do nothing more than describe the inertial frame in rotating coordinates.  Of course, one can always define a coordinate t' in that way, but that coordinate has no relativistic significance, just as the Galilei transform t' = t doesn't give you a real time for an inertial frame.  Obviously you have to use the Lorentz transform t' = γ (t − vx/c2) + constant to do that.  That's the whole point of special relativity!

The unfortunate fact is that many in the precision-timing world think that creating a set of rotating coordinates that describe an inertial frame actually produces a relativistic rotating frame.  This misconception highlights the increasing need for a proper awareness of time on a rotating platform in modern physics, as our (rotating) world's timing requirements become ever more exacting.  Some of the many papers written in response to the 2012 OPERA neutrino experiment's suggestion of faster-than-light neutrinos discussed the time measurements that may or may not have been involved.  Whatever their conclusions, these discussions showed that an understanding of UTC in high-precision experiments can be lacking in cases where its importance should be paramount.  This is not a fault of experimenters; rather, it's a fault of the modern language of precision timing, which can easily suggest that UTC time is a valid time for the ECEF.

In discussions of rotation that relate to spinning disks, there are some who would represent spacetime on the edge of such a disk by a cylinder (which is certainly valid as far as simple pictures go), then cut the cylinder on a line parallel to the time axis, flatten the cylinder out, draw lines through events, and call those "lines of simultaneity".  But this procedure mis-applies standard one-space-dimension ideas of simultaneity to two space dimensions, and that is not a meaningful thing to do; it's certainly not consistent with ideas of simultaneity already established by the Lorentz transform.  We don't have the freedom to re-arrange spacetime into pieces that suit any arbitrary definition of simultaneity such as this one.  Instead, the cylinder should be cut by a plane of simultaneity, and that plane certainly doesn't intersect the cylinder in a helix.  Simultaneity's definition and its application (such as using this plane) was established by Einstein in accordance with the principles of relativity.  And while in principle it's straightforward to analyse on a rotating platform, in practice the details are difficult.