# Is this a relativistic paradox?

In general relativity is postulated that $m_\mathrm{inertial} = m_\mathrm{gravitational}$. Suppose one has a very very intense uniform gravitational field parallel with the $z$-axis of a 3-dimensional Cartesian system. Suppose we also have 2 very light particles, of equal mass, initially at rest with respect to the Cartesian system, and initially with the same $z$ coordinate, but separated by a very large distance D (in the $xy$-plane), such that the gravitational interaction between them is very weak. Then the 2 particles are released simultaneously in the gravitational field. According to the general relativity, the reference frame associated with the freely falling particles is an inertial one, and in this frame, the particles barely move towards each other due to the weak gravitational attraction between them. However, for an observer associated with the initial Cartesian system which is at rest, these particles appear to be very massive in a very short time (since they accelerate very fast due to the very strong gravitational field they are freely falling in).

So, for a high intensity field, after a very short time (as measured in the Cartesian reference frame), the particles would collide due to the very intense gravitational attraction between them, generated by their huge dynamical masses. The higher the initial uniform gravitational field, the faster the collision. But a collision is a spacetime point and it also has to occur simultaneously in the frame of reference that is freely falling where it seems to occur only after a very very long time interval. The collision is due to the motion in the $xy$-plane in both frames of reference and it should not be affected by the motion along the $z$ axis. Is this a paradox? What goes wrong?

• "accelerate very fast due to the very strong gravitational field they are freely falling in" - spot the contradiction? If they are in free fall they are by definition not accelerating. It's the co-ordinate frame that is accelerating in resistance to the intense gravity. Dec 28 '16 at 22:04
• "Cartesian system which is at rest with respect to the particles" - the one that's not inertial because it ignores gravity? Dec 28 '16 at 22:04
• Not sure Jan, there are various inertial ways to resist gravity. Being on a planetary surface, for example. An orbit. Using inertial thrusters. Dec 28 '16 at 22:10
• @JMLCarter those are not inertial ways, are they? On a planet, the planet surface pushes you, giving you weight. In orbit, you are in free-fall. Not sure what inertial thrusters are, but normally thrusters are something that gives you acceleration. Dec 28 '16 at 22:20
• Just take the orbit frame then (for these purposes). It doesn't resist gravity per se, but it can be considerd at rest relative to the intense gravity field and used as a frame of reference. You could also define the ref frame a great distance from the intense field at this point the effects of gravity due to the field would be negligible, wouldn't they? Again it's not resisting gravity, but it could be at rest w.r.t. the particles and the source of the intense gravity field. Dec 28 '16 at 22:36

If "the initial Cartesian system which is at rest" is not free-falling along with the particles, then it must be accelerating upward very quickly to resist the very strong gravitational field, so it is a highly noninertial frame. An observer in this frame would feel strong acceleration, which would be equivalent to being deep in a strong gravity well resisting the force of gravity. They would therefore experience very strong time dilation, and the observer's clock would run much slower than the free-falling particles' clocks. The observer would therefore see the free-falling particle's time as running very quickly, and the very slow attraction that the free-falling particles observe would get "sped up" into very fast attraction from the noninertial observer's perspective.

• Where to begin. 1) Is the cartesian system with the sun at it's origin having to accelerate upwards to resist gravity? Can not the postion of the earth be measured in this frame. 2) How does it make sense to talk about accelerating a frame? Objects can experience acceleration, and a frame can be tied to an object. 3) The gravitational time dilation of the observer in the ref frame would be less than the time dilation of the particles, which fell closer to the mass causing the gravitation field. (more) Dec 28 '16 at 23:33
• 4) Time dilation is in effect in proximity to a mass such as a black hole. Lower objects are more time dilated. 5) The observer would see the particles time as running relatively slower. 6) Time dilation is an effect of acceleration due to gravity - we seem here to be considering that time-dilation can alter apparent mass and gravitational attraction - it is confusing cause and effect. Dec 28 '16 at 23:37
• 1) No, that frame is (almost perfectly) inertial, but I don't see the relevance to my explanation. 2) I didn't say that the frame was being accelerated, I said that it was accelerating. This is a slight but completely standard abuse of language that means that the frame is noninertial, and so any object tied to that frame feels an acceleration. Apparently I got my point across to the OP. 3) and 4) Proximity to a black hole is not the only possible source of time dilation. By the equivalence principle, strong acceleration is locally equivalent to resisting a strong gravitational ... Dec 29 '16 at 4:44
• ... field, which in turn is equivalent to being very close to the event horizon of a black hole but firing boosters to avoid falling it, which we know causes strong time dilation. 6) I am certainly not claiming that time dilation directly affects mass or gravitational attraction - I'm saying that it causes two attracting masses to take longer to come together, offsetting the increased attraction that comes from their kinetic energy contribution to the stress-energy tensor. Dec 29 '16 at 4:48

Relative to each other the two particles are not moving, and observe no mass increase.

• I think the question was how to get this picture in line with the assumed behavior in the Cartesian reference frame. Dec 28 '16 at 22:07
• to determine the gravitational attraction between two objects its necesary to consider their relative motions. The increase in mass in observed in the cartisan frame only impacts interactions between objects in that frame and the masses. Dec 28 '16 at 22:16
• The observer at rest in the initial Cartesian frame is accelerating, against the gravitational field. But either way, the two particles, in any coordinate system, will follow geodesics that will maintain them at the same distance from each other, obviously they will not get closer or further from each other since the gravitational field is uniform. They will not attract each other any more than before. Their momentum will increase, but their rest masses are always the same. You can calculate in the 'inertial' frame and get the result, or calculate in their rest frame and then transform(easier) Dec 29 '16 at 1:41
• I didn't see any inconsistancy between my entries and yours (- are you just elaborating?) Dec 29 '16 at 1:49
• I up voted your answer. I'm not sure on the comments, too much unnecessary discussion of time dilation, and you mentioned mass increase, there is no increase in mass in any frame, even if they were moving wrt each other. Mass is rest mass, the rest is momentum, and of course total energy which is not invariant. But your answer is perfect, in my opinion. Dec 29 '16 at 8:16