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Metrics and forces

  1. Aug 14, 2006 #1
    I need help, please, in understanding two extreme but quite different situations, say 1 and 2 below:

    1. Suppose that an observer falls freely and radially towards a neutron star. As he approaches the star he will begin to detect, by observing test particles in his local inertial frame (set up far from the star), increasingly more apparent tidal phenomena. For instance test particles he releases from rest along a line through the centre of the star will be measured to accelerate and separate from each other.

    In fact this observation must lead him to conclude that his inertial frame is getting too big for its boots, as it were, and that he must restrict it to a volume in which such tidal phenomena remain imperceptible. The extent of a local inertial frame is of course subjective and depends on circumstances.

    Setting aside this caveat, the observer will find that if he ties the particles together with string before releasing them, the string will eventually break. This he will attribute to a tidal force, if he adopts a Newtonian perspective instead of explaining such phenomena in terms of the Schwartzchild metric.

    2. Consider the same observer (somehow surviving) in an inflating flat FRW universe that begins to expand exponentially rapidly after he has set up his local inertial frame. Suppose he again releases two test particles from rest in this frame. What happens as the scale factor, and its derivatives with respect to time, change exponentially with time?

    Am I correct in assuming that nothing at all happens, and that his local inertial frame, with its test particles at rest, remains undisturbed despite the extreme "stretching of space" that takes place everywhere as his universe inflates? (I believe that this is the view taken by cosmologists.)

    Or am I wrong, and will a string connecting these particles break?
    Last edited: Aug 14, 2006
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  3. Aug 14, 2006 #2
    I just want to point out that the two situations are not exactly opposites.

    While "time" in a gravitational field has a non-linear relationship with distance from the center, in a FRW model the expansion is linear with regards to time.
    If this impacts the "coupling" of space-time with regards to EM-forces I do not know.
    Last edited: Aug 14, 2006
  4. Aug 14, 2006 #3


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    In GR, the tidal force is one of the components of the Riemann tensor.

    The tidal force in the radial direction works out to be just 2GM/r^3 in a GR "frame field", so the result is very similar to the Newtonian result. (There area compressive tidal forces as well, just like the Newtonian case).

    frame field:

    I think the author (probably Chris Hillman from the way it reads) addreses the GR "tidal tensor" and mentions that it's the same as the Newtonian version.

    This blows up as one approaches the singularity (r->0). That's why singularities are singular.

    If the matter density is low enough not to affect the expansion, (case A) the expansion will be exponential and the tidal force per unit length in a "frame field" will be constant for a "De-sitter" universe.


    If the matter density is high enough, it will attempt to "fight" the expansion for a while, the expansion will be slower than exponential. As the matter density thins, this will eventually become an exponential expansion like case A.

    So the tidal force won't increase indefinitely in a "De-sitter" universe with a cosmological constnat, the amount of tidal force / unit length will basically be set by the value of the cosmological constant.

    I hope this is isn't too unclear
  5. Aug 15, 2006 #4
    Yes, the point I am interested in is whether in the case of the FRW model there is or there isn't "coupling" of space-time with regards to EM-forces, as you put it.

    In cosmology folk may have mixed views, from what I've read. Some seem to assume there is no coupling at all, while others talk of expansion exerting (in the case of the less-extreme expansion of the Hubble flow) a weak "force" that tends to disrupt EM and/or gravitationally-bound objects.

    I'm hoping that the experts in this specialised forum will be able to clarify the matter. Thanks for your help, MJ.
  6. Aug 15, 2006 #5
    I follow your reply to the first toy situation I described. Thanks very much for the explanation, which I follow and agree with.

    In your discussion of the second situation (the de Sitter FRW case), though, I didn't quite understand some of the points you made.

    First, you seem to accept that there is indeed a "tidal force" that can "fight the expansion". Is there really and truly a "force" of this kind in the de Sitter case? If so, exactly how does it arise from a dilation or expansion that is isotropic, in a homogeneous universe? This is actually the central question I'm seeking an answer for -- the string-breaking question, as it were!

    Second, I didn't understand what you meant by "tidal force/unit length ", which sounds a bit like surface tension in a liquid. But this is rather a minor point.

    I hope I'm not just being stupid about the whole thing.
  7. Aug 15, 2006 #6


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    Sorry, when I was writing it I could tell I wasn't being very clear, but I wasn't sure what to do to fix it.

    Consider two masses.


    Without a cosmological constant, and with matter present, B always accelerates towards A.

    This is why the expansion of a universe slows down (decelerates) in cosmologies without a cosmological constant.

    With a cosmological constant, without matter present, B always accelerates away from A. (Well, it depends on the sign of the constant, actually. I should say, with a "dark energy" sort of cosmological constant, B always accelerates away from A).

    When you have both (matter, and "dark energy") the two effects fight each other.

    Note that the directon of expansion doesn't have anything to do with the direction of acceleration. While all FRW models without a cosmological constant have B acclerating towards A, the universe still expands in these models. (B moves away from A, but acclerates towards it, i.e. B slows down as it gets further away from A).

    If you stick a big meter stick in a matter-free de-Sitter universe (well, it still has the meter stick in it! But no other matter) the meter stick will experience a tidal force. You can calculate the magnitude if you know the Riemann curvature tensor from the geodesic deviation equation. The force will be a "pull it apart" sort of force as per the diagram. In formal language, B and A both follow geodesics, but the geodesics accelrate away from each other (geodesic deviation).

    The accleration between A and B is proportional to the distance. Thus, force / distance.

    If A and B are twice as far apart, the relative acceleration (aka "force") is twice as great.
  8. Aug 15, 2006 #7
    It's all starting to make sense to me, thanks to you. Give me a short while, and I may yet end up understanding expansion properly.

    One further point: since a meter stick experiences a "tidal force" in an (otherwise) matter-free expanding universe, which as you say is a "pull it apart" sort of force, what happens during the extreme expansion of inflation?

    In situations (such as near a neutron star or sun-sized black hole) where tidal forces are extreme, material objects would literally be pulled apart. Objects even as small as nuclei would eventually be disrupted if they approached close enough to the singularity of a a black hole.

    How do elementary entitities (electrons? quarks?) survive inflation? Is it perhaps the high initial mass/energy density that mitigates inflation's extreme disruptive "tidal" effect, as you said:

  9. Aug 15, 2006 #8
    Upon reflection I see that the solution I suggested above for this question is sheer nonsense. My apologies.

    The helpful analysis you gave, Pervect, should I think be viewed from a perspective where the word "force" is banned. In a sense general relativity describes gravitational effects kinematically, i.e. without invoking the concept of "force". This is why discussing the other forces of nature and gravity in the same context makes one speak with a forked tongue, as it were!

    So, when it comes to inflation, one should regard the universe's expansion by many orders of magnitude in an infinitesimal instant as a kinematic given. Of course this involves enormous kinematic relative accelerations between separated objects which, as you say, increases with their separation.

    But I can't then understand the survival of even elementary bits of matter !
  10. Aug 15, 2006 #9


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    It helps ease the headache if one thinks that space between particles expanded and the particles felt no force or acceleration. That is, if there were any material particles during inflation proper - perhaps just photons; I don't know!
  11. Aug 16, 2006 #10
    Yes, I do need something like Grandpa Headache Powders for all this!

    But the idea you express that "space between particles" expands and that "the particles felt no force or acceleration" is exactly the delusion I'm trying to unravel! Cosmology is full of the nonsense notion that "space expands", when in fact nobody even knows what "space" is. The better texts (e.g. Peebles' Principles of Physical Cosmology ) carefully avoid this trap.

    The particles you mention, like the meter stick mentioned in the quote below will, if they are finite in size, "try to stretch" with expansion against whatever cohesive forces they are endowed with.

    I disagree here only with the use of the word "force".

    But you are right that during inflation there may not even be such things as material particles --- who knows. The question I asked may therefore be moot.
  12. Aug 16, 2006 #11


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    Been a busy day, and then PF crashed.

    I probably should fix a few glitches in my explanation :-(. If you have two particles A and B, and they are both following geodesics in a DeSitter space, or near the Earth, or near a black hole, tidal forces will tend to make them accelerate relative to each other.

    It is only when external forces (such as the electromagnetic forces in a meterstick) act on A and B that they maintain a constant separation. It turns out that the force needed to keep them a constant distance apart is just their mass times the acceleraton they would have if you let them follow geodesics. So when you put a meter stick in space, the ends of the meter-stick are not following geodesics, but we can still use the geodesic equation to calculate the force being applied to keep the two particles at a constant distance.

    The tidal forces in De-sitter space are conceptually the very same tidal forces as those due to a black hole, or due to the Earth.

    In GR, we just calculate the magnitude of the components of the Riemann tensor (easy with software, not so easy to do by hand) in a frame-field, and we have the "tidal force" resulting from a metric, regardless of whether that metric is a black hole, De-sitter space, or whatever.

    One thing that you may not appreciate is how tiny they are. The magnitude of the force in geometric units turns out to be H^2, where H is the hubble constant.

    The MKS units are (meters / sec^2 ) / meter = 1/sec^2. So there isn't any big unit conversion issue, you just need to express H in units of inverse seconds.

    To get a number for the tidal force for two objects 1 km apart in m/s^2 I'd need an estimate of what H was during the inflation era, and I don't have a clue. (But I can tell you that it would be 1000/H^2, where H was expressed in seconds).

    Space Tiger might (or might not) know the value of the Hubble constant during inflation (Ned Wright says that it's constant, but gives no value) - or have some insight into whether or not it's even knowable.

    There's also the issue of whether we need to correct the formula I mentioned (for empty De-sitter space) for the presence of matter - however since expansion during inflation is usually described as being exponential, I think that implies that the gravitational effects of matter were totally overwhelmed during inflation and can be safely ignored.
  13. Aug 16, 2006 #12
    Well, I certainly don't feel stretched by the present expansion rate of expansion! But had I been around during inflation, I might:

    Although H (say (71 Km per sec) per MPc, or about 3 x 10 ^ (-19) per sec), is now very small, it must have been vastly bigger during inflation.

    Inflation caused a fractional increase in linear scale of the Universe of about 10 ^ 43 and lasted for about 10 ^ (-34) sec (Andrew Liddle, Introduction to Modern Cosmology, p. 106). Averaging H out (to get a rough estimate) then estimates H at about 10 ^ 77 per sec during inflation, which is 10 ^ 96 times bigger than today's value, unless I'm being too simple minded and have got it all wrong.

    Big enough accelerations to disrupt a meter stick and perhaps much else!

    You have sorted out my muddled thinking (the quote below) very nicely, thanks Pervect, and I hope the points you raised do get clarified in this thread.

  14. Aug 16, 2006 #13
    Sorry Pervect I do not follow you at all here.

    The general theory of relativity postulates that rods get contracted due to space-time curvature and the special theory of relativity postulates that rods get contracted when observed in relative motion due to a rotation in space-time. So in other words, EM-forces follow space-time curvature. But now we a led to believe that it does not apply to the FRW metric or a De Sitter space.

    What bothers me is the lack of (theoretical) criticism of those kind of cosmological models. Really we are just making it match our observations rather than presenting a sound theory.
    Don't get me wrong there is nothing wrong about that but then treating it as something that magically rolled out of "the Einstein equations" is not really giving a right representation IMHO.
  15. Aug 17, 2006 #14


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    I can see that you're not following me. I'm not sure what to do about it.

    Modulo some issues regarding rotating frames, "tidal forces" can be regarded as measures of certain components of the Riemann curvature tensor, and vica versa.

    This is one of the simplest ways to start appreciating the Riemann in "physical" terms, rather than as a mathematical abstraction. Parallel transporting vectors around closed curves is all well and good, and is one way to define the Riemann, but a tidal force is something that one can measure, directly, and it turns out that these tidal forces ARE (in non-rotating frames, anwyay) equal to specific comonents of the Riemann.

    As far as refrences go, all I can suggest is to read up on the geodesic deviation equation.


    Armed with this information, we can find the tidal forces near a black hole, or the tidal forces in a De-sitter universe, simply by evaluating the Riemann tensor and finding the appropriate components.

    The calculation itself is rather technical, so you'll just have to trust me when I say that the components of the Riemann are 1/H^2 in a "frame-field" in a De-sitter metric.
  16. Aug 18, 2006 #15
    I've been having some second thoughts about my understanding of what you wrote, Pervect.

    In the quote below I've shown the puzzles I have in bold -- they are mostly niggles about units and suchlike, but I'd like to get them straight so that I can combine what you wrote and my estimate of H during inflation (10 ^ 77 --- is this correct?) to estimate stretching forces in Newtons. Apologies for bothering you again!

    Last edited: Aug 18, 2006
  17. Aug 20, 2006 #16


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    Let's see if I can reply to this without it crashing before I put in a longer reply....
  18. Aug 20, 2006 #17


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    units where c=G=1/(4 pi epsilon_0) = 1

    In terms of the scale factor a(t), H = (da/dt) / a , which is equal to
    (da/a) / dt, so I think this is basically correct. I'd replace "between any two points" with "between any two points with constant co-moving coordinates". Points with constant co-moving coordinates are points that are moving with the Hubble flow

    The metric is (assuing spatial flatness, which I assumed BTW)

    ds^2 = -dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2)

    where x,y, and z are "comoving" coordinates.


    Oops, 1000*H^2, 1000 for the 1000 meter (1 km) separation

    As far as the magnitude of H goes, 10^77 * sec^-1 seems awfully high to me. This would imply that the universe doubled in size every ln(10)*10^-77 seconds.

    i.e. a(t) = exp(Ht), da/dt = H exp(HT)

    so when 1/H = 10^-77 sec, you have a doubling of the scale factor a about every 2*10^-77 seconds.
  19. Aug 20, 2006 #18
    I should have looked at Wikipedia straight off. But thanks for this URL. That's fine.

    Yes, it does seem very high to me too. I wouldn't be surprised to find that I'm wrong by several tens of orders of magnitude. But whichever way you slice it, inflation is a very extreme process and I suppose one can expect extreme tidal accelerations as a matter of course here. I hadn't appreciated this feature of inflation until you pointed me in the right direction.

    In another branch of physics, namely the elasticity of solids, the Hubble constant is exactly analagous to the dilation strain rate. In this context the dilation strain rate is usually small, say intermediate between the miniscule presently observed Hubble constant and its enormous and constant value during inflation.

    One could view inflation in a Newtonian way, and very simply, as an (extreme) explosive dilation of the "cosmic fluid", driven by the mysterious tranformation of false to true vacuum. But it's not appropriate to talk of such matters as fluid explosives right now!

    Thanks for your help in clarifying my understanding of such dilations.
  20. Aug 20, 2006 #19
    And how exactly does this explain the claim that expansion of the universe does not apply to small objects like a meter stick? :confused:

    So we are led to believe that while gravity reduces a volume of a sphere, expansion does not increase it and that that is all logical due to the Rieman curvature tensor?

    Really, so do you claim that the inter-relationship between the time and space dimensions as defined in GR in a gravitational field are handled exactly the same as expansion in a De Sitter space or a FRW model?
    It seems to me that time has simply a linear relationship in a De Sitter and FRW model and is fundamentally treated differently than time in GR.
    Last edited: Aug 20, 2006
  21. Aug 21, 2006 #20
    I have always found expansion much more difficult to understand than the (it turns out unecessary) curvatures of space sections so beloved of those who write cosmology texts. You may be having similar troubles, MJ.

    Nevertheless I think that you can rely on what Pervect says, and do as he suggested:


    But let me try and help --- I'm quite simple minded about such matters. Perhaps too simple minded. Does the following make sense to you?

    The point is often made that the present expansion of the universe doesn't make objects like a meter stick expand. Let me construct a "straw man description" to explain why this is so.

    Consider a planet exploding during a Star-Wars battle. Once the explosion has endowed the planet fragments with radial velocity it provides no further driving force -- it's all over and done with in an instant. But as the fragments (meter sticks and suchlike) fly apart they lose speed as the gravity of "interior" debris pulls them back. (By interior I mean debris closer to the centre if the explosion than the piece of debris being considered).

    The fragments also experience tidal forces from the interior debris. They will be distorted by these forces to an extent that depends on how "stiff" they are. Some meter sticks may be stretched by these forces. (Others will be compressed --- it depends on their alignment with respect to the centre of the explosion).

    But there is no uniform expansion of fragments. Why should there be?. Here my "straw man description" and the universe's expansion are well matched.

    Nor can the universe's uniform expansion produce any tidal forces. Because such expansion has high symmetry, in that it is isotropic and the same everywhere, it is impossible to pinpoint a "centre of the explosion", or to define "interior debris" that can produce tidal forces. So there are none!

    Cosmologists and general relativity folk have a horror of the kind of "straw man analogy" that I have simple-mindedly given. But as Hermann Bondi in Cosmology explained long ago, more simple-minded (such as Newtonian) treatments of the universe's expansion give very similar answers to the vastly more sophisticated machinery of general relativity.

    And this is true of the "inflationary scenario" as well.

    Fred Hoyle in a disparaging mood coined the now-universally-adopted name "Big Bang" for the universe's origin, a denigration initially resisted by cosmologists. But later they invented inflation, a violent process near "the beginning" that involves a sudden huge release of energy (in 10 ^ -34 sec) and a vast expansion (by a factor of 10 ^ 43).

    If this isn't appropriately described as an explosive "Big Bang, what is? And during this process, while the release of energy was proceeding, there must have been huge tidal forces, originating as Pervect described. Unlike the "coasting" present expansion of the universe, influenced only by gravity (if you ignore the recently discovered acceleration of expansion), inflation is continually driven by energy release which maintains the exponential expansion and produces tidal forces.

    Before this thread I hadn't appreciated this aspect of inflation, and suspect that many cosmologists haven't either.
    Last edited: Aug 21, 2006
  22. Aug 21, 2006 #21


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    Using Baez's & Bunn's "The meaning of Einstein's equation" approach, the volume of a sphere of coffee ground decreases if the sphere contains mass. I assume that's what you are referring to.

    If you have a universe containing matter and no cosmological constant and a sphere of "test particles", the test particles are attracted to the center of the sphere by the enclosed mass.

    This shows that matter never causes expansion to speed up, it always causes expansion to slow down.

    Furthermore, the tidal forces will always be compressive in a universe composed of matter with no cosmological constant. If you imagine the ends of the bar moving on geodesics, they'd "naturally" move towards the center. It's only the pressure in the bar that keeps them apart. But the magnitude of the forces involved are very small.

    So, looking at the normal universe case, we have an expanding universe, but tidal forces that work to oppose the expansion. The expansion slows down as a result of these forces. And we have compression in a bar, due to the gravitational attraction of the ends of the bar to the center. This assumes the bar is in a region that has the "normal" amount of matter, that the matter is smoothly and continously distrbuted throughout the universe. We know that this is true on the average by the so-called cosmological principle (which is basically isotropy).

    It's a bit backwards to think of the tidal forces as being due to the expansion of the universe. It's not a "force due to expansion". Rather, the tidal forces are due to the matter in the universe, and they slow down the expansion of the universe, because test particles are attracted to the center of the sphere.

    We've ignored the region external to the sphere , but this isn't an issue, as it has no net effect by Birkhoff's theorem, similar to the way that Newtonian gravity has no effect in the inside of a spherical shell.

    You might expect the same of an expanding gas cloud with Newtonian physics - gravity acts to oppose the expansion.

    Oldman has made very similar points, I see.

    This is quite consistent with the Riemann tensor approach, too. It's just a different way of viewing the same result.

    Things change when we add a cosmological constant. Things start to get weird. The vacuum contains a positive energy, but a negative pressure.

    Assuming that the universe is unchanging with time (as it will be if there is no matter in it), the effect of this negative pressure and positive energy denisty is a sort of anti-gravity. The enclosed mass (Komar mass) in the sphere is equal to volume * (density + 3*pressure) (and because the pressure is negative and equal to the density, the enclosed mass in a sphere is negative.

    This is how a cosmological constant can fuel an accelerating expansion. Particles in a sphere start to accelerate away from the sphere, because empty space has a negative Komar mass.

    The Komar mass formula isn't particularly intuitive, but it's one of the simpler formulas for mass in GR, and applies as long as the system is static. You can read the Wikipedia article if you like, but it's a bit advanced.


    BTW, this is a fairly recent addition that I recently wrote :-).

    The tidal forces due to the cosmological constant cause tension, rather than compression, because the bar ends want to move away from each other.

    This may seem a bit weird, but that's how it works.

    I don't follow you. De Sitter and FRW models are both GR models.
    Last edited: Aug 21, 2006
  23. Aug 21, 2006 #22
    Well at least something we can agree on. :smile:

    To me it looks like a plain old patching to make the theory match experimental results.

    This whole notion of a cosmological constant is preposterous to me, but I realize that for others it is like an obvious piece of Gospel that is both perfectly explanable and logical.

    Let's for the sake of argument assume what you say is correct, then explain to me how it is consistent to have a linear dependency on time in such a anti-gravitational field (for instance the scale factor in a FRW metric) but a non linear dependency on time in regular gravitational field?

    If you think I am wrong prove me wrong by expressing -a(t) as a scaling factor in a Schwarzschild geometry similarly as done in the FRW metric.
    Last edited: Aug 21, 2006
  24. Sep 6, 2006 #23
    I've taken some time off to think about what I learned from your kind replies in this thread, Pervect, and I've come across one point that still bothers me quite badly. Please forgive me for being so slow and obtuse.

    I gather that in many (but not all) respects general relativity gives the same answers as Newtonian gravity. Indeed, remarks you've made in the post below (I've picked them out in bold) seem to support this view.

    I have difficulty reconciling Newtonian gravity with your remarks about tidal forces.

    Newtonian tidal forces occur only in the presence of gravitational gradients. They do not occur when a uniform gravitational field accelerates (or decelerates) an extended object. Specifically, when in a uniform gravitational field (approximated by the field we live in), an arrow is shot vertically upwards it decelerates, but experiences no tidal forces.

    The same is true of a linear array of "coffee grounds" projected simultaneously upwards with identical initial velocities (in a laboratory vacuum). The initial separations of grains remain constant throughout flight. (This situation mustn't be confused with sequential stroboscopic photos of a projected ball, common in elementary physics texts, where the sequential separations decrease).

    In the case you were considering -- of a matter-only universe -- while gravity of course acts to slow expansion, it would seem to me that the "tidal forces" you mention in this homogeneous and uniform situation have no analogue in Newtonian gravitation. I am perturbed by this and can't help wondering if in this case you have the wrong end of the meter stick (as it were!) when you state above that:

    "... the tidal forces will always be compressive in a universe composed of matter with no cosmological constant. If you imagine the ends of the bar moving on geodesics, they'd "naturally" move towards the center. It's only the pressure in the bar that keeps them apart. But the magnitude of the forces involved are very small."

    In fact I doubt if there are such tidal forces. But I've been wrong so often. Could you clarify this apparent dichotomy, please?
    Last edited: Sep 6, 2006
  25. Sep 6, 2006 #24

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    I don't know if the following has any relevance.

    Using Newtonian gravity, consider small spherical subset of a large, uniform, spherically symmetric distribution of matter. The large and the small spheres don't have to be concentric. Consider test particles moving in this small sphere that don't interact with the matter in the sphere - in your case an arrow shot out in any driection (along a negligibly small tunnel) from the centre of the small sphere.

    The matter outside the small sphere has, to a good approximation, no tidal gravitational influence on the arrow. The tidal force of gravity (due to stuff inside the small sphere) inside the tunnel is, I think, proportional to the distance from the small sphere's centre, so the business end of the arrow experiences more tidal force towards the centre of the small sphere than does than the feathered end.

    Newtonian tidal forces compress the arrow.
  26. Sep 6, 2006 #25
    Thanks for this help, George. But I find your argument difficult to apply to a uniform and homogeneous universe.

    In the case you consider all "small spheres" on whose surfaces the arrow lies produce gravitational forces (with gradients) directed towards their centres. These forces cancel out because of spherical symmetry, and there is no net gravitational and tidal force!

    Maybe I'm missing your point?
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