I read Einstein’s book on Relativity last year. It’s divided into two parts:
The Special Relativity section demonstrates that physical laws hold fast regardless of what Newtonian (non-accelerating) reference frame is used.
The General Relativity section demonstrates that physical laws hold fast regardless of what reference frame is used, be it an accelerating one or not.
If one can follow the arguments in the book, one will conclude that it is impossible to tell a gravity field from a linear acceleration field. But what about a radial acceleration field? This is the active principle at the heart of my question (to follow):
In terms of orbital mechanics, what is the difference between A) a planet orbiting a star at a constant radius, and B) a planet orbiting a star at a constant radius, and the star is also rotating at a rate to exactly match the orbit of the planet. In other words, from both the star and from the planet, it appears as though there is no movement. An observer on either the star or on the planet would understandably be confused, as the two bodies will appear not to have a relative motion nor a gravitational attraction to each other. Yet, the two bodies will remain at a constant distance from each other. How can our observer justify this observation and yet adhere to the known laws of physics?
Assume the planet always shows the same face to its star, and that the “fixed stars” are not visible from this system (they have no telescopes, or it is always daytime, or it is always hazy, or for whatever reason you can imagine).
In the second case, the observer would postulate a “gravity field” that exactly mimics the effects of the centrifugal force one would expect in a rotating reference frame. This “centrifugal gravity” would exactly balance out the effects of the spacetime curvature, and the planet would experience no net force.
The moon doesn’t orbit the Earth; they both orbit around a point between them but because of the difference in mass, that point is about 1700 km beneath the Earth’'s surface.
Have you looked at Coriolis forces? If you’re in any sort of non-linear accelerated system you will be able to do do measurements telling you the nature of this acceleration.
In your example the local scientists could hypothesise whatever odd laws they wanted, but satellites orbiting either the star or the planet would show that they were in fact rotating. (Or the scientists could go on an contrive a reason for orbital speeds being different depending on direction of the orbit.)
“the barycenter.” I keep reading that in online summaries of orbital mechanics. However, a center is a point, and I cannot measure tangential speed with respect to a point, so this makes no sense to me.
I cannot look at a point and tell you how quickly I am moving around it. Someone must choose another point outside of the barycenter and extend a line through the two. Then I can tell you how many seconds it takes me to get back around to that reference line; if we have my distance from the barycenter, I can even give you a tangential speed.
It’s this second reference point that I am looking for. It can’t be a point on the surface of the earth. This would mean that:
A geosynchronous orbit would have an orbital speed of zero; and we all know orbital mechanics would dictate an orbiting distance of zero (or infinity) for this case.
The orbital speed required would be drastically different for a polar pass than it would be for an equatorial pass.
So how is our second point decided? How do we measure orbital speed?
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Have you looked at Coriolis forces?
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This is the key. A planet that is tidally locked is also rotating once an orbit. Many of the extrasolar planets that have been detected have very short orbital periods, on the order of 24 hours or less; these planets will also be rotating once every orbit, and could have rotation periods shorter than that of the Earth.
Coriolis effects will be very detectable on these planets, especially if they have an atmosphere; hurricanes, Hadley cells, and so on. A tidally locked planet (or star) is in a very different state to one which is not rotating at all.
I guess I’m confused by your confusion. Usually, we assume that we have some set of coordinates (for time & space) that we’ve set down in advance to describe our problem; the setup that’s usually thought of in Special Relativity is a “rods-and-clocks” grid that allows you to measure distances not at your location. If someone sitting at the barycenter of the system set up such a coordinate grid, they could measure the satellite’s velocity easily.
This question really isn’t about general relativity as such (in general relativity a rotating star behaves differently gravitationally to a non-rotating star, unlike Newtonian gravity), it’s more about the difference between rotating and non-rotating frames.
One way in which the difference can be seen in terms of the behaviour of test particles (i.e. objects of negligible mass). The behaviour of test particles launched by the observer on planet is completely different in the rotating and non-rotating cases and by conducting experiments in which the paths of test particles are observed, the observer on the planet could determine the rotation of the planet, the mass of the planet, the mass of the star, etc.
The non-rotating reference frame of the rest of the universe. Even if we, as in your hypothetical, couldn’t see it, we could deduce the existence of a non-rotational universe with the orientation it has, from observations of local orbital mechanics.
Actually, that’s not true. It’s not impossible but very difficult. If you’re on a space ship that’s constantly accelerating at 1g and you drop two objects they will fall ***exactly ***parallel to each other. However in the gravity field of something like the Earth, since they’re both traveling towards the same point (the Earth’s center) they’re not falling exactly parallel, so the distance between them will decrease ever so slightly as they fall. In this scenario however the amount is so tiny it would be extremely difficult to measure.