Something I struggle with: orbital orientation

[JoeSchmuckatelli]

Presuming no atmospheric drag or active control / guidance / gyroscopes / spin, etc... how would a 'dead' satellite in orbit behave?

 

Does either Fig 1 or Fig 2 show the correct way a dead or otherwise unguided, unstabilized object in orbit would behave?  If neither... can someone explain what would really happen and why?

 

(For now... I'm guessing the answer is Fig 2, and willing to be corrected / educated! )

[Terwin]

In most cases it will be Fig 2 over the short term, but if one side of the satellite is heavier than the other, then it will slowly change to Fig 1.  The relative scale of the difference will determine how quicly that happens.

 

For example, the moon is unbalanced, and so it is Fig 1, with one side always facing the earth.

This is because the heavier side will be pulled more strongly than the lighter side when it is not facing either directly towards or directly away form the planet, causing rotation to either speed up or slow down until the rotation exactly matches the orbital period.

 

Edited March 24, 2020 by Terwin

[p1t1o]

28 minutes ago, JoeSchmuckatelli said:

Presuming no atmospheric drag or active control / guidance / gyroscopes / spin, etc... how would a 'dead' satellite in orbit behave?

 

Does either Fig 1 or Fig 2 show the correct way a dead or otherwise unguided, unstabilized object in orbit would behave?  If neither... can someone explain what would really happen and why?

 

(For now... I'm guessing the answer is Fig 2, and willing to be corrected / educated! )

Figure 2 is exactly right. The satellite will only rotate if it is given a rotational impulse. Whilst in orbit, it is in freefall, with no external influence, its attitude (angle in space, if you like) will be entirely independent of the planet. 

 

There is a caveat though - tidal effects.

Gravity weakens with distance. So if an object is large enough, the gravity at the part furthest away will be less than the gravity felt at the part closest to the planet. This effect manifests as a tension between the two ends, acting to pull them apart. With the lower-altitude part of the large object feeling more gravity, this part is puled towards the planet more then the more distant parts, like a pendulum (Note that the centre of gravity is still in orbit, the large satellite does not lose altitude due to tidal effects, but it feels the tension). This is how moons become "tidally locked" and why our moon always has the same face facing us. (Your figure 1 is showing a "tidally locked" satellite)

This of course works with an object of any size, but with human-scale object like a satellite, the effect is infinitesimal in the extreme and will be massively drowned out by larger forces acting on the satellite such as light pressure, drag or solar wind etc.

 

Further reading: Google the "Roche limit" - depending on several factors, with large satellites (like moons) the tidal effects can be large enough to break them apart entirely, this is one of the ways planetary rings form.

Edited March 24, 2020 by p1t1o

[mikegarrison]

15 minutes ago, p1t1o said:

This of course works with an object of any size, but with human-scale object like a satellite, the effect is infinitesimal in the extreme and will be massively drowned out by larger forces acting on the satellite such as light pressure, drag or solar wind etc.

There have been attempts to test "gravity-gradient stabilization" (ie. inducing a sat to be tidally locked). Most of them have failed, according to what I have read, but a light satellite with a relatively strong bimodal mass distribution on either end of a long tether or beam should be somewhat self-stabilizing.

[JoeSchmuckatelli]

2 hours ago, p1t1o said:

Figure 2 is exactly right. The satellite will only rotate if it is given a rotational impulse. Whilst in orbit, it is in freefall, with no external influence, its attitude (angle in space, if you like) will be entirely independent of the planet. 

 

There is a caveat though - tidal effects.

Gravity weakens with distance. So if an object is large enough, the gravity at the part furthest away will be less than the gravity felt at the part closest to the planet. This effect manifests as a tension between the two ends, acting to pull them apart. With the lower-altitude part of the large object feeling more gravity, this part is puled towards the planet more then the more distant parts, like a pendulum (Note that the centre of gravity is still in orbit, the large satellite does not lose altitude due to tidal effects, but it feels the tension). This is how moons become "tidally locked" and why our moon always has the same face facing us. (Your figure 1 is showing a "tidally locked" satellite)

This of course works with an object of any size, but with human-scale object like a satellite, the effect is infinitesimal in the extreme and will be massively drowned out by larger forces acting on the satellite such as light pressure, drag or solar wind etc.

 

Further reading: Google the "Roche limit" - depending on several factors, with large satellites (like moons) the tidal effects can be large enough to break them apart entirely, this is one of the ways planetary rings form.

Thanks!

 

So... this might sound weird, or be too esoteric for this forum... but why is the orbit behaving in a Newtonian fashion?  (Feeds into a larger thing I'm struggling with: why we can use Newtonian physics to sling a probe past Pluto, but we need Einstein to describe the precession of Mercury)  In other words, if gravity is merely a reflection of curved space-time, why isn't Fig 1 correct?

[mikegarrison]

1 minute ago, JoeSchmuckatelli said:

Thanks!

 

So... this might sound weird, or be too esoteric for this forum... but why is the orbit behaving in a Newtonian fashion?  (Feeds into a larger thing I'm struggling with: why we can use Newtonian physics to sling a probe past Pluto, but we need Einstein to describe the precession of Mercury)  In other words, if gravity is merely a reflection of curved space-time, why isn't Fig 1 correct?

Because the differences between Newton and Einstein are very, very small. GPS sats need timing so precise they do need to adjust for relativity, but Fig. 1 versus Fig. 2 does not require such precision.

[Bill Phil]

1 hour ago, JoeSchmuckatelli said:

Thanks!

 

So... this might sound weird, or be too esoteric for this forum... but why is the orbit behaving in a Newtonian fashion?  (Feeds into a larger thing I'm struggling with: why we can use Newtonian physics to sling a probe past Pluto, but we need Einstein to describe the precession of Mercury)  In other words, if gravity is merely a reflection of curved space-time, why isn't Fig 1 correct?

Actually I believe we do use relativistic corrections for certain trajectories provided the need is present.

With low enough masses and velocities the differences are small enough to be ignored.

A satellite will retain its orientation/rotation relative to the “fixed stars” (which aren’t fixed but we say they are when it’s convenient) or the frame of reference unless acted upon by a torque or other influence. I believe this is the case whether or not you use Newtonian physics or GR. 

Gravity can apply torque though to large enough objects. 

[magnemoe]

10 hours ago, mikegarrison said:

There have been attempts to test "gravity-gradient stabilization" (ie. inducing a sat to be tidally locked). Most of them have failed, according to what I have read, but a light satellite with a relatively strong bimodal mass distribution on either end of a long tether or beam should be somewhat self-stabilizing.

Yes, note that most satellites doing communication or earth observation is set to rotate once every orbit like figure 1, same is true for the IIS however here drag is also an major issue. 
The problem with gravity stabilization is that You need an odd shaped satelite who would add weight and be harder to launch, you could use an tether but that would make adjusting orbit harder. 
As you can keep the rotation and just adjust with reaction wheels with a bit of RCS its not practical. 

However as other point out this rotation will not stay fixed on an dead satellite. But I don't think the satellite will stop rotating like in figure 2 either.  My guess is an rater chaotic rotation. 

[kerbiloid]

15 hours ago, JoeSchmuckatelli said:

why we can use Newtonian physics to sling a probe past Pluto, but we need Einstein to describe the precession of Mercury

At least because 1 Pluto orbital period ~= 1000 Mercury's turns,
while Mercury is 100 times closer to the Sun than Pluto,
so the relativistic effects are by 1..2 orders of magnitude greater, and any angular error accumulates 1000 times faster.

[p1t1o]

18 hours ago, JoeSchmuckatelli said:

Thanks!

 

So... this might sound weird, or be too esoteric for this forum... but why is the orbit behaving in a Newtonian fashion?  (Feeds into a larger thing I'm struggling with: why we can use Newtonian physics to sling a probe past Pluto, but we need Einstein to describe the precession of Mercury)  In other words, if gravity is merely a reflection of curved space-time, why isn't Fig 1 correct?

Newtonian physics and Einsteinian physics are both approximations of the same thing - physical reality.

The question "Why is the orbit behaving in a newtonian fashion?" is the wrong question - it has no meaning.

The orbit is behaving in a real, physical fashion and we use Newtonian physics to describe and predict it.

Nature doesnt follow our theories, our theories follow nature.

 

For regimes below relativistic (say, the problem involves no velocities above 0.1c), both Einsteinian and Newtonian physics are extremely accurate - but Einsteinian physics is harder and more complex, whilst Newtonian calculations can mostly be done in your head, so we would always use Newtonian here.

When velocities approach relativistic regimes, Newtonian physics loses its accuracy (like trying to measure the temperature of the core of the sun, with a mercury thermometer, the results - vast expansion of mercury vapour - make no sense to the scale printed on the side of the glass tube) so we would use Einsteinian.

[JoeSchmuckatelli]

57 minutes ago, p1t1o said:

Newtonian physics and Einsteinian physics are both approximations of the same thing - physical reality.

The question "Why is the orbit behaving in a newtonian fashion?" is the wrong question - it has no meaning.

The orbit is behaving in a real, physical fashion and we use Newtonian physics to describe and predict it.

Nature doesnt follow our theories, our theories follow nature.

 

For regimes below relativistic (say, the problem involves no velocities above 0.1c), both Einsteinian and Newtonian physics are extremely accurate - but Einsteinian physics is harder and more complex, whilst Newtonian calculations can mostly be done in your head, so we would always use Newtonian here.

When velocities approach relativistic regimes, Newtonian physics loses its accuracy (like trying to measure the temperature of the core of the sun, with a mercury thermometer, the results - vast expansion of mercury vapour - make no sense to the scale printed on the side of the glass tube) so we would use Einsteinian.

I love this part of your response - thanks for the reminder! 

So - if we're looking at something big, or something fast - or both like the gravitational lensing of light, the better approximation is Einstein?  That I should see his work as mathematicaly descriptive rather than physically descriptive? 

Some of my reading has led me to understand his argument about lensing is that the light is not curved by the gravity of the lensing object - but rather it travels in a straight line through spacetime that has been curved.  Trying to grasp curved spacetime as a reason for orbits /gravity as opposed to a field (~gravity being akin to magnetism) makes sense when you consider the feather and hammer drop at the same rate on the moon... 

This in turn led me to ask if this is a true description of gravity - why isn't fig 1 curving space to keep the satellite facing the planet? I. E. The satellite isn't pulled by some gravitational field originating from the planet... But is rather trying to go straight in spacetime that has been curved by the planetary mass. 

 

But then I've also seen that orbits actually work like fig 2... And from what I can see - fig 2 is at odds with curved space but would work if gravity were a field. 

Which is why I struggle to balance the two concepts. 

 

(side note - if anyone can suggest some good books explaining this, I'd love to see them!)

EDIT - I just finished Ron Cowen's Gravity's Century which is a great read... but it hasn't answered the question I'm struggling with 

Edited March 25, 2020 by JoeSchmuckatelli

[Nightside]

9 minutes ago, JoeSchmuckatelli said:

This in turn led me to ask if this is a true description of gravity - why isn't fig 1 curving space to keep the satellite facing the planet? I. E. The satellite isn't pulled by some gravitational field originating from the planet... But is rather trying to go straight in spacetime that has been curved by the planetary mass. 

 

But then I've also seen that orbits actually work like fig 2... And from what I can see - fig 2 is at odds with curved space but would work if gravity were a field. 

Does the orientation of the satellite have any consequence to it's following the curvature of space-time?  

I would argue that Figures 1 & 2 are exactly the same, if we are considering the sat to be a point mass.  For a point mass, we can ignore gravity-gradient due to mass eccentricity and angular momentum.

If we consider those things, then Fig 1 is just showing a sat rotating at 1 rev/orbit, and fig 2 is showing a sat rotating at 0 rev/orbit, how do these different angular momenta figure into a path through curved space? (I guess momenta is the plural of momentum, according to spellcheck - something they don't teach you in school!).  

 

[JoeSchmuckatelli]

9 minutes ago, Nightside said:

Does the orientation of the satellite have any consequence to it's following the curvature of space-time?  

I would argue that Figures 1 & 2 are exactly the same, if we are considering the sat to be a point mass.  For a point mass, we can ignore gravity-gradient due to mass eccentricity and angular momentum.

If we consider those things, then Fig 1 is just showing a sat rotating at 1 rev/orbit, and fig 2 is showing a sat rotating at 0 rev/orbit, how do these different angular momenta figure into a path through curved space? (I guess momenta is the plural of momentum, according to spellcheck - something they don't teach you in school!).  

 

Well - that's kinda the crux, isn't it?  

Viewing as a point mass is good for the maths... But I'm not trying to plot it out mathematically... I'm trying to understand what the math is saying about how gravity works. 

And at the risk of exposing my great ignorance, I'll expand: 

If we take away all rotation or spin or any active control of the object / satellite wouldn't it behave differently if one of the current descriptors were correct? 

 

If gravity is best described as merely an effect of curved spacetime, the object should follow Fig 1... but if its a field should it not follow Fig 2?

[Nightside]

I have another comment, but could you first clarify why you expect different results in these cases?

16 minutes ago, JoeSchmuckatelli said:

If gravity is best described as merely an effect of curved spacetime, the object should follow Fig 1... but if its a field should it not follow Fig 2?

 

[JoeSchmuckatelli]

5 hours ago, Nightside said:

I have another comment, but could you first clarify why you expect different results in these cases?

 

I'll try.  Again, I'm acknowledging profound ignorance of the maths and trying to picture what is going on.  FWIW - I'm a reader of physics and cosmology, but not a student or scholar... in fact 'frozen caveman lawyer' would be closer to the truth!

 

My curiosity is driven by several seemingly inconsistent things I've read about (in no particular order):

• Photons are presumed to be massless, and thus unaffected by gravity; except we've photographs of gravitational lensing - used as one of many proofs of Einstein's theories, ergo gravity is an effect of mass curving spacetime - but here, where an outside observer might think the path of the photon is actually being turned... the answer is that from its frame of reference it's still going straight, but spacetime is curved near the lensing mass; thus no 'gravitational field' acted upon the photon
• The search for the Higgs (& etc.) is a quest to find a particle to help rectify the Standard Model with GR... but if they succeed, won't that mean accepting gravity has field like properties; and if so... how does a massless photon get turned by a mass?
• The folks who launched New Horizons past Pluto used Newtonian physics to perform this incredible feat - they did not have to 'get down into the weeds' with Einstein; meaning Newton's physics is a pretty damn good approximation - and while it doesn't demand that gravity be a field... it doesn't refute this possibility either.
• The precession of Mercury isn't well described using only Newtonian physics, but is described well by Einstein's theories... so at some point Newton's approximation isn't sufficient and Einstein's is better... but why?
• Our planet experiences seasons: the axial tilt keeps our Northern Hemisphere toward the sun during summer and away during winter... but why does it behave differently than a photon?  I.e. if a fast moving photon's trajectory compared to the background stars can be bent by curved spacetime near a large enough mass... why doesn't the earth's path follow the same 'straight line' idea (I.e. moving in a straight line but through space that has been curved by the mass of the sun to keep it in an orbit... which in this case might keep the Northern Hemisphere always toward the sun rather than generally toward Polaris)…  which, I get that the spin stabilization is an enormous factor; but that seems inconsistent (to me) with curved spacetime - I.e the planet is affected differently than the photon (read; it's not 'still traveling in a straight line through curved space) -- and this, to me makes more sense if gravity is a field; the spin stabilization keeping the axis aligned with the fixed stars 'just works' better referencing a field produced by the mass of the sun attracting the mass of the earth (or each, the other) than does curved spacetime
• Tidally locked bodies could be due to either a 1 rotation to 1 orbit, or because they're not uniformly balanced and thus the denser part will be closer to the mass it orbits (like a Weeble will wobble but not fall down)... but then a tidally locked body also looks like one behaving like a photon; from its frame of reference it's following a straight line, but the outside observer sees it as travelling a circular path defined by the mass it orbits warping spacetime.

All of which probably helps you understand my confusion less than before I tried to explain.  (Insert rueful grin)

 

But it's like this; the coin dropped into that funnel at the zoo starts out in a straight line, but is then captured by the curve of the funnel, which spins it down to the donation bin.  The coin, in effect, acts like the photon; its path would otherwise be straight except for the fact that the surface it is traveling on is curved - and the axis of its rotation is all over the place.  IF that is a decent analogy to the curvature of spacetime defining the path of an object around another massive object... why isn't the earth's axis of rotation all over the place as well?

Edited March 25, 2020 by JoeSchmuckatelli

[magnemoe]

9 hours ago, p1t1o said:

Newtonian physics and Einsteinian physics are both approximations of the same thing - physical reality.

The question "Why is the orbit behaving in a newtonian fashion?" is the wrong question - it has no meaning.

The orbit is behaving in a real, physical fashion and we use Newtonian physics to describe and predict it.

Nature doesnt follow our theories, our theories follow nature.

 

For regimes below relativistic (say, the problem involves no velocities above 0.1c), both Einsteinian and Newtonian physics are extremely accurate - but Einsteinian physics is harder and more complex, whilst Newtonian calculations can mostly be done in your head, so we would always use Newtonian here.

When velocities approach relativistic regimes, Newtonian physics loses its accuracy (like trying to measure the temperature of the core of the sun, with a mercury thermometer, the results - vast expansion of mercury vapour - make no sense to the scale printed on the side of the glass tube) so we would use Einsteinian.

Or if you need extreme accuracy because of reasons like GPS and the way it works. You probably need  to calculate for Mercury drift if you send an probe to intercept it especially if you need multiple flybys. 
But for normal orbital mechanics it don't 

[mikegarrison]

4 hours ago, JoeSchmuckatelli said:

I'll try.  Again, I'm acknowledging profound ignorance of the maths and trying to picture what is going on.  FWIW - I'm a reader of physics and cosmology, but not a student or scholar... in fact 'frozen caveman lawyer' would be closer to the truth!

 

My curiosity is driven by several seemingly inconsistent things I've read about (in no particular order):

[large snip]

You are still making the mistake @p1t1o mentioned. Plus, you are not actually doing the math so you are not seeing that the predictions from the Newton model and the Einstein are *very* close.

There are 1,296,000 arc-seconds in a full circle. The Newtonian calculation for the precession of Mercury was off by 43 arc-seconds/century. It was enough to be spotted in very careful astronomical observations, but this means it would take three million years for the extra precession to go all the way around the sun full circle.

If I say a Subway sandwich is a foot long, is it? Well, it's about a foot long. The more precisely I measure it, the more likely I am to find that it is not exactly 12 inches long. I might say that the "visual model" says it's a foot long, the "ruler model" says it's 11-7/8 inches long, and the "laser measurement" model says it is 11.8264 inches long. But these are not really contradictory.

[JoeSchmuckatelli]

5 minutes ago, mikegarrison said:

You are still making the mistake @p1t1o mentioned. Plus, you are not actually doing the math so you are not seeing that the predictions from the Newton model and the Einstein are *very* close.

There are 1,296,000 arc-seconds in a full circle. The Newtonian calculation for the precession of Mercury was off by 43 arc-seconds/century. It was enough to be spotted in very careful astronomical observations, but this means it would take three million years for the extra precession to go all the way around the sun full circle.

If I say a Subway sandwich is a foot long, is it? Well, it's about a foot long. The more precisely I measure it, the more likely I am to find that it is not exactly 12 inches long. I might say that the "visual model" says it's a foot long, the "ruler model" says it's 11-7/8 inches long, and the "laser measurement" model says it is 11.8264 inches long. But these are not really contradictory.

Grin - as I typed my response above, his words kept niggling the back of my mind; clearly I have a few days of thinking ahead to try to parse out why I've been so pigheaded about this!  

 

-- and thanks for the arc-second bit; I've read around that several times but never had it put so succinctly.  

[Nightside]

17 hours ago, JoeSchmuckatelli said:

Photons are presumed to be massless

It depends on who is doing the presuming... don’t tell Bill Nye and his merry band of Lightsailors that photons gave no mass.

On the other hand, in my line of work, geotechnical engineering, we assume air is massless for some calculations. Obviously this is false, but the mass of any air in the soil is so much less than the margin of error in the weights of solids or water, that it can be ignored.

For most observations there are innumerable factors that are difficult to measure and calculate, but it isn’t always clear why. You are absolutely right to try figure out why things are calculated one way or another.

But back to gravity...

[mikegarrison]

Photons have no *rest* mass. But photons are never at rest, and they have momentum.

[K^2]

On 3/25/2020 at 9:08 AM, JoeSchmuckatelli said:

If gravity is best described as merely an effect of curved spacetime, the object should follow Fig 1... but if its a field should it not follow Fig 2?

In both models, the object can follow either orientation, depending on how fast it's spinning. If the satellite spins at 1 revolution per orbit, it will follow orientation in Fig 1. (Or five revolutions per orbit! That's called aliasing. But now I'm off on a tangent.) However, if we insist that the object is not spinning, it will be closer to Fig 2. Now, the reason I'm saying "closer" is because in Newtonian physics, there's an exact notion of what it means for object to not spin, and then we get Fig 2 and that's that. In General Relativity things get a little more complicated.

In Newtonian Physics, we assume that motion is relative. You can't answer the question of how fast you are going. Only how fast you are going relative to something else. But you can always answer a question of how rapidly you are rotating. In Newtonian Physics, there is a globally inertial frame of reference. You can pick a direction and always know that direction, and if you keep facing in that same direction, you aren't spinning. Simple! And so a non-spinning satellite always faces the same direction, and we have Fig 2. I'm sure that's clear, but I wanted to walk through this to set things up for General Relativity.

Well, in GR, things get complicated. Imagine that you are standing on the surface of Earth at the equator. You start out pointing your hand directly North. Now you walk North until you reach the pole. (I don't know, water's frozen, or something.) You are trying to keep pointing in the same direction, but along the surface, and nothing really changes until you make it to the pole. You then start going directly to your right. So you're moving South, but 90° of latitude from where you started. If you kept pointing your hand in the same direction, you are now pointing it East. And once you're at equator, you can move directly West until you're back where you started from. You always tried to point in the same direction, but somehow, you started out pointing North, and ended up pointing East. That's result of curvature.

Things get further more complicated because time is a dimension and so, just like space, is subject to curvature. That's why gravity pulls on objects that aren't moving - they are still "moving" through time. And so in general, just like you can't be sure of direction you're pointing when moving through curved space, even if you aren't going anywhere, keeping track of what's the same direction as you age through time is also impossible. So how can we tell if something is spinning if we can't say if the orientation an object is pointing at now is same or different than orientation a few minutes later. Well, we still have concept of parallel transport. Just like you could keep pointing in the "same" direction as you moved along the surface of the Earth, you can maintain the "same" direction. And so while there is no global inertial frame, there is always a local inertial frame. And because of that, a notion of whether an object rotates local to that object. For a satellite, that's a local free-falling frame along that satellite's orbit. And relative to that, we can absolutely say if the satellite is turning or not. But in general, you, standing on the ground, might disagree with measurements on the satellite itself.

This is the part where I would have to drag in a lot of heavy math to derive these things, so I hope you take my word for it. (Or ask someone else who understand the math and whom you trust more than stranger on the internet!) If Earth was a perfect sphere and was not rotating itself, nor had significant electric charge, the curvature it would produce is described by Schwarzschild Metric. It's one of the earliest exact solutions to Einstein Field Equations that we know. It's also why we knew to look for black holes and made many other discoveries. If our satellite was completely irrotational relative to its local free-falling frame of reference in Schwarzschild Metric, it would also perfectly follow Fig 2. The reason satellite moves in a circle but the chosen "forward" direction doesn't curve with it is related to why a much faster satellite would curve less. Even light moves through time as well as space, and what we need for a reference is a straight line in space only, and that does not curve in this particular metric.

But Earth isn't a perfect sphere and isn't even electrically neutral. But it's Earth's rotation that throws a biggest wrench into our assumptions. There's another exact solution to Einstein's Field Equations that describes space-time near a massive rotating object. It's called a Kerr metric. And Kerr metric is weird. It doesn't get too wild until you have something really heavy with a lot of angular momentum, like a supermassive black hole, but technically, it's still a better description of what happens near Earth. And in that metric, there's something called frame dragging. It's named so because it's almost as if rotating object was pulling space-time itself along, creating a bit of a twist. And that twist does mean that an object orbiting above, one that's perfectly steady relative to its own local frame of reference, will appear to rotate ever so slightly from perspective of someone standing on the surface. Of course, from perspective of the satellite, it's the rest of the universe that'd be drifting around ever so slightly.

And just to be clear, I don't expect that effect to be remotely measurable. There are too many other things going on. There are tidal forces that make measurement of rotation much harder. There is the fact that Earth isn't a sphere, and gravitational field will vary, generating various torques that you'll have to exclude. And even external forces due to light pressure and what little of air resistance there is making it impossible to keep things steady enough. But mathematically speaking, there should be a tiny amount of drift.

[JoeSchmuckatelli]

On 3/28/2020 at 4:00 AM, K^2 said:

... .

Grin

 

I love this stuff.  Thanks for the long reply! 

 

I really appreciate people taking the time to write responses to my questions, and I learn something new every time.  e.g. I've read about frame dragging, but never heard of the Kerr metric until today (now I have more reading ahead of me!). 

 

Back with more questions after I've had a chance to read more and scratch my head a bit!