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The principle of Maximum ageing

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Hello all.
The principle of Maximum ageing says a stone (with wristwatch) in free fall takes the path which accumulates the greatest proper time on the stone's wristwatch.
Time runs slower at at sea level than at the summit of Mt Everest.
When an apple falls from a tree, its path is down to the ground. Yet, this free fall path would slow the accumulation of the apple's proper time. What am I misunderstanding?

 

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1 hour ago, crowman said:

When an apple falls from a tree, its path is down to the ground. Yet, this free fall path would slow the accumulation of the apple's proper time. What am I misunderstanding?

As I recall the proper time will not change as the object falls in a gravitational well.  A second will still be a second in proper time.  The coordinate time will be different.

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The principle concerns the path from A to B (two given events on a world line), it doesn't tell you where B must be. Other things tell you that. Eg. drop a stone from rest while standing on Earth, and it will fall downward. Or throw it upward, it will fall upward for some time. B will generally be different in these cases. Either way, if you take two points A and B on the stone's world line, the freefall path between A and B has the greatest proper time of all possible (including non-freefall) paths between A and B.

For example, if A is some point on Earth, and B is the same location a few seconds later, a stone thrown upward so it passes through A and B in freefall will age the most. A fly that takes off from A and lands again at B will age less than the stone. A clock sitting on the ground at A (and B) will age less than the stone.

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Seems redundant but, proper time always moves at 'proper' time.
( has no dependence on frames or relative position in a gravitational well )

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12 hours ago, crowman said:

Yet, this free fall path would slow the accumulation of the apple's proper time. What am I misunderstanding?

Proper time is the geometric length of a test particle’s world line in spacetime - not just in space, and not just in time, but in spacetime. What we find is that in spacetime, the longest possible trajectory between two events is always a geodesic, which is a world line where the test particle does not feel any acceleration at any point. If you start off near a massive object, the only trajectories that avoid you feeling proper acceleration in your own frame are generally those that bring you closer to the massive object (free fall, in the classic sense), since you would need acceleration to do anything else (unless of course you already come in very fast, e.g. in a slingshot manoeuvre).

So think about it geometrically - very loosely speaking, world lines near a massive object tend to be longer as compared to similar ones far away, because spacetime there is “stretched out”, particularly in the time direction (that is why e.g. a radar signal passing by a massive planet takes longer to get to its receiver - because its world line through spacetime is longer). Or you can think of it this way - if you sat on a clock somewhere near a massive object, and look back at another reference clock that is somewhere far away, then the far-away clock would appear to go faster. This is due to gravitational time dilation between your own frame, and the far-away frame. So actually, more time is accumulated (very loosely speaking) near a massive object, as compared to anywhere else.

Note that this can either be a maximum or a minimum, depending on how you choose the signs in your metric. That’s why it is more generally called the principle of extremal ageing.

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11 hours ago, Markus Hanke said:

Or you can think of it this way - if you sat on a clock somewhere near a massive object, and look back at another reference clock that is somewhere far away, then the far-away clock would appear to go faster. This is due to gravitational time dilation between your own frame, and the far-away frame. So actually, more time is accumulated (very loosely speaking) near a massive object, as compared to anywhere else.

How is more time accumulated near the massive object? You've just stated that the far away clock appears to go faster. "Accumulation of time" would refer to proper time.

 

Does the principle imply that two objects in different freefall orbits that intersect at two events, must age the same amount between the two intersections? One could not have maximal aging along one freefall path between the two events, yet have the other age more, right?

If so, then you could have one clock orbiting a massive object several times at a fixed radius, while another clock orbits once, starting at the same radius but traveling far away from the mass before returning. Both are in freefall, both can start and end together. The "escaping" clock would need a faster initial speed, and would "age less" due to SR time dilation, but would also "age more" while having higher gravitational potential. Do they necessarily age the same between events where they meet? Or are there other caveats or restrictions to the principle? My intuition is that you could make the eccentric orbit so far away and so slow at aphelion that it would have to age more, but the principle seems to say that's wrong.

Edited by md65536

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15 hours ago, md65536 said:

How is more time accumulated near the massive object? You've just stated that the far away clock appears to go faster. "Accumulation of time" would refer to proper time.

The total proper time accumulated on a clock between two events in spacetime is equivalent to the geometric length of the clock’s world line C that connects these events, hence:

\[\tau=\int _{C} ds=\int _{C}\sqrt{g_{\mu \nu } dx^{\mu } dx^{\nu }}\]

Just to clarify - I mentioned the comparison to a far away clock only as a pedagogical aid to illustrate a principle; such a comparison is of course not physically precise, because these two clocks would connect a different set of events. Come to think of it, I should not have brought this up at all, as it only confuses the issue further. Please consider it retracted. You can’t really compare such clocks, since there is no good synchronisation convention you could use for this.

15 hours ago, md65536 said:

Does the principle imply that two objects in different freefall orbits that intersect at two events, must age the same amount between the two intersections?

Not necessarily, because the total proper time generally depends on the path that is taken as well as the metric (see line integral above). For example, in an equatorial orbit around a rotating black hole, you would get different readings depending on whether your orbit the black hole in the direction of its rotation, or in the opposite direction.

15 hours ago, md65536 said:

One could not have maximal aging along one freefall path between the two events, yet have the other age more, right

Apologies, I don’t quite understand what you mean with this...?

15 hours ago, md65536 said:

Do they necessarily age the same between events where they meet?

Possibly they could age the same, but not necessarily - again, because the total time accumulated depends on the metric and the path that is taken in spacetime. So it really depends on how they travel, and what the geometry of the underlying spacetime is like. There is really no generally valid answer to this, it depends on the specific case.

15 hours ago, md65536 said:

My intuition is that you could make the eccentric orbit so far away and so slow at aphelion that it would have to age more, but the principle seems to say that's wrong.

I am personally finding that, even though my intuition about all things GR has gotten far more accurate over the years, it is still a slippery slope and sometimes leads me to conclusions that later turn out to be wholly wrong. In cases like the above, really the only way to be sure is to do the math, and evaluate the above integral for the specific case at hand.

In very general terms, we can say that the longest possible path through spacetime between two fixed events is always a geodesic of that spacetime. You can take the above integral, fix the start and end points (i.e. decide on two fixed events), and then vary it between all possible paths between these events, finding the extremum - what you will find is that you end up with the geodesic equation. Because this is a differential equation, its solution is determined by initial and boundary conditions - so if you have a test body coming in at high speed, you might get a free fall geodesic that is entirely different from the one you’d get if you start off at rest. The dynamics are really quite complex, especially in non-trivial spacetimes, so intuition often fails here. It’s best to just sit down and calculate.

For your scenario with highly elliptic orbits, it is also very important to remember that orbital distance and orbital speed are not good indicators of how long the world line of the test particle is in spacetime. You definitely have to do the maths here to tell the whole story.

Edited by Markus Hanke

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I think it is really important to emphasise that in spacetime, you can only meaningfully compare clocks that start together at rest, and end together at rest, i.e. which connect the same two events. This is what the principle of extremal ageing applies to - if you have a number of different paths which connect the same two events, the principle tells us that that path which represents the longest proper time (i.e. the greatest geometric interval) is physically realised.

Schwarzschild spacetime admits a number of Killing vector fields (which is to say it has certain symmetries), so you can extend this principle a little bit in order to compare setups like the one in question here. I think the relevant scenario would be if you have two events A (near a central mass) and B (further from a central mass) which are situated along a radial line with the centre of the mass.
How would a test particle move - will it fall from B to A, or would it go from A to B? If you do the maths, you’ll find that - even though the purely spatial distance is the same in both cases -, the world lines A-B and B-A nonetheless have different geometric lengths in spacetime. The length of B-A is longer than the one of A-B, so a clock freely falling from B-A records more proper time than a clock riding a rocket propelling itself from A-B. This is because the path A-B involves proper acceleration in the frame of the clock, which always implies extra time dilation (ref equivalence principle) on top of the background curvature of spacetime; the free-fall path B-A does not. So B-A will accumulate more proper time. Technically speaking, the curve B-A is a geodesic (segment) of Schwarzschild spacetime, whereas A-B isn’t. The presence of proper acceleration always means a shortening of a test particle’s world line.

So to make a long story short - test particles tend to remain in free fall, because that is how they accumulate the most proper time on their clocks => principle of extremal ageing. Any external force implies local proper acceleration, which in turn implies time dilation, leading to less proper time recorded between the same set of events. This is why test particles which start off at rest always fall towards a central mass, never away from it - because the two world lines aren’t the same ones, even though the spatial trajectory may appear to be.

Obviously, if you start off not at rest, but with some initial momentum, much more complicated dynamics may result. Intuition quickly fails for such scenarios, and it’s really a matter of doing the maths then.

Edited by Markus Hanke

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3 hours ago, Markus Hanke said:

Apologies, I don’t quite understand what you mean with this...?

I'm describing cases where you have two freefall paths that pass through the same pair of events, A and B. A trivial example would be two particles in similar circumpolar orbits leaving together above the north pole and meeting again above the south pole.

A more useful example is two particles in eccentric orbits of different sizes, and they meet at one's perihelion and the other's aphelion, and the particle in the smaller orbit makes two orbits for every one of the larger. Since the particle in the larger orbit makes an orbit at lower speeds (in the gravitational mass's reference frame, say) than the one in the smaller orbit, and is also at a higher gravitational potential, it must age more than the one in the smaller orbit.

Therefore the principle of maximal aging cannot truthfully say "If a particle traveling between events A and B is in free fall, then its aging is greater than any other path between A and B."

I've tried finding the actual definition of the principle, and found several variations, including many like the above which I think are false. I've also seen, "the aging is greater than any other nearby path" which is true, and "The path of maximal aging between A and B is a geodesic", which is true.

A possible problem is they're assuming that a geodesic between A and B is unique, when really they can only assume that it is locally unique?

The closest to definitive I can find is from Taylor and Wheeler's "EXPLORING BLACK HOLES Introduction to General Relativity Second Edition":

Quote

DEFINITION 2. Spacetime patch
A spacetime patch is a region of spacetime large enough not to be limited
to differentials but small enough so that curvature does not noticeably affect
the outcome of a given measurement or observation on that patch.

[...]

DEFINITION 3. Principle of Maximal Aging (Special and General
Relativity)
The Principle of Maximal Aging says that a free stone follows a worldline
through spacetime (flat or curved) such that its wristwatch time (aging) is a
maximum across every pair of adjoining spacetime patches.

That certainly excludes my example. It seems that free fall aging is maximal among "nearby" paths, and the caveat is necessary. A lot of web pages describing the principle of maximal aging are leaving it out and mislead me to the incorrect conclusion that any freefall paths between events A and B will have maximal aging among all possible paths between A and B.

23 minutes ago, Markus Hanke said:

I think it is really important to emphasise that in spacetime, you can only meaningfully compare clocks that start together at rest, and end together at rest, i.e. which connect the same two events. This is what the principle of extremal ageing applies to - if you have a number of different paths which connect the same two events, the principle tells us that that path which represents the longest proper time (i.e. the greatest geometric interval) is physically realised.

Yes they must pass through the same events to compare them, but they definitely don't have to be at rest. If two world lines intersect, that's a single event, regardless of the objects' velocities. If a world line passes through a given event, it does so in every frame of reference.

Edited by md65536

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4 minutes ago, md65536 said:

Therefore the principle of maximal aging cannot truthfully say "If a particle traveling between events A and B is in free fall, then its aging is greater than any other path between A and B."

That is not the definition I would use, since the possible paths connecting events depend on initial and boundary conditions as well (specifically - whether your test particle starts off at rest, or has initial momentum). It is better to say that the principle implies that any free-fall path connecting two given events must be a geodesic of spacetime. In singly-connected spacetimes, this choice is unique, since for a given set of appropriate boundary conditions, the geodesic equation has precisely one unique solution.

Technically you could have spacetimes the topology of which is multiply-connected; in those cases you can then have more than one geodesic connecting the same two events. I don’t know though if it would be possible to have more than one geodesic of the same length connecting the same events. That is an interesting question, but I’d have to think about that first before attempting an answer. I certainly couldn’t think of a physically realisable example right now.

14 minutes ago, md65536 said:

I've tried finding the actual definition of the principle

Well, the technical definition is that you start with the integral I quoted earlier, hold the start and end points as fixed, and then vary the paths between these points. The principle simply states that the path physically taken is the one that is an extremum of this proper time functional, i.e. the longest one.

Note that the principle of extremal ageing is a specific example of a more fundamental principle, the principle of least action. That is the fundamental principle that underlies all field theory frameworks, specifically quantum field theory, and hence the Standard Model. Also GR itself follows from it - applying the principle of least action to the Einstein-Hilbert action gives you the Einstein field equations.

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2 hours ago, md65536 said:

Yes they must pass through the same events to compare them, but they definitely don't have to be at rest. If two world lines intersect, that's a single event, regardless of the objects' velocities. If a world line passes through a given event, it does so in every frame of reference.

Indeed, you are right, I didn’t quite think this detail through. What’s more, if we want to be dealing with free fall world lines, then the clocks can’t really be at relative rest at the final event, as this would imply acceleration.

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9 hours ago, Markus Hanke said:

Indeed, you are right, I didn’t quite think this detail through. What’s more, if we want to be dealing with free fall world lines, then the clocks can’t really be at relative rest at the final event, as this would imply acceleration.

"Rest" and acceleration are frame-dependent. A stone thrown upward momentarily comes to rest without proper acceleration.

I disagree with the general characterization of the principle you're using. It's not something that applies only in the simplest cases, it always applies. So that includes a free-falling particle with a world line billions of years long, falling past countless moving masses. It can come to rest many times. The principle doesn't say anything about initial conditions, and it doesn't have to because it still applies in all cases.

The only restriction is it can't be applied to "distant" (non-adjoining) spacetime patches. To try to paraphrase Taylor/Wheeler ("The Principle of Maximal Aging says that a free stone follows a worldline through spacetime (flat or curved) such that its wristwatch time (aging) is a maximum across every pair of adjoining spacetime patches."): The principle only applies without restriction in "flat enough" spacetime, but it can be applied to an arbitrarily complicated (curved?) free-fall world line by dividing the world line into small enough sections that pass through flat-enough spacetime patches, and applying the principle to each of those sections. If it's complicated enough, there may be other paths that involve greater aging (such as the multiple orbits examples I've given above), but those necessarily involve paths across spacetime patches that are not adjoining a patch through which the world line in question passes. (I think that's what it's saying.)

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14 hours ago, Markus Hanke said:

Note that the principle of extremal ageing is a specific example of a more fundamental principle, the principle of least action.

That needed to be mentioned, Markus. +1

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7 hours ago, md65536 said:

"Rest" and acceleration are frame-dependent. A stone thrown upward momentarily comes to rest without proper acceleration.

“Rest” is indeed frame dependent, but proper acceleration (as opposed to coordinate acceleration) is not - it’s something that all observers agree on.

7 hours ago, md65536 said:

I disagree with the general characterization of the principle you're using. It's not something that applies only in the simplest cases, it always applies. So that includes a free-falling particle with a world line billions of years long, falling past countless moving masses. It can come to rest many times. The principle doesn't say anything about initial conditions, and it doesn't have to because it still applies in all cases.

I think you may have misunderstood what I was attempting to say, possibly I didn’t make my thoughts clear enough. Of course the principle always applies - it is a fundamental principle of physics, derived from the principle of least action, and there are no exceptions to its validity in the classical world. However, calculating a specific free-fall geodesic is an operation that must depend on initial and boundary conditions, since the underlying equations are differential equations. You cannot obtain a specific, unique solution to a differential equation without imposing some form of initial/boundary condition. So this is how different specific geodesics of the same spacetime are distinguished - they are just different solutions to the same equation (i.e. the same principle of extremal ageing).

So you can have different free fall world lines around the same central mass, all of which naturally adhere to the principle of extremal ageing; but they also represent different sets of physical boundary conditions, which you’d need in order to physically realise such trajectories. So the uniqueness of solutions to the geodesic equation is preserved. Hence, the geometry of spacetime along with the principle of extremal ageing determine its geodesic structure, and boundary conditions pick out the specific geodesic that is being realised.

Does this makes more sense now?

 

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I hope you chaps don't mind me not saying anything in replies, but I'm reading your replies and learning new things.

All best

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8 hours ago, crowman said:

I hope you chaps don't mind me not saying anything in replies, but I'm reading your replies and learning new things.

 Feel free to bring us back to the original examples etc., I've kind of gone off into possibly irrelevant details.

19 hours ago, Markus Hanke said:

So you can have different free fall world lines around the same central mass, all of which naturally adhere to the principle of extremal ageing; but they also represent different sets of physical boundary conditions, which you’d need in order to physically realise such trajectories. So the uniqueness of solutions to the geodesic equation is preserved. Hence, the geometry of spacetime along with the principle of extremal ageing determine its geodesic structure, and boundary conditions pick out the specific geodesic that is being realised.

Does this makes more sense now?

No. I don't know if you're trying to say, paraphrased, "A given free particle follows a path of maximum ageing, and a given particle has initial conditions." Which is obviously redundant. Or, are you saying "A given free particle follows a path of maximum ageing among all possible nearby paths with the same initial conditions." I can very roughly prove that this is redundant if you doubt it. Either way, the principle doesn't say anything about initial conditions and it doesn't imply that its application depends on initial conditions.

I feel like we're having a conversation like this: P1: Given some initial conditions, the velocity of a particle is dx/dt. P2: That doesn't depend on initial conditions. P1: Do you understand that a specific particle that has velocity will have initial conditions?

Otherwise, I'm missing the point of why you're talking about initial conditions, especially since limiting the initial conditions will exclude some of the applications of the principle already discussed in this thread.

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2 hours ago, md65536 said:

"A given free particle follows a path of maximum ageing, and a given particle has initial conditions." Which is obviously redundant. Or, are you saying "A given free particle follows a path of maximum ageing among all possible nearby paths with the same initial conditions."

Both of these are true statements, and actually express the same thing. The first statement reflects the formulation of the principle in terms of the geodesic equations

\[a^{\mu } =0\]

whereas the second statement expresses it in terms of the variational principle (i.e. variation of the integral I quoted earlier, with fixed start and end points). Since variation of the proper time functional yields the above differential equations, these statements are physically and mathematically equivalent.

2 hours ago, md65536 said:

Otherwise, I'm missing the point of why you're talking about initial conditions, especially since limiting the initial conditions will exclude some of the applications of the principle already discussed in this thread.

The principle of extremal ageing applies always, everywhere, and in all cases where free fall takes place; it basically tells us that any free fall world line that is physically realised must be a geodesic of the underlying spacetime. A geodesic is a curve which parallel-transports its own tangent vector, which physically means it is a trajectory on which proper acceleration is exactly zero at all instances. So, we write down the equation above. This is a system of partial differential equations, since acceleration is the second covariant derivative of the position function wrt whatever quantity it has been parametrised with. In order to find a specific, unique solution to the equations, we must impose initial/boundary conditions; that is the nature of a differential equation.

So, I am talking about initial/boundary conditions, because these are needed to select a specific geodesic, amongst all geodesics within the region of spacetime in question. This does not at all limit the applicability of the principle in any way - it’s just the procedure we need to follow to mathematically determine a unique, specific geodesic. 

In terms of calculus, we are dealing with segments of curves; since world lines have to be smooth and differentiable everywhere, we need to ensure that this remains the case at the start and end points of the segment as well; this physically corresponds to our boundary conditions, and selects the specific geodesic.

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Alright, I have no further dispute and I think I'm getting hung up on details I don't understand.

But back to the original problem, which is that a stone fixed on the top of a mountain will age more than some others, even though it isn't in freefall.

However, another stone thrown straight upwards from the top of the mountain, with any velocity that keeps it nearby (no escape velocity etc.), will return to the same spot. Having a freefall path, it will have aged more than the fixed stone did between the throw and the landing.

 

So you can vary the initial conditions and get different geodesics and let that determine the events A and B between which you're comparing different path lengths. Or you can fix A and B and get a locally unique geodesic and unique initial conditions. Another thing the principle doesn't mention is the notion of the two events A and B. All of these details are not aspects of the principle, they're just things we're using to properly specify a particular case that we're applying the principle to.

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10 hours ago, md65536 said:

But back to the original problem, which is that a stone fixed on the top of a mountain will age more than some others, even though it isn't in freefall.

 

10 hours ago, md65536 said:

However, another stone thrown straight upwards from the top of the mountain, with any velocity that keeps it nearby (no escape velocity etc.), will return to the same spot. Having a freefall path, it will have aged more than the fixed stone did between the throw and the landing.

The above two statements appear to contradict one another - which one do you mean to say ages more?
Generally speaking, the fixed stone is subject to constant proper acceleration, so it will experience a time dilation commensurate to that.

10 hours ago, md65536 said:

So you can vary the initial conditions and get different geodesics and let that determine the events A and B between which you're comparing different path lengths.

Yes, indeed.

10 hours ago, md65536 said:

Or you can fix A and B and get a locally unique geodesic and unique initial conditions.

Yes, also correct.

10 hours ago, md65536 said:

Another thing the principle doesn't mention is the notion of the two events A and B. All of these details are not aspects of the principle, they're just things we're using to properly specify a particular case that we're applying the principle to.

Yes, absolutely right!
Specifying two events A and B is one possible example of initial/boundary conditions.

10 hours ago, md65536 said:

Alright, I have no further dispute and I think I'm getting hung up on details I don't understand.

I personally think you understand the principle of extremal ageing very well :) In fact, you have a much better grasp on world lines, geodesics and extremal ageing than many other amateurs would, at least that is the impression I got on this thread. There was only some slight confusion on the role initial/boundary conditions play in this, but that is understandable.

If you like I could work through an actual example for you - something extremely simple, such as geodesics on a flat 2D manifold. Obviously this is a completely trivial example, but it would nevertheless show where and how initial/boundary condition come into this. Let me know if you think that would be of benefit for you!

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22 hours ago, Markus Hanke said:

The above two statements appear to contradict one another - which one do you mean to say ages more?

The thrown stone is in freefall and ages the most among any nearby paths.

What I meant by the first statement is that you can find other arbitrary non-freefall paths that age less than the stone sitting on top of the mountain, even one that includes sections of freefall as per OP's example. Eg. if you dropped a stone off the mountain in freefall, but then brought it back up so that you could make a definite comparison of their proper times, it could have aged less than the stone on the mountain, depending on how you do it (but this example involves a non-freefall path and doesn't violate the principle of max ageing).

Also... technically nothing's stopping you from comparing two distant (in space and/or time) worldlines any way you want to, and coming up with different answers, but that's not going to violate the principle either.

 

22 hours ago, Markus Hanke said:

If you like I could work through an actual example for you - something extremely simple, such as geodesics on a flat 2D manifold. Obviously this is a completely trivial example, but it would nevertheless show where and how initial/boundary condition come into this. Let me know if you think that would be of benefit for you!

If it's worth it for you; exposure to more maths would help me.

 

One thing I was stuck on is the idea and meaning of globally non-unique geodesics. A wiki page cites Misner/Thorne/Wheeler's Gravitation, p. 316 and I looked it up and it seems to contradict a couple of things from this thread. However I don't want to bring up the details until I understand it better, which I might not do. I doubt it affects the basic understanding of all this.

Edited by md65536

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1 hour ago, md65536 said:

(but this example involves a non-freefall path and doesn't violate the principle of max ageing).

Yes, one needs to be careful to compare like with like; the principle does not really say anything about non-geodesic paths.

1 hour ago, md65536 said:

One thing I was stuck on is the idea and meaning of globally non-unique geodesics.

Do you mean different geodesics connecting the same pair of events?

1 hour ago, md65536 said:

A wiki page cites Misner/Thorne/Wheeler's Gravitation, p. 316 and I looked it up and it seems to contradict a couple of things from this thread.

I own a copy of Misner/Thorne/Wheeler, it’s the text from which I learned a large junk of my GR knowledge. Feel free to ask away! I had a quick look at page 316, but couldn’t spot anything that contradicts this thread. It gives the integral I wrote earlier (up to a sign), and points out that geodesics are extrema (as opposed to always being maxima) of that variation, as I did earlier as well. 

1 hour ago, md65536 said:

If it's worth it for you; exposure to more maths would help me.

Suppose we want to determine free fall world lines on a flat 2D manifold (like a piece of flat paper). For simplicity, let’s choose a simple Cartesian x-y coordinate system on our manifold, like the ones we all used in high school; the world lines we are looking for can then be written in the form \(y(x)\). I’ll be very sloppy with notation here, since I only want to show the basic blueprint of how this is done.

First, we apply the principle of extremal ageing. It tells us that free fall world lines - regardless of what kind of manifold we are on - must be geodesics of that manifold; so in other words, the principle tells us that, among all possible world lines, the ones representing free fall must be geodesics of the manifold. That means that proper acceleration must vanish at all points of such paths - the principle can thus be mathematically stated as 

\[a^{\mu}=0\]

Since we are in 2D, the index \(\mu \) can take the values (0,1), so we have a system of two equations. But since they are not independent, in the sense that the world line we are looking for is parametrised as \(y(x)\), we need only one of these equations to find our solution. We remember also that acceleration is the second covariant derivative wrt time, which leaves us with the equation

\[\frac{d^{2} y^{0}}{dx^{2}} +\Gamma ^{0}_{\mu \nu }\frac{dy^{\mu }}{dx}\frac{dy^{\nu }}{dx} =0\]

We know that we are on a flat manifold, so all metric coefficients are constants, meaning the Christoffel symbol in the above equation vanishes, leaving us with simply

\[\frac{d^{2} y}{dx^{2}} =0\]

Integrate once:

\[\frac{dy}{dx} =A=const.\]

Integrate again:

\[y=Ax+C\]

Of course, we recognise this immediately as the equation of a straight line. Which is what one would expect - the geodesics on a flat 2D manifold are straight lines. But note that the above isn’t any specific line - it’s actually a 2-parameter family of lines, which is parametrised by two constants A and C. So what the principle of extremal ageing does is reduce the set of all possible world lines down to the set of just those world lines that are geodesics. But there are still infinitely many of them, corresponding to the infinitely many possible choices of A and C.

Here is where boundary conditions come in - to reduce the set of all geodesics down to one specific, unique geodesic, we impose boundary conditions. In this case we have two undetermined constants, so we have to supply two boundary conditions to uniquely determine a geodesic. For example, that could be two events such as (0,1) and (2,2), which yields the geodesic

\[y( x) =\frac{1}{2} x+1\]

So there you go, this is the general workflow. In curved GR spacetimes, the specific maths become much more complicated, since the Christoffel symbols don’t vanish, and you generally have more than one equation remaining in the system - but the general outline is still the same. 

Is this helpful in any way? 

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