# Gravitational Lag Thought Problem

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Hypothesis:

If gravity is not instantaneous then any two masses travelling parallel to one another will not only converge, as one would expect, but they will also decelerate against the axis of their mutual forward motion, violating the law of conservation of momentum. (Their CM frame will slow down, even though there are no external forces on the objects in that frame).

Hypothetical reason:

Both objects will be "experiencing the arrival of" some gravity from the other object, but it will have been "time-lagged" (i.e. delayed in it's arrival by the constraint that gravity is not instantaneously acting at a distance--that it, for instance, propagates a the speed of light).

Therefore, each object will be vectored not toward the other object's "present" location, but toward some "previous" location of it (time has passed during the transmission of the force, after all).

The force will unexpectedly draw the objects toward their "previous positions" that are "behind their present positions", and thereby retard their forward velocity (in addition, of course, to drawing them toward one-another as we expected).

Question:

How does the theory of relativity cope with this paradox?

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a) your assumption about the gravitational field of the two objects and their reaction to this field is just that: your assumption. It is not entirely clear that a proper treatment would yield the scenario you envisage (though it might).

b) the gravitational field can in principle carry energy and momentum. This is also true in simpler field theories like the electromagnetic interaction in which it is widely known and accepted that electromagnetic waves carry energy and momentum.

c) I'm not entirely sure under which conditions momentum is conserved in general relativity at all. The standard expression for the conservation of energy and momentum is a local property, which can be "expanded" to a global property in a flat space. But in a non-flat space it is not clear (to me) that you still have conservation of momentum.

My personal guess is towards problem a). Replace your masses with the electromagnetic field two electrical charges, and the same argument should hold (unless I missed an important part of your argument). But in that case you can go to a coordinate system where the charges are at rest (except for the motion towards another), and see that they just attract each other normally, there. Since momentum is conserved in this center of mass system, so it is in the original coordinate system. I would imagine the gravity case to work equivalently.

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@timo:

I think you are right to assume that if there were a problem with my reasoning, it would reside in "a)". It would have to be. It certainly does not reside in the later steps, which are airtight.

But your b) is not right, in my mind. Regardless of what energy (although I do not raise the energy question) and momentum (I do raise that) the gravitational field can "carry" there is NEVER any reason for a CM frame (or any system) to change its net momentum absent forces that are completely external to that frame and it's constituents. No such external forces are allowed in this thought problem.

As for your c) I would state that momentum is in fact big-time conserved in general relativity, 100%.

As you state, I think my "argument" does in fact hold equally well for travelling charges, if they are opposite. (If they are alike, then the CM frame should speed up, as a "backward repulsive force" (==forward force) would replace a "backward attractive force" (==gravity-lag drag).

Not only do I think that your "travelling charges problem" maps extremely well onto this apparent paradox, but also we could consider it in terms of visible light, which is equivalent and much more intuitively satisfying. With these two masses advancing in parallel to one another, let's say they are ships with windows. They look back at the two base-stations that shot them out, and those stations are exactly in the middle of their rear windows. But when they look at eachother out of their side windows, are they seen to be exactly 90 degrees to the side? I believe: NO. They each see the other's previous position. How lagged that position is, relates to how much time has passed since the photons came off the other object. That places the other object visibly "behind its actual, current position".

If you don't think that non-instantaneous gravity exerts drag on two objects moving in parallel, then wouldn't you also have to concede that looking out your side window you would see the other object directly to the side and not lagging behind a little bit? Just like your travelling charge scenario, this one too is analogous, yes?

Edited by DieDaily
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As for your c) I would state that momentum is in fact big-time conserved in general relativity, 100%.

Stating it is a whole lot easier than making it precise and proving it.

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

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@Dr Rocket. Absolutely, and I invite the precision that will be required to solve this "paradox". This is a "flat space-time" problem. There are no external gravitational effects, no super-massive neighbors, no hyper-relativistic velocities, etc., and we're not talking about whether "the entire universe conserves...this or that..."; merely the local system. Really, it's a very simple, very clear, flat-space local problem. I don't imagine anyone will need to hedge, hum or haw, or try to divert the discussion to larger, less relevant things in order to solve it. It's clearly stated...knock it down! And, if you like, I would welcome a little of that precision. Especially if it's right on the stated target!

Edited by DieDaily
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Hypothesis:

If gravity is not instantaneous then any two masses travelling parallel to one another will not only converge, as one would expect, but they will also decelerate against the axis of their mutual forward motion, violating the law of conservation of momentum. (Their CM frame will slow down, even though there are no external forces on the objects in that frame).

Hypothetical reason:

Both objects will be "experiencing the arrival of" some gravity from the other object, but it will have been "time-lagged" (i.e. delayed in it's arrival by the constraint that gravity is not instantaneously acting at a distance--that it, for instance, propagates a the speed of light).

Therefore, each object will be vectored not toward the other object's "present" location, but toward some "previous" location of it (time has passed during the transmission of the force, after all).

The force will unexpectedly draw the objects toward their "previous positions" that are "behind their present positions", and thereby retard their forward velocity (in addition, of course, to drawing them toward one-another as we expected).

Question:

How does the theory of relativity cope with this paradox?

You're assuming something that Relativity denies, that there is such a thing as "absolute motion". Essentially you are saying that there would be a method of determining whether or not the objects are "moving" or not. Relativity states that no such test is possible.

This is the same as the "light clock" example. Start off by bouncing a light back and forth between the objects. As seen from either object, the light passes directly between the two. Now accelerate the objects equally. repeat the experiment. The light will still bounce back and forth between the objects, and neither will see the light as coming "from behind" the other object. You can't tell any difference between before and after acceleration.

The same would be true of gravity. Two objects "moving" behave the same as two "at rest". Mainly because "moving" and "at rest" are just arbitrary choices based on the frame of reference you decide to use.

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@Dr Rocket. Absolutely, and I invite the precision that will be required to solve this "paradox". This is a "flat space-time" problem. There are no external gravitational effects, no super-massive neighbors, no hyper-relativistic velocities, etc., and we're not talking about whether "the entire universe conserves...this or that..."; merely the local system. Really, it's a very simple, very clear, flat-space local problem. I don't imagine anyone will need to hedge, hum or haw, or try to divert the discussion to larger, less relevant things in order to solve it. It's clearly stated...knock it down! And, if you like, I would welcome a little of that precision. Especially if it's right on the stated target!

There is no such thing as gravity in flat spacetime. Gravitation is a manifestation of curvature.

Edited by DrRocket
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c) I'm not entirely sure under which conditions momentum is conserved in general relativity at all. The standard expression for the conservation of energy and momentum is a local property, which can be "expanded" to a global property in a flat space. But in a non-flat space it is not clear (to me) that you still have conservation of momentum.

Generally a space-time may not have enough symmetries to make clear sense of a global conservation law. However, in general relativity we do indeed have conservation of energy-momentum at the local level. Understanding why we have to discuss energy-momentum and not energy and momentum separate is clear; we in general have no unique, nice and clear separation of space-time into space and time.

There is no such thing as gravity in flat spacetime. Gravitation is a manifestation of curvature.

The closest thing, while staying in the context of general relativity is to linearise. This maybe very useful here as the analogy with electromagnetism as suggested by Timo would be closer.

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Hypothesis:

If gravity is not instantaneous then any two masses travelling parallel to one another will not only converge, as one would expect, but they will also decelerate against the axis of their mutual forward motion, violating the law of conservation of momentum. (Their CM frame will slow down, even though there are no external forces on the objects in that frame).

Hypothetical reason:

Both objects will be "experiencing the arrival of" some gravity from the other object, but it will have been "time-lagged" (i.e. delayed in it's arrival by the constraint that gravity is not instantaneously acting at a distance--that it, for instance, propagates a the speed of light).

Therefore, each object will be vectored not toward the other object's "present" location, but toward some "previous" location of it (time has passed during the transmission of the force, after all).

The force will unexpectedly draw the objects toward their "previous positions" that are "behind their present positions", and thereby retard their forward velocity (in addition, of course, to drawing them toward one-another as we expected).

Question:

How does the theory of relativity cope with this paradox?

Just to clarify; the objects are at rest with respect to each other?

EDIT: which I have just read is what Janus said.

Edited by between3and26characterslon
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DieDaily:

If you consider the moving charges a good equivalent example (most physicists would indeed consider it being equivalent in the limit of weak gravitational fields), then the problem with your inherent assumptions is perhaps best highlighted by the question where the magnetic field is. The electromagnetic field of a moving charge is not simply the superposition of a lot of electromagnetic fields of non-moving charges along its trajectory. Same should go for the gravitational field.

Generally a space-time may not have enough symmetries to make clear sense of a global conservation law. However, in general relativity we do indeed have conservation of energy-momentum at the local level. Understanding why we have to discuss energy-momentum and not energy and momentum separate is clear; we in general have no unique, nice and clear separation of space-time into space and time.

I don't see what you're trying to say there. What I said is that I don't know under which conditions one can sensibly talk about global conservation of momentum (*). This does, and I should probably have said that explicitly, include the question whether one can sensibly define such a thing as global momentum in the first place. You seem to be saying that the existence of global conservation of momentum is not clear in general, which seems to be the same statement. Did I miss your point?

(*) and to be honest, I don't care as much as to work it out myself.

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I don't see what you're trying to say there. What I said is that I don't know under which conditions one can sensibly talk about global conservation of momentum (*). This does, and I should probably have said that explicitly, include the question whether one can sensibly define such a thing as global momentum in the first place. You seem to be saying that the existence of global conservation of momentum is not clear in general, which seems to be the same statement. Did I miss your point?

I think we are saying the same thing. Global conservation law are a tricky thing in general relativity.

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This is the same as the "light clock" example. Start off by bouncing a light back and forth between the objects. As seen from either object, the light passes directly between the two. Now accelerate the objects equally. repeat the experiment. The light will still bounce back and forth between the objects, and neither will see the light as coming "from behind" the other object. You can't tell any difference between before and after acceleration.

But how do you accelerate the objects at the same time according to all observers? You can't.

If I have a viewpoint from which the 2 objects are symmetrical, and I synchronize the start of their acceleration, then they'll always be symmetrical to me.

But each of the objects will see that they appear to have had a head start vs the other.

Assuming gravity waves behave exactly like light, an object's gravitational pull on me will always appear to come from exactly where the object appears to be.

So let's restate the problem with a different example:

Imagine 2 objects P and Q at the start of a race.

Imagine a very long start line, many light seconds long, with the objects separated by 1 lightsecond.

A light signal equidistant to P and Q starts the race. Suppose P and Q instantly accelerate to some significantly fast speed (so we consider only 2 inertial frames: at rest relative to the start line, and moving relative to it).

From P's perspective, it started the race 1 second ahead of Q. P won't see Q start the race for 1 second, and vice versa.

The resolution to the paradox is this: If we imagine any photons moving through space, we can imagine them moving along with whatever inertial frame we choose to consider, correct? So, imagine photons emitted from Q a fraction of a nanosecond after Q starts to move (assume it is essentially at the starting line) and traveling along the start line, perpendicular to the velocity of P and Q. From P's moving inertial frame, these photons will "move along with P" and remain incoming from a perpendicular direction.

This must mean that the start line appears to curve "forward" according to P. When it switches frames, it sees the start line stretching out to the side but now stretching slightly forward of perpendicular. It appears as if Q is "already ahead" of it. After 1 second, it appears to catch up to Q laterally at the same time that Q appears to start moving. They would remain "side by side" except for that first second (and if they stopped at the end line, Q would appear to be behind P for only that last second).

I read in Carl Sagan's Cosmos that traveling near the speed of light would warp things so that things that were behind you would squish into your forward cone of vision, but I never really made sense of that until now.

Edited by md65536
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I'm gratified by the thoughtful, smart responses. But, if I may, I would like to "narrow down" what I am talking about, by drawing directly upon your well-considered responses. I thought about a more rigorous (long and boring) definition of this thought experiment, but went for conciseness, trying to cut to the heart of the issue. I think I should now be more rigorous in defining it, because my previous decision seems to have been in error.

@Janus: who states:

You're assuming something that Relativity denies, that there is such a thing as "absolute motion". Essentially you are saying that there would be a method of determining whether or not the objects are "moving" or not. Relativity states that no such test is possible.

I don't follow your reasoning. First, in no way did I intend (nor, I think, have I in any way suggested) any notion whatsoever of "absolute motion". More rigorously: the problem is defined such that there exist two base stations in some far reach of empty, inter-supercluster space (let's say for simplicity that these stations are effectively massless, or "extremely low mass", but ultimately it doesn't matter whether they are...it's merely a slightly more complex version of the exact same problem which resolves in the exact same way). These two base stations sight each other. They each rotate their launchers exactly 90 degrees from the sight-line of the other station. They then each eject a mass into a "flat space-time" (meaning simply that there are no other nearby NET sources of gravity--i.e. we don't care that there exists some huge ambient distribution of masses in the universe because these masses are both remote and, let's say, equally distributed along every direction/range--such that there is no ambient net force on our local system). Obviously, I hope, you can see that I don't mean to propose that these two test masses themselves fail to "bend space-time".

So, these two test masses initially set out along exactly parallel paths at some velocity (non-relativistic, for our purposes, but that does not really matter either...the same will likely hold for any velocity up to and including "c"). We know that they are moving away from the base stations (the launchers) because, let's say, there is a radar transceiver on each station, bouncing waves off each projectile and these pings are taking longer to bounce back with each successive ping, which is, say, iterated once per second. We know that they are not initially convergent due to the fact that they were launched at precisely 90 degrees to the sight-line of the other station.

Furthermore, let's talk about their simultaneity. We suppose a midpoint probe/observer that is equidistant from the two base stations...in fact it is directly between them, let's say. When the two base stations launched their test masses, that midpoint probe noticed and registered these launch events. It registered the EMR "launch signals" that were released from the launchers of each base station at the time of launch. It also visually noticed that each test mass left "at the same apparent time" and it noticed that as it bounces it's own radar signals off of the two base stations at an interval of once every second, that the reflected signals always return simultaneously (confirming that it was really and truly equidistant between them, especially given how flat space-time is here). Furthermore, it notices, throughout the experiment, that radar pings that are aimed at the two test objects keep arriving simultaneously. Last, it is noticed that the radar pings from the test objects have increasing red shifts as time goes on (and their velocity becomes less orthogonal to the pings as they get more distant, thus ever increasing the red-shift.) So, purely from instrumentation, we know that: the objects are moving away, and that they are travelling (initially) parallel. As a reminder, we expect that they will converge due to mutual gravitational attraction. What we DO NOT expect is that there will be CM-frame drag due to the time-lag between when "gravitons are emitted" and when "gravitons are received". Please, let's not get side-tracked by the word "gravitons". It's an identical problem absent discrete, finite-speed, force-carrying particles. The key here is the "finite speed" of the mechanism of propagation of gravitational information...NOT the specific nature of that information.

What we have here is not some spooky "entire universe". It is not subject to the vagaries of entire-universe integrals wherein we wring our hands about what is or is not conserved GLOBALLY. GLOBAL need never enter into this. In fact, it's quite the opposite. It is merely a clearly defined, effectively isolated local system being proposed here. When I say that "space time is flat" in this area, I do not mean that each of the two objects is not attracting the other one. That was, I hoped, abundantly obvious and clear--in fact it's the essential issue around which the whole exercise is defined and built. Obviously, the two objects attract each other. As I stated, the whole point of this exercise obviously rests on this. What I AM saying is that all of the local curvature of space-time is exclusively whatever curvature the two objects themselves impose on it (all other curvature, the ambient curvature, being direction/range invariant and therefor NOT capable of any NET force on our local system...we don't care if we are amid some local minima or maxima because we assume here that it is directionally invariant...a good assumption in deep inter-cluster space...at the very least, in real-universe terms, these ambient effects are too many orders of magnitude too small to matter for our argument).

So when ajb states:

There is no such thing as gravity in flat spacetime. Gravitation is a manifestation of curvature.
I would respond that this should go without saying. Obviously the two masses are themselves curving space time. While this is a true no-brainer, I definitely do need to apologize for not making it explicit...I should have done so. Even more explicitly: THE TWO OBJECTS CURVE SPACE TIME (AND ARE THEREFORE GRAVITATIONALLY ATTRACTED TO ONE ANOTHER) BUT NOTHING ELSE DISCERNIBLY DOES SO.

Now that I have clarified things, in my mind the following evasions are no longer permissible:

1. That I have implied "absolute motion". I certainly have not. The two test masses move relative to the massless base stations and they slow down due to my conjectured "gravitational-lag drag" relative to them also. "Absolute motion" has nothing to do with anything, least of all this thought experiment.

2. Space-time is not flat here, but it is completely flat OTHER THAN the curvature introduced by the two test masses. This is the whole point of the exercise...to isolate two masses travelling in parallel in order to decide whether gravitons with non-infinite speed would actually act as a generalized drag force on parallel forward motion. (For the sake of the thought experiment, I assert that they would indeed do so and invite you resolve: WHY DON'T THEY?.)

3. Changes in the space-time curvature "caused" by these masses, propagate across space non-instantaneously, for instance: no faster than the speed of light.

4. Therefore the "perceived" curvature (gravitational acceleration) of one test object reflects the "stale" state of the curvature "caused/initiated" by the other mass. In effect, there is this dilemma: either the other mass instantaneously and at a distance "updated" the first object as to it's "real and present" location (and therefore its "real and present" space-time gravity well) IN WHICH CASE THE TWO OBJECTS ARE DRAWN DIRECTLY TOWARD ONE ANOTHER (violating nothing) or else, as I suggest, they must respond instead to "stale data". They must be drawn toward the "previous location" of each other, clearly resulting in rearward drag.

IF THIS IS NOT SO, THEN WHAT IS THE MECHANISM WHICH "UPDATES" THE "PERCEIVED" SPACE-TIME CURVATURE CAUSED BY ONE OBJECT IN REAL TIME AT THE LOCATION OF THAT DISTANT OTHER OBJECT? Another equally firm way of putting it is to imagine SCENARIO B: infinitely fast (instantaneous) updating (across any distance) of gravitational force/curvature. Surely even the most jaundiced can imagine, for arguments' sake, a universe in which gravitons have infinite speed. Ok, if you can do that, then take a look at the DIFFERENCE between that scenario and the finite-speed graviton scenario. ARE THEY IDENTICAL? IF NOT, HOW SO?

I hope this thought problem is more clearly stated now. I sense that in each response there is a retreat to the "general, whole-universe" case, and also the notion that "the problem is ill-posed". Guys and gals, I've presented a whole universe here. There are two massless launchers (or if that troubles you, let's say that these two objects happened to find themselves, for whatever reason, travelling apace and in parallel for some unknown reason but minus any launchers). There are two masses, which for all intents and purposes, are the only two masses in the universe. Third, we have another frame consisting of a detector midway between the launchers (and therefore equidistant from the test masses at all initial and subsequent times) which confirms via radar that those masses are really receding from the launchers...or, if you like, eliminate the launchers and assume that some probe is merely anywhere that is equidistant between the masses. (Does this imply "absolute motion"? Hell no, there is no such thing as absolute motion...there are no privileged frames...and I do not propose a privileged frame! Gawd, no!)

So let's get to the heart of the matter. Take lagged gravity. Take gravity that is instantaneous over any distance. Is there a difference between them in terms of how things would play out? If you can argue that lagged gravity does not cause motion-damping in this case, then how now with instantaneous gravity? Would that, equally mysteriously, accelerate them? Would it leave them alone to proceed undamped? Would the two be the same in every way?

Edited by DieDaily
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The pilot in any one of the projectiles consider himself and his projectile to not be moving, when he looks out through his window he can see that the other projectile is not moving either and it is therefore effectively at rest with him. When he measures the direction of its influence of gravity on him it points exactly towards where he sees it.

Since both projectiles are not moving they can not observe any lag in space neither for gravity nor EM radiation.

Note also that it was DrRocket and not ajb that made the statement of gravity and curvature.

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Look, what I am saying is that regardless of how we pose this question:

1. Gravitational force (space-time curvature)

2. Opposite electrical charges force (definitely analogous)

3. Inspection of the direction of incident photons (hopefully completely analogous)

We must (?) admit that the incident gravitational(1.)/electromagnetic(2.)/visual(3.) information is "lagged"--that the information that one test particle receives has undergone some lag--a time interval has elapsed between the emission time and reception time. Isn't NOT admitting this notion tantamount to admitting that the information "traveled" at an infinite velocity? If it was finite, then WHY does one particle NOT see (attract to) the "past state", rather than the "present state" (which could only be discernible via faster-than-light information transfer)?

I don't see how this is in any way frame-dependent. I don't see how I am proposing some privileged-frame scenario.

The pilot in any one of the projectiles consider himself and his projectile to not be moving, when he looks out through his window he can see that the other projectile is not moving either and it is therefore effectively at rest with him. When he measures the direction of its influence of gravity on him it points exactly towards where he sees it.

@Spyman: If this is true, then the "pilot" looks out his back window and sees his launch station directly behind (180 degrees) and out the side window he sees the other object at precisely 90 degrees (or 270 for the other one). If you are willing to agree to this, then we can go forward, but are you sure that you want to agree to that? [Edit follows:] But I would certainly not hesitate to add that you have hit on the crux of the matter--the rest frame--and that this frame, in this thought experiment, would undergo a mysterious deceleration that would occur absent external forces. Also, very importantly, I never argued that he would not experience a gravitational force in the direction "that he sees it"...that is actually integral to my argument...it's the basis upon which I rest my claim that scenarios 1., 2., and 3. above are analogous. Last, what you state: "Since both projectiles are not moving they can not observe any lag in space neither for gravity nor EM radiation." is tantamount to stating that any radar pings from one object to the other (and back) will occur INSTANTANEOUSLY. You said it, not I. But, is this really true? Remember, you have just stated that there is no EM lag. If so, there is no travel time for signals that are either emitted or reflected (bounced back) between the two. Are you sure of that?

Edited by DieDaily
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The glib answer is that the nearly massless base stations are the things that move (they would be if they were massless) and the two massive projectiles remain virtually stationary, the two base stations do not attract each other as they are massless and the two projectiles converge with no shenanigans in the momentum. But I am being silly - I understand your problem - it seems very nicely formulated and I await a better explanation than mine.

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The glib answer is that the nearly massless base stations are the things that move (they would be if they were massless) and the two massive projectiles remain virtually stationary, the two base stations do not attract each other as they are massless and the two projectiles converge with no shenanigans in the momentum. But I am being silly - I understand your problem - it seems very nicely formulated and I await a better explanation than mine.

Thanks! Much appreciated! And I too am awaiting the clear, concise death-blow to my thought problem! As per the massless launchers...let's ignore them. We could assume that the test masses had internal propulsion, such as chemical rockets, in which case we don't even need to stipulate that the launchers recoiled. Let alone recoiled infinitely quickly due their infinitesimal masses! But good observation!

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Di Daley (good welsh name that) - the more I read of GR (and incidently QM as well) the more I think that quite often simple problems do not admit to simple answers. And it isn't lack of knowledge or inability to explain on the half of the physicists - it's just a mound of learning, maths and mindset that is not easily transferred.

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@imatfaal, well, thank you sir. I am, in fact, Welsh/Irish of decent (my real last name starts with "Mc", lol) as well as an equal amount of German genes also! Let me console you: I do have that learning you speak of and perhaps I even have that mindset you refer to also. I am a professional Physicist associated with a leading department (not that that really matters). Yet, like you, I notice contrary viewpoints (in my case amongst my students)...hence this post. I am delighted to be here in this forum; it's one of the few pseudo-academic sojourns I've ever made into the wonderful collective madness of the Internet. I must confess that, in the coffee room, my associates and I have a little bet going (for money)...about whether the people who visit this site will react dogmatically (insensate animals) or sentiently (doubting investigators or erudite explainers). No value judgement there...just recapping the supposedly fundamental precepts of truly scientific skepticism. Thus far, my colleagues and I are still at odds (cash-wise), because nobody has substantively answered in either way. Your response, however, we feel must be tallied within the margin of the latter category, although there was quite a bit of argument along the lines of whether or not you are entitled to have a valid viewpoint. I'm afraid that there was insufficient specific math in your comment to definitively decide this question. So, you may have cost me some cash! We (I, mostly) were primarily expecting two things: anger and incoherence. So far, I'm pleasantly disappointed.

Edited by DieDaily
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Hypothesis:

If gravity is not instantaneous then any two masses travelling parallel to one another will not only converge, as one would expect, but they will also decelerate against the axis of their mutual forward motion, violating the law of conservation of momentum. (Their CM frame will slow down, even though there are no external forces on the objects in that frame).

Hypothetical reason:

Both objects will be "experiencing the arrival of" some gravity from the other object, but it will have been "time-lagged" (i.e. delayed in it's arrival by the constraint that gravity is not instantaneously acting at a distance--that it, for instance, propagates a the speed of light).

Therefore, each object will be vectored not toward the other object's "present" location, but toward some "previous" location of it (time has passed during the transmission of the force, after all).

The force will unexpectedly draw the objects toward their "previous positions" that are "behind their present positions", and thereby retard their forward velocity (in addition, of course, to drawing them toward one-another as we expected).

Question:

How does the theory of relativity cope with this paradox?

My understanding: In the reference frame you are using the gravitational vector points toward the current position (assuming no acceleration other than that of their convergence)

That way the effect would be consistent in all frames. IIRC Swansont pointed this out to me in a thread a few years back.

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@J.C.MacSwell. There no accelerations (other than that of their expected convergence, [and possibly also that of my thought-conjectured "lag-drag"]). So, PLEASE tell me more about how and why? What did this IIRC Swansont say a few years back? What is his counter-argument or explanation? Please note that if my hypothesis is correct then the CM frame is obviously non-inertial.

Edited by DieDaily
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I don't see how this is in any way frame-dependent. I don't see how I am proposing some privileged-frame scenario.

I am not an expert on Relativity but AFAIK you have to be in the observers frame of reference when interpreting his experiences.

But I would certainly not hesitate to add that you have hit on the crux of the matter--the rest frame--

This is what Janus was pointing out, there is no absolute "rest" frame, in the view of the pilots they are the ones at rest.

--a time interval has elapsed between the emission time and reception time.

Yes, a time interval has elapsed between the emission time and reception time, there is a time lag but since they are not moving there will not be any space lag, i.e. the signals will not appear to come from a distance behind the objects because without movement there is no behind.

If this is true, then the "pilot" looks out his back window and sees his launch station directly behind (180 degrees) and out the side window he sees the other object at precisely 90 degrees (or 270 for the other one). If you are willing to agree to this, then we can go forward, but are you sure that you want to agree to that?

No, gravity will be acting on the projectiles even before launch and the only thing keeping those angles is the physical pressure from the launch tubes pushing on the projectiles while the engines on the base stations are compensating for gravity and keeping the launch tubes parallel.

As soon as the projectiles leaves the launch tubes gravity will start to pull them together and change the angles.

When the pilot looks out through the side window he will se the other projectile perpendicular to the back-front direction of his projectile but when he looks out through the back window the base station will no longer be directly behind him.

Also, very importantly, I never argued that he would not experience a gravitational force in the direction "that he sees it"...

I did not think you did either, sorry if I was unclear and appeared to argue that you did.

Last, what you state: "Since both projectiles are not moving they can not observe any lag in space neither for gravity nor EM radiation." is tantamount to stating that any radar pings from one object to the other (and back) will occur INSTANTANEOUSLY. You said it, not I. But, is this really true? Remember, you have just stated that there is no EM lag. If so, there is no travel time for signals that are either emitted or reflected (bounced back) between the two. Are you sure of that?

No, I said no lag in "space", there will of course be lag in "time", so there is no instantaneous action.

If the pilot sends a radar ping bouncing off the other projectile, he aims perpendicular to his back-front direction and gets the ping back exactly perpendicular to his back-front direction. The forth and back trip of the ping takes time, but since they are not moving it can not come from any other direction than straight from the other projectile which is located perpendicular to his back-front direction.

As per the massless launchers...let's ignore them. We could assume that the test masses had internal propulsion, such as chemical rockets, in which case we don't even need to stipulate that the launchers recoiled.

Ok, so lets remove the base stations and make the projectiles true spacecrafts with engines. They are located in parallel directions a distance apart and at rest with each other. At exactly the same moment they accelerate equally and then turn off their engines and coast in freefall.

Before the launch one pilot looks out through his side window, did he see the other spacecraft perpendicular to his back-front direction?

If the spacecrafts are convertibles and the pilots are holding a rope taut between them, what will happen when they ignite their engines?

If they start exactly at the same time, accelerate equally and also turns off their engines exactly at the same time, so that they can be considered to be at rest with each other as they were before the launch, then why would they appear to be at different locations after launch?

Edited by Spyman
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Ok, so lets remove the base stations and make the projectiles true spacecrafts with engines. They are located in parallel directions a distance apart and at rest with each other. At exactly the same moment they accelerate equally and then turn off their engines and coast in freefall.

Exactly the same moment according to whom? This is why we need an observer equidistant between the two.

Before the launch one pilot looks out through his side window, did he see the other spacecraft perpendicular to his back-front direction?

Yes... but what about as they are accelerating initially? Each would see himself blasting off first.

If the spacecrafts are convertibles and the pilots are holding a rope taut between them, what will happen when they ignite their engines?

If they start exactly at the same time, accelerate equally and also turns off their engines exactly at the same time, so that they can be considered to be at rest with each other as they were before the launch, then why would they appear to be at different locations after launch?

Again... all of this 'exactly at the same time' stuff must be relative to some observer... if it is relative to the observer equidistant between the two ships, then the ships will see something different.

Here is a topic I started years ago that deals with, essentially, the same issue. This could be the thread J.C. was referring to.

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Exactly the same moment according to whom? This is why we need an observer equidistant between the two.

But the synchronization was not part of the question, they are assumed to be synchronized.

Yes... but what about as they are accelerating initially? Each would see himself blasting off first.

During acceleration they are out of sync, but each would also see himself turning off the engines first.

if it is relative to the observer equidistant between the two ships, then the ships will see something different.

Yes, but also not part of the question.

Edited by Spyman
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I read in Carl Sagan's Cosmos that traveling near the speed of light would warp things so that things that were behind you would squish into your forward cone of vision, but I never really made sense of that until now.

I believe that my post fully resolves the paradox but no one has acknowledged or refuted it.

If two rockets are separated by one lightsecond and are launched at the same time in their rest frame, they will each see themselves start to move one second before the other does.

But length contraction will make perpendicular lines that are in the launchers' frame appear to curve forward.

The other rocket will appear ahead of me (on one of those curved lines) until I catch up to it exactly at the point where it reaches the same speed as me. While traveling with the same velocity they are relatively at rest, and will be exactly perpendicular to each other, even though they each witnessed having a head start over the other.

In short: The effects of length contraction ensure that nothing impossible happens. My previous post explains more.

The principle is essentially this: http://en.wikipedia.org/wiki/Aberration_of_light

I'm assuming that gravitons and photons behave identically.

Edited by md65536

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