Mechanics behind curved time causing gravitation/geodesics

[tachi158]

TL;DR Summary

Do particles veer towards objects because they encounter uneven clock rates, and they must maintain 4-velocity; since time is slower the space axes must compensate with acceleration. This causes path to veer towards decreasing clock rate.

Warning: Long post, apologies beforehand.

So science/physics isn't my field of study or work, but am always fascinated by it (looking back, perhaps I should have went down that route). In any event, a few months ago I went to finally learn more about relativity after reading a discussion of time dilation on the show, The Expanse. Actually, that curiosity led me to some pop-sci videos and a commenter said that the speed of light, really was just the speed of "causality." Never heard that term before so down the deep rabbit hole I went, trying to understand the basics of gravity without knowing the math.

I think I finally have a surface "general" understanding of SR and GR as I was completely blind before. But one thing I've been trying to figure out, is the mechanics(?) behind apparent gravitational attraction. I'm trying to clarify a video referenced on these forums, about a beaver with a clock on its head and feet. I've seen it mentioned that this was not accurate (point particles for example). So from what I understand (please correct me if I'm wrong, especially if I use the wrong terms).

Objects travel along a a straight worldline if no forces act upon it. In flat spacetime, it appears to be a straight line along the time axis(?) But in curved spacetime, it's a geodesic. It is mass/stress-energy bends spacetime. But it is the curvature of time that is mostly responsible for apparent gravitational attraction or gravitation. The curvature of space only comes into play with really massive objects and/or high velocity. Time dilation, gravitational redshift, is what I've read in sources (found on this forum) to describe the time curve (and how similar the results are to Newtonian gravity). And that curvature of time vs curvature of space is an artificial consideration. The curvature is of spacetime.

So my question: Imagine a point particle, traveling on it's worldline, in flat spacetime. No forces acting on it, no curved spacetime. Let's say it is moving though, in direction x. Let's say it travels in that direction for a sufficiently long time and it approaches a massive object ("MO"). A planet, sun, whatever. The MO is 45 degrees to the right. If the MO was not there, the particle's coordinate path is straight ahead (inertial frame). Because the MO bends spacetime, the inertial frame is now curved to the right, towards the object. Depending on the particle's relative velocity and position to the MO, it could just curve right and fly past it, go into orbit, or eventually crash into the MO. We all travel through our 4-velocity, which must maintain a constant magnitude (the more you travel through space, the less through time, etc.).

So a few things I have read: Geodesics is the longest proper time path between two points in spacetime. Or more accurately extremal, but in the case of objects it's usually maximal. And maximum proper time equates to the straightest path in 4D. But objects curve time, such that the closer you get to the MO, the less proper time there is, relative to farther distances from the MO. Gravitational time dilation. I've been trying to reconcile the two concepts which appears contradictory.
(1) Geodesics are the longest proper time path between two objects; (2) Geodesics of objects curve towards massive objects and increasingly travel through the area of slower time.
But perhaps I understood it wrong. A geodesic is just the path between two events, and this "longest proper time" thing has nothing to do with the curvature and apparent attraction. Massive object curves spacetime, particle moves towards the MO, and the path it takes will be the path of maximal proper time thereafter. Almost like that rule is ancillary, and not determinative.
An analogy I read from A.T. on this thread states

The rock advancing trough spacetime steers towards the area of slower time(~=denser spacetime), similar to a light ray advancing trough a medium with varying optical density also steers towards the area of higher optical density.

That seems similar so the two clocks analogy, but perhaps instead of on it's head and feet, it's above the head, and below the feet. But why does it steer towards the area of slower time? If objects like to linger in areas of faster proper time, why does the object veer down to areas of slower time?

One explanation I found, which I find plausible, but want to double check was (my own words):

The time dilation/redshift/curvature of time causes uneven flow of time the further away you get from the mass. In flat space the flow of time is even on either side. An object at "rest" is still going through the time axis. You just cruise along on our 4-velocity headed straight up the time vector. When spacetime is curved, the time axis is no longer orthogonal to the space axes. There is a mixture. You really can't be at rest any more even if you tried. Why? The 4-velocity must be conserved as the speed of light c. So as time goes slower, the space axes of the 4-velocity must compensate to maintain it.

So when you encounter uneven clock-rate gradients, you get pulled in and must accelerate to maintain the 4-velocity. This acceleration has the direction of decreasing clock rate.

Sounds similar to "veering towards slower time/denser spacetime." Like a plane that is in flat spacetime, approaches that MO from earlier. The right wing encounters the slower clockrate first, gets pulled in, must accelerate to maintain 4-velocity, so on and so forth and depending on velocity and trajectory, is now crashing into earth (or orbiting, etc).

Is this accurate?

 

[Nugatory]

TL;DR Summary: Do particles veer towards objects because they encounter uneven clock rates, and they must maintain 4-velocity; since time is slower the space axes must compensate with acceleration. This causes path to veer towards decreasing clock rate.

No. For a much better explanation, search this forum for the short, excellent, and often recommended video by our member @A.T.

In flat spacetime, it appears to be a straight line along the time axis(?) But in curved spacetime, it's a geodesic.

It's always a geodesic, and the choice of the time axis is completely arbitrary with no more physical significance than the way "left" and "right" point in different directions as you choose to face in different directions. It just so happens that in flat spacetime it is usually convenient to choose the time axis so that it lines up with the geodesic followed by some object in free fall (typically a "stationary observer") so that's how it's usually drawn. See it drawn that way often enough and you might be forgiven for concluding that that's the only way.

One explanation I found, which I find plausible, but want to double check was (my own words):

Without knowing the source and broader context of this quote it's hard to say, but it sounds somewhere between just plain wrong and a marginally OK explanation for the non-serious layperson.

 

[A.T.]

Is this accurate?

Note that all those analogies boil down to the same result: gravity is along the gradient of gravitational time dilation.

The geodesic visualization:

 

[tachi158]

Note that all those analogies boil down to the same result: gravity is along the gradient of gravitational time dilation.

The geodesic visualization:

Yes I've seen your video and it took me a while to get it, but I understand it more now. Watched it maybe a dozen times and learn something new each time.
I get that gravity is along the gradient of gravitational time dilation. What I'm trying to understand is, what is behind the apparent physical mechanism that causes the object to veer towards the center of the mass? Why is the gradient not repulsive, instead of (apparently) attractive. I guess what I'm wondering is, there is a reason the gradient is not repulsive. So what is it, that makes the gradient turn in, vs out?

Let's say I'm a particle. I'm traveling along, and on my left and right, "clock time" is the same. So nothing is curving me left or right. If I travel long enough, I encounter a massive object, to my right. Now, clock time on my right is slower, and clock time on my left is faster. Why does my geodesic curve right, and not left?

 

[Ibix]

I get that gravity is along the gradient of gravitational time dilation.

Be aware that this model has severe limitations - primarily that time dilation can only be defined in static or stationary spacetimes. These do not exist outside of simple models, although a lot of realistic circumstances are "close enough" for the concept to be useful.

What I'm trying to understand is, what is behind the apparent physical mechanism that causes the object to veer towards the center of the mass?

If you mean "why does mass and energy curve spacetime" we don't know. Einstein's field equations provide a relationship but there's no explanation in terms of particle exchange or something. Quantum gravity might provide one, but we don't have a working theory yet.

If you mean "given some spacetime, why are the paths of objects in free fall what they are" then the answer seems to be "because that's the path that extremises their proper time". It's analogous to asking why things travel in a straight line on a Euclidean plane - that's the path that extremises the path length. But there isn't really a good answer beyond that.

As always in science, the basic justification for anything is "when we use this model we make precise predictions". It doesn't really answer "why" questions.

 

[A.T.]

Yes I've seen your video and it took me a while to get it, but I understand it more now. Watched it maybe a dozen times and learn something new each time.

For more context see chapter 9 & 10 here:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n153/mode/2up

I get that gravity is along the gradient of gravitational time dilation. What I'm trying to understand is, what is behind the apparent physical mechanism that causes the object to veer towards the center of the mass? Why is the gradient not repulsive, instead of (apparently) attractive. I guess what I'm wondering is, there is a reason the gradient is not repulsive. So what is it, that makes the gradient turn in, vs out?

GR doesn't provide any underlying mechanism beyond the concept of geodesics in curved spacetime. This is a postulate of the theory, not something that follows from the theory.

This is analogous to the postulates of Newtonian physics, where force free objects advance along straight lines in a space-time diagram. To get to GR you redefine "force free" as "free falling" (gravity doesn't count as a Newtonian force) and generalize "straight lines" to geodesics, which are "locally straight" lines on curved manifolds. You then introduce a space-time distortion that matches the gravitational time dilation and gravity automatically follows from the geodesics on that distorted manifold.

Let's say I'm a particle. I'm traveling along, and on my left and right, "clock time" is the same. So nothing is curving me left or right. If I travel long enough, I encounter a massive object, to my right. Now, clock time on my right is slower, and clock time on my left is faster. Why does my geodesic curve right, and not left?

To see why, you need to have a more intuitive grasp of geodesics.

The online book linked above also has a subsection to help with this intuition (book pages 182-183, document pages 195-196 ):
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n193/mode/2up

See also chapter 2 (Fig 2.6) of this thesis:
http://www.relativitet.se/Webtheses/tes.pdf

To model geodesics you can also stick adhesive tape to curved surfaces (as straight as possible with minimal tear and folds at the edges):
http://www.relativitet.se/einstein_eng.pdf

Note that what you often see presented as a "mechanism" are just analogies to make the concept of geodesics more intuitive. Like the analogy to light rays in media with varying optical density discussed here (book page 160-163, document page 173-176):
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n171/mode/2up

 

[pervect]

TL;DR Summary: Do particles veer towards objects because they encounter uneven clock rates, and they must maintain 4-velocity; since time is slower the space axes must compensate with acceleration. This causes path to veer towards decreasing clock rate.

This is basically on an interpretation of an APPOXIMATION to Genral Relativity, known as the Newtonian limit. It does not recover the full theory.

Mathematically, it rises from an approximate metric, as in the rather short wiki article https://en.wikipedia.org/wiki/Newtonian_limit.

To understand the significance of this remark about an "approximate metric", one needs to understand what a metric is. If one picks "smooth" coordinates which are otherwise completely arbitrary, the metric is just an expression for the "distance" (actually the space-time interval) between two nearby points. The geodesic equation in GR can then be regarded as the means for finding the shortest paths between two points in an arbitrary and possibly curved geometry. Note that there are theories other than GR which complicate this simplification, but that's mostly beyond the scope of my post.

As I write this, I realize that there is a further pre-requisite. One needs to understand the space-time interval. This comes from special relativity in treatments such as Taylor and Wheeler's "Space-time physics".

So, if you're in the mood to do some further reading, I'd suggest studying special relativity with the textbooks "Space-Time physics" or some similar geometrically oriented text to understand the geometrical notion of the space-time interval. I believe Taylor has a copy of the first edition of the standard text "Space-time physics" on his website, https://www.eftaylor.com/. This is the first edition, later editions used as textbooks are not free.

"Exploring black holes" from the same author is a gentle introduction to extend the geometrical notions to include curvature and more. I believe there are links to this on Taylor's website as well.

The Newtonian approximation to GR is good enough to be useful, but it won't predict some phenomeon in the full theory such as gravitational waves, or (I think) the precession of Mercury's orbit.

Without the background of what a metric is, it may be difficult to appreciate how the short wiki article on the Newtonian limit can be loosely thought of as arising from a time dilation caused only by a Newtonian potential $\Phi$. But that's basically what you're rather long post describes.

The overall path I am suggesting to go beyond the Newtonian limit starts with the "geometrical" view of space-time. The geometrical view starts with the flat space-time of special relativity, and can be expanded (with significantly more effort) to the curved space-time geometries of General relativity.

At a popular level, the difference between special and general relativity is the difference between navigating on a plane, and navigating on the (almost spherical) surface of the Earth.

 

[PeterDonis]

The geodesic equation in GR can then be regarded as the means for finding the shortest paths between two points in an arbitrary and possibly curved geometry.

For timelike geodesics, i.e., the worldlines of objects, it is the longest path, not the shortest--the path with the longest proper time. (There are plenty of caveats here but this will do for now.)

 

[A.T.]

The geodesic equation in GR can then be regarded as the means for finding the shortest paths between two points in an arbitrary and possibly curved geometry.

For timelike geodesics, i.e., the worldlines of objects, it is the longest path, not the shortest--the path with the longest proper time. (There are plenty of caveats here but this will do for now.)

Putting aside all the caveats, I don't even think this condition for a geodesic is helpful to the OPs question. The starting point of the question is just one point and the initial direction, not two points that need to be connected.

 

[pervect]

Putting aside all the caveats, I don't even think this condition for a geodesic is helpful to the OPs question. The starting point of the question is just one point and the initial direction, not two points that need to be connected.

I was trying not to introduce the extra complexity of discussing a connection. And I'll point to "Curving" , by E.F. Taylor, as my inspiration, https://www.eftaylor.com/pub/chapter2.pdf, as my inspiration. I believe this was from an earlier edition of "Exploring black holes", I am not quite certain. It's not from the current web-only edition of "Exploring black holes", which has the same chapter title, but different content. (The second edition is also available on the WWW from Taylor's website).

Distances” Determine Geometry

Describe an object with a table of distances between points.
Describe spacetime with a table of intervals between events.

Nothing is more distressing on first contact with the idea of curved space-
time than the fear that every simple means of measurement has lost its
power in this unfamiliar context. One thinks of oneself as confronted with
the task of measuring the shape of a gigantic and fantastically sculptured
iceberg as one stands with a meterstick in a tossing rowboat on the surface
of a heaving ocean.
Reproduce a shape
using nails and string.
Were it the rowboat itself whose shape were to be measured, the proce-
dure would be simple enough (Figure 1). Draw it up on shore, turn it
upside down, and lightly drive in nails at strategic points here and there
on the surface. The measurement of distances from nail to nail would
record and reveal the shape of the surface. Using only the table of these
distances between each nail and other nearby nails, someone else can
reconstruct the shape of the rowboat. The precision of reproduction can be
made arbitrarily great by making the number of nails arbitrarily large.

Introducing a connection is the standard approach in differential geometry, and then one insists that the connection be both metric compatible and torsion free. So one needs to understand not only the metric, but also the connections, and possibly even torsion.

These are useful concepts, but I think that understanding the concept of interval is sufficient to go a very long way.

This was perhaps a bit of digression, as the main thing I wanted to point out is that the OP's ideas have been echoed by many other posters, and I believe they can be traced back to the Newtonian limit of General Relativity, as I mentioned. The Newtonian limit is a useful idea, but it's not the full theory of General Relativity.

 

[tachi158]

Thank you every one. I have a lot more to read. I was away in Las Vegas (so no response for a few days) but was doing some digging this morning. And I found reference (from a Youtube video lol) to an article by Roy R Gould in the American Journal of Physics (May 2016, "Why does a ball fall?: A new visualization for Einstein's model of gravity" https://aapt.scitation.org/doi/10.1119/1.4939927)
Unfortunately the article isn't open access, but in it he says, "

D. Warped time causes the ball to fall

At first glance, the distortion of time seems just as insignificant as the distortion of distance. The scale of time (scale_t) varies by the factor:

scale_t = (1- ((2MG/R²c²)x)^1/2

between waist-height and floor level, again a change of only 1 part in 10-¹⁶.This tiny variation with height is much too small for our senses to detect, although it has been measured using atomic clocks,6 providing direct proof of Einstein's claim that the scale of time varies with location.

Why does this minuscule warping of time influence the ball's trajectory so dramatically? The reason is that even a small interval of time corresponds t o a large distance. When time is measured in meters. Imagine redrawing Fig. 2 with the time ax.is in meters rather than seconds. The half-second it takes the ball to fall corresponds to about 150 x 10⁶m, so if both axes were drawn at the same scale, the spacetime diagram would stretch halfway to the Moon! Then we would see that the ball's trajectory in spacetime is very nearly a straight line. The distortion of time is indeed tiny. Yes, the ball falls I m to the floor. but it has to travel through 150 x I 06 m of time to get there!

We conclude that the distortion of time-not space causes a ball to fall. In essence, an object is deflected towards the region in which time flows more slowly. In this way, the invisible dimension of time makes its presence palpable in our three-dimensional world.

The bolded italicized statement is what I'm trying to understand. Why are objects deflected towards regions in which time flows more slowly? Especially since geodesics tend to prefer areas where time flows faster? Maybe it's unrelated.

It gets thrown out a lot, objects are deflected towards regions in which time flows more slowly. So I was trying to understand the math behind it? I think this goes back to the 4-velocity statement I made in the beginning. The magnitude of an objects four velocity vector must stay constant. If the temporal component changes, the spatial component must change to compensate. I'm not asking why the universe has deemed it so, just wondering what the math is.

So when an object encounters uneven time gradients, the temporal component of the four velocity changes, and its spatial component must compensate (more motion through space, less through time). And then you get the deflection towards the massive body. Unless this is completely wrong, this is how I'm viewing it?

Also read some stuff about the conservation of energy, etc. that I'll explore later

 

[PeterDonis]

The bolded italicized statement is what I'm trying to understand.

First a misstatement in what comes before needs to be corrected. The ball's trajectory in spacetime is not "nearly" a straight line. It is exactly a straight line. It is a geodesic, and that's what a geodesic is.

With that understood, the word "deflected" in the italicized statement is also misleading. It is not that the object gets "deflected". It's that spacetime is curved. The object is following a straight trajectory through curved spacetime.

when an object encounters uneven time gradients, the temporal component of the four velocity changes, and its spatial component must compensate

This is not a good way to view it, because the components are frame dependent. Actual physics is never frame dependent; the laws are the same in all frames.

 

[PeterDonis]

A new visualization for Einstein's model of gravity

Unfortunately this is only a visualization for one particular solution of the Einstein Field Equation. But the general model of gravity that Einstein came up with is the field equation--not any particular solution of it. The field equation is the general law of gravity. Particular solutions are specialized models for particular situations. You would be much better served by looking at a presentation of the general law--the field equation itself--rather than of one particular solution. You might try this article by John Baez:

https://math.ucr.edu/home/baez/einstein/

 

[Ibix]

Unless this is completely wrong, this is how I'm viewing it?

If you are content with only studying the Schwarzschild solution then it'll do. But the approach leaves a lot to be desired - it only really works in the Schwarzschild case because the spherical and time symmetry means that everything must depend on $r$ and $r$ alone, so all the gravitational phenomena seem directly connected. This is not generally true, however. For example in a rotating black hole the gravitational time dilation depends only on the radius and latitude, but free fall paths curve in the direction of changing longitude.

 

[PeroK]

The bolded italicized statement is what I'm trying to understand. Why are objects deflected towards regions in which time flows more slowly? Especially since geodesics tend to prefer areas where time flows faster? Maybe it's unrelated.

IMO, you can't really understand physics through this half-hearted approach. If you pick up a serious textbook on GR it will specify the assumption/postulate that particles move on worldlines of maximal proper time (and light moves on null worldlines).

This is a special case of a more general principle that appears across all physics, where certain key quantities are maximalised or minimalised by nature. For example, Newton's laws of motion have an equivalent formulation in terms of minimalising the so-called Lagrangian.

 

[A.T.]

Why are objects deflected towards regions in which time flows more slowly?

I think you are stuck in trying to find a "mechanism" by which the time dilation gradient "causes" the geodesic deflection. But that is the wrong way to think about it, as they both are effects of the space-time geometry. They are related, but relation is not causation.

To understand how space-time geometry affects geodesics see the links in post #6. But the animation with the apple and the conical space-time patch shows the core of it.

 

[tachi158]

First a misstatement in what comes before needs to be corrected. The ball's trajectory in spacetime is not "nearly" a straight line. It is exactly a straight line. It is a geodesic, and that's what a geodesic is.

With that understood, the word "deflected" in the italicized statement is also misleading. It is not that the object gets "deflected". It's that spacetime is curved. The object is following a straight trajectory through curved spacetime.This is not a good way to view it, because the components are frame dependent. Actual physics is never frame dependent; the laws are the same in all frames.

Regarding the misstatement, that's not what it says, he says an objects trajectory in just the temporal portion is nearly a straight line. Referencing Figure 2 (which you obviously can't see), he says if it was drawn to actual scale, and written in meters rather than seconds, it would extend halfway to the moon. Figure 2 was just a flat spacetime diagram with two axes, t and x. A nearly straight line. He lays out that the change of only 1 part in 10 to the negative 16. So a line that is halfway to the moon, with a change of only 1 part in 10-¹⁶. Pretty sure that is nearly a "straight line" but not a straight line (speaking in Euclidian terms).

As for deflection, he says "in essence." The article is directed towards Physics teachers looking for concepts suitable to introduce brand new physics students (high school) to Einstein's model of gravity, so simpler terms are used. Yes, the object is not actually deflected. But it's a term a 9th grader will understand. My fault for not providing more context on the article.

(Aside: the author is lamenting the fact that Einstein's model is largely absent many introductory physics classes, and hopes to simplify it by (1) recasting Newton's interpretation of the falling ball in simple geometric form, which enables the teacher to show that Newton's and Einstein's models make mutually incompatible predictions about whether a force acts on a ball, and whether the ball's trajectory through spacetime is curved or straight, (2) introduce non-Euclidian geometry of Einstein's model by using an object familiar to students (an ordinary wall map), and (3) focusing on the physical concept of scale, INOW varying scale of distance and time, to sidestep the mathematical abstraction of curved spacetime (since curvature of time can be split into two parts, a stretching of distance and time, and a twist which is much more difficult for a new physics student to grasp, and needed primarily for relativistic phenomena such as frame dragging). The author argues it is useful to avoid curved spacetime at the introductory level because it requires a lot more mathematics but also because most illustrations used just confuse the students and brings up the rubber sheet which makes zero sense).

As for the "frame dependent" not sure what you mean there? Just trying to understand how the curvature of time causes the attractive gravitation we are all familiar with. I'm on a spaceship and I'm flying by a massive object. I've got burners and I'm resisting the gravitation effect from the planet. If the object is on my left, that means time runs a tiny bit slower on the right, than on the left. That's still one frame, no? I mean it's not a point particle. But it's no bigger than the frame used in train thought experiments. I guess it is two frames, if you consider left and right of a point particle.

Unfortunately this is only a visualization for one particular solution of the Einstein Field Equation. But the general model of gravity that Einstein came up with is the field equation--not any particular solution of it. The field equation is the general law of gravity. Particular solutions are specialized models for particular situations. You would be much better served by looking at a presentation of the general law--the field equation itself--rather than of one particular solution. You might try this article by John Baez:

https://math.ucr.edu/home/baez/einstein/

I'll read it for sure. But I generally just wanted to know how specifically, the curvature of time is responsible for most of what we see as gravity. I understand it's all a curvature of spacetime. If that's all I knew, I would have been happy. It was only after reading that the curvature of space really doesn't come into play until you get to massive speeds, and therefore it's the curvature of time that is the cause of the apparent effects of gravity we are most familiar with. Curvature of space is easy to understand. The hill points towards Earth, you ride along, you encounter the slope, you go towards Earth. Then I read here and everywhere else, it's actually not the curvature of space, but the difference in time that causes the apple to fall back to Earth, that cause the moon to orbit, etc.

Baez did write "The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where

. This allows a slight amount of spacetime curvature to have a noticeable effect." which is similar to what Gould wrote in that other article.
I don't have the knowledge to dive into the maths but I see if I can understand the conclusions. Thank you for the article I'm reading it now.

If you are content with only studying the Schwarzschild solution then it'll do. But the approach leaves a lot to be desired - it only really works in the Schwarzschild case because the spherical and time symmetry means that everything must depend on $r$ and $r$ alone, so all the gravitational phenomena seem directly connected. This is not generally true, however. For example in a rotating black hole the gravitational time dilation depends only on the radius and latitude, but free fall paths curve in the direction of changing longitude.

I'm totally fine with that. As said above, I knew it was a curve of spacetime, but when I read it was MOSTLY the curvature of time that caused the apparent attraction laypeople are all familiar with, I had to close that loop. No way I could walk away without at least getting a cursory explanation. Curvature of space is easy to get. Analogies are numerous, intuitive to understand. Curvature of time? Harder to understand but can be digested. Curvature of time causing the effects we were all led to believe were caused by the curve of space (rubber sheet analogy). Mind blown. I had to understand more, without majoring in physics (45 years old, trained in law, not physics!).

IMO, you can't really understand physics through this half-hearted approach. If you pick up a serious textbook on GR it will specify the assumption/postulate that particles move on worldlines of maximal proper time (and light moves on null worldlines).

This is a special case of a more general principle that appears across all physics, where certain key quantities are maximalised or minimalised by nature. For example, Newton's laws of motion have an equivalent formulation in terms of minimalising the so-called Lagrangian.

I've come across most of these concepts, again I'm in a major physics rabbit hole you normally don't find lawyers in! Purely for entertainment purposes and a serious textbook on GR might take up more time than I have. But finding some text on wordlines and extremal time (which can mean, maximal, null, and stationary I think?) threw me for a loop. I just wanted a cursory explanation of how curvature of time is responsible for apparent gravitation. Not a full understanding of GR.

I think you are stuck in trying to find a "mechanism" by which the time dilation gradient "causes" the geodesic deflection. But that is the wrong way to think about it, as they both are effects of the space-time geometry. They are related, but relation is not causation.

To understand how space-time geometry affects geodesics see the links in post #6. But the animation with the apple and the conical space-time patch shows the core of it.

I was perhaps using the wrong terms. I was more or less asking why do geodesics "curve" towards mass, and not away? More specifically, why do wordlines generally bend towards areas of less proper time, despite the postulate that particles move on wordlines of maximal proper time as referenced above?

Let's say I make the argument: a test particle approaching a massive object will continually encounter areas of decreasing proper time. Because particles like to linger in areas with relative proper time, its wordline will curve away and "appear" to be deflected from the massive object.
This is wrong. And I've spent the past few weeks months trying to figure out why? I've encountered the 4-velocity explanation multiple times by multiple authors, physicists, professors. And its intuitive for a problem that seems unintuitive.

 

[PeroK]

I've come across most of these concepts, again I'm in a major physics rabbit hole you normally don't find lawyers in! Purely for entertainment purposes and a serious textbook on GR might take up more time than I have. But finding some text on wordlines and extremal time (which can mean, maximal, null, and stationary I think?) threw me for a loop. I just wanted a cursory explanation of how curvature of time is responsible for apparent gravitation. Not a full understanding of GR.

I often quote Roger Bacon on this. It's amazing that he wrote this nearly 800 years ago:

"Whoever then has the effrontery to study physics while neglecting mathematics must know from the start that he will never make his entry through the portals of wisdom."

Roger Bacon (1214-84)

He was right. You don't have to learn the whole subject, but I do think you have to accept that GR is (in one sense) mathematics. It cannot really be explained by getting the right words in the right order (distortion, veer, tends to prefer, time flows faster).

The understanding is first and foremost mathematical. The nice wordy explanation comes later. You have to be prepared to learn some mathematics.

Here's a thing. If we are talking about an apple falling from a tree or a tennis ball going in an approximate parabola, the explanation of why it accelerates is quite clear. It's not "really" accelerating! It's everything on the ground that is accelerating. An object on the ground has a real, measurable upward force from the ground. The object in freefall does not. From that perspective, the curved path has no mystery. It's only curved from the perspective of something that is itself being accelerated (in the opposite direction).

Now, of course, that only shifts the question. But, it does show, I hope, why you may be barking up the wrong tree entirely by looking for an explanation for the curved path for the object in free fall. Or, at least, it highlights some of the subtleties involved.

In fact, GR and curved spacetime are only an essential part of the explanation when you look at the wider, global picture. E.g. two objects falling on opposite sides of the Earth. Or, one object falling from close to the Earth's surface and another (with less acceleration) from much higher up.

At the very least, I would open your mind to much broader explanations for GR than the overly classical view that something must be forcing or causing an object in freefall to veer off some hypothetical path. The objects in freefall are on a "natural" (geodesic) path through spacetime. Unlike an object on the ground, they are not being forced onto an unnatural path.

 

[PeterDonis]

he says an objects trajectory in just the temporal portion is nearly a straight line.

And that is wrong. It isn't "nearly" a straight line. It is exactly a geodesic. "Just the temporal portion" makes no sense; you have to include at least one spatial dimension to draw a diagram at all.

Figure 2 was just a flat spacetime diagram with two axes

And spacetime around the Earth is not flat, so a flat diagram will be distorted. Perhaps the line looks only "nearly" straight in the diagram, but that's a distortion of the diagram. The actual line in actual spacetime is exactly straight.

The article is directed towards Physics teachers looking for concepts suitable to introduce brand new physics students (high school) to Einstein's model of gravity

This context is helpful, but I'm not sure making misstatements is the best way to do that. However, it does appear to be a common teaching philosophy, to tell students wrong things that are simple, and later try to make corrections by adding complications. I personally couldn't stand that as a student, but perhaps I'm an outlier.

 

[PeterDonis]

As for the "frame dependent" not sure what you mean there?

It means dependent on which frame (or coordinate chart) you choose.

Just trying to understand how the curvature of time causes the attractive gravitation we are all familiar with.

Then you are on a fool's errand, because it doesn't. As @A.T. pointed out in post #16, the cause of the gravity we are all familiar with is spacetime curvature, not "curvature of time". "Curvature of time" is an effect of spacetime curvature in the vicinity of an isolated gravitating mass like a planet or star, just like the gravity we are all familiar with.

I'm on a spaceship and I'm flying by a massive object. I've got burners and I'm resisting the gravitation effect from the planet. If the object is on my left, that means time runs a tiny bit slower on the right, than on the left. That's still one frame, no?

You can use a frame in which the spaceship is at rest. But then all of the statements you are making about "curvature of time" are false, because they are only true in the rest frame of the planet, not of the spaceship. But correct statements about spacetime curvature are valid in any frame.

It was only after reading that the curvature of space really doesn't come into play until you get to massive speeds, and therefore it's the curvature of time that is the cause of the apparent effects of gravity we are most familiar with.

This is a common pop science description, but it doesn't really help because it is frame dependent (it's only valid in the rest frame of the planet, as above), and it doesn't generalize to other situations.

Curvature of space is easy to understand.

As the saying goes, all complex questions have simple, easy to understand wrong answers.

Baez did write "The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where View attachment 322644. This allows a slight amount of spacetime curvature to have a noticeable effect." which is similar to what Gould wrote in that other article.

Similar perhaps, if you ignore that Baez talks about spacetime curvature, not time curvature. And much less similar if you consider the overall context, since Baez's overall description is very different from the one you referenced.

I was more or less asking why do geodesics "curve" towards mass, and not away?

Because that's how the spacetime curvature works out for that particular solution of the Einstein Field Equation (the vacuum region surrounding an isolated gravitating mass like a planet or star).

why do wordlines generally bend towards areas of less proper time

"Generally" they don't. They happen to in this particular situation, if you restrict attention to the rest frame of the planet, and if you interpret "less proper time" in a particular way--the proper time of stationary observers (i.e., ones who are not freely falling towards the mass).

despite the postulate that particles move on wordlines of maximal proper time as referenced above?

Maximal proper time compared to nearby possible worldlines. That's a different comparison from the comparison of "less proper time" as compared to stationary observers at higher altitudes.

All this illustrates why the "time curvature" explanation causes more problems than it solves; it is leading you down a rabbit hole of questions that you wouldn't even need to ask at all if you focused on spacetime curvature instead.

Let's say I make the argument: a test particle approaching a massive object will continually encounter areas of decreasing proper time. Because particles like to linger in areas with relative proper time, its wordline will curve away and "appear" to be deflected from the massive object.
This is wrong. And I've spent the past few weeks months trying to figure out why?

Unfortunately, your "answer" is causing as many problems as the original question. See above.

 

[A.T.]

I was perhaps using the wrong terms. I was more or less asking why do geodesics "curve" towards mass, and not away? More specifically, why do wordlines generally bend towards areas of less proper time, despite the postulate that particles move on wordlines of maximal proper time as referenced above?

The maximal-proper-time-rule applies only locally and effectively just means that they are "locally straight" (= geodesics).

Beyond that, the maximal-proper-time-rule is not helpful to answer your question, because you don't have two points in space-time that you need to connect with a geodesic. You have a initial point and a initial direction in space-time, and want to know why this worldline direction changes with respect to the space & time axes. This follows from the "locally straight" property and is easiest to understand on a cone surface (see animation in post #3). See also the links in post #6.

The confusion you have is very similar to one that people have about optical refraction: "Why does light bend towards the medium with slower propagation speed, if light always takes the locally quickest path between two points." Here again, when you don't have two points, but a initial point and a initial direction, the quickest-path-rule is not useful, because you have different constraints.

 

[pervect]

The "maximal (more generally, extremal) proper time rule" is sufficient to solve for the trajectory of a test particle given an initial position and velocity.

Specifying some set of coordinates and using the maximal (more strictly, extremal) time approach, one gets a second order differential equation of motion, parameterized by proper time. I.e. if the coordinates are x,y,z,t one gets a system of second order differential equations in $x(\tau), y(\tau), z(\tau), t(\tau)$. Here $\tau$ is the proper time, which is the sort of time a clock measures, and t is the coordinate time, an agreed on system of labeling events to say "when" they occured.

Finding the motion is then a matter of solving the differential equation of motion, i.e. the geodesic equation, with a specified initial value for position and rate of change of position (proper velocity).

MTW uses this approach, and IIRC Taylor does also in "Exploring Black Holes", which I've previously mentioned.

The formulation in terms of position and velocity as boundary conditions at a single point as the boundary conditions of the differential equation guarantees a unique solution from the theory of differential equations. Specifying two points does not necessarily guarantee a unique solution. (Digression: It does turns out that a neighborhood exists for which the solution is unique.)

I am a bit surprised that there appears to possibly be some question about this. Possibly I have misunderstood the objection? In any case, I'll give a reference from "Gravitation" by Minser, Thorne Wheeler henceforth MTW. See section $13.4, pg 315, "Geodesics as world lines of extremal proper time".

Note that MTW takes the approach of using an arbitrary parameter to parameterize the worldline, then notes that the formula becomes simplified when chooses proper time as the parameter. Of course, this is somewhat specific to timelike geodesics, but the technique can be extended to spacelike and null geodesics.

 

[A.T.]

Possibly I have misunderstood the objection?

There is no objection to using the extremal-proper-time-rule in a formal derivation of the geodesic as you describe. What I meant by "not usefull/helpful" refers to gaining the correct intuition and avoiding the apparent paradox the OP sees:

... why do wordlines generally bend towards areas of less proper time, despite the postulate that particles move on wordlines of maximal proper time ...

To resolve this on an informal level, I think it is useful to first translate the locally-extremal-path-rule into "locally straight path", which is simpler to grasp and visualize intuitively.

 

[tachi158]

The maximal-proper-time-rule applies only locally and effectively just means that they are "locally straight" (= geodesics).

Beyond that, the maximal-proper-time-rule is not helpful to answer your question, because you don't have two points in space-time that you need to connect with a geodesic. You have a initial point and a initial direction in space-time, and want to know why this worldline direction changes with respect to the space & time axes. This follows from the "locally straight" property and is easiest to understand on a cone surface (see animation in post #3). See also the links in post #6.

The confusion you have is very similar to one that people have about optical refraction: "Why does light bend towards the medium with slower propagation speed, if light always takes the locally quickest path between two points." Here again, when you don't have two points, but a initial point and a initial direction, the quickest-path-rule is not useful, because you have different constraints.

Got it. I did question myself in the first post, whether the maximal proper time rule even applied to my question, hence me asking this question in the first place.

And that is wrong. It isn't "nearly" a straight line. It is exactly a geodesic. "Just the temporal portion" makes no sense; you have to include at least one spatial dimension to draw a diagram at all.

I don't think you're grasping the context of that straight line. He isn't referencing a geodesic. He's referencing a line, on a diagram. I attached a screenshot. He says if this diagram were drawn to scale, and time was measured in meters. In the half second the ball fell from waist-height to the floor, if the diagram were to scale, it would stretch 150 x 10⁶m to the right, and 1 meter down. That is "nearly" a straight line, and I don't know how else to explain that.

And spacetime around the Earth is not flat, so a flat diagram will be distorted. Perhaps the line looks only "nearly" straight in the diagram, but that's a distortion of the diagram. The actual line in actual spacetime is exactly straight.

Yes, nobody is objecting there. But this is in reference to a diagram.

This context is helpful, but I'm not sure making misstatements is the best way to do that. However, it does appear to be a common teaching philosophy, to tell students wrong things that are simple, and later try to make corrections by adding complications. I personally couldn't stand that as a student, but perhaps I'm an outlier.

It's not a misstatement though, your attributing his statement to something he isn't referencing.

 

[PeterDonis]

He isn't referencing a geodesic. He's referencing a line, on a diagram.

Yes, I understand that. What I am saying is that that diagram misrepresents the situation. The line that looks straight on the diagram, isn't actually straight; it's curved. And the line that looks curved on the diagram, isn't actually curved; it's straight.

 

[tachi158]

Then you are on a fool's errand, because it doesn't. As @A.T. pointed out in post #16, the cause of the gravity we are all familiar with is spacetime curvature, not "curvature of time". "Curvature of time" is an effect of spacetime curvature in the vicinity of an isolated gravitating mass like a planet or star, just like the gravity we are all familiar with.

I understand it's a curvature of spacetime, but it is specifically the curvature of time component that has the largest effect on slow moving objects (weak field). I mean, this is not an unknown concept. Sean Carroll, whom you have cited, uses it
https://www.preposterousuniverse.com/blog/2009/10/12/a-new-challenge-to-einstein/

Given that extremely reasonable assumption, GR makes a powerful prediction: there is a certain amount of curvature associated with space, and a certain amount of curvature associated with time, and those two things should be equal. (The space-space and time-time potentials φ and ψ of Newtonian gauge, for you experts.) The curvature of space tells you how meter sticks are distorted relative to each other as they move from place to place, while the curvature of time tells you how clocks at different locations seem to run at different rates. The prediction that they are equal is testable: you can try to measure both forms of curvature and divide one by the other. The parameter η in the abstract is the ratio of the space curvature to the time curvature; if GR is right, the answer should be one.

There is a straightforward way, in principle, to measure these two types of curvature. A slowly-moving object (like a planet moving around the Sun) is influenced by the curvature of time, but not by the curvature of space. (That sounds backwards, but keep in mind that “slowly-moving” is equivalent to “moves more through time than through space,” so the curvature of time is more important.) But light, which moves as fast as you can, is pushed around equally by the two types of curvature. So all you have to do is, for example, compare the gravitational field felt by slowly-moving objects to that felt by a passing light ray. GR predicts that they should, in a well-defined sense, be the same.

This is a common pop science description, but it doesn't really help because it is frame dependent (it's only valid in the rest frame of the planet, as above), and it doesn't generalize to other situations.

Don't really need to generalize it to other situations, just wanted to understand it in this situation.

All this illustrates why the "time curvature" explanation causes more problems than it solves; it is leading you down a rabbit hole of questions that you wouldn't even need to ask at all if you focused on spacetime curvature instead.

Unfortunately, your "answer" is causing as many problems as the original question. See above.

I mean, the standard you are setting is even higher than someone like Sean Carroll sets for himself, so perhaps you are right. I am on a fools errand.

A slowly-moving object (like a planet moving around the Sun) is influenced by the curvature of time, but not by the curvature of space.

https://www.physics.ucla.edu/demowe..._and_general_relativity/curved_spacetime.html

Lewis Carroll Epstein in his book Relativity Visualized has developed several marvelous illustrations curved spacetime. Art has a copy of the book and model transparencies that you can curve and flatten out on the overhead projector to show:

1. -- how the "curvature of time" causes objects to fall downward near the surface of the earth and causes time to run slower in the basement than on the top floor of the building.

All I wanted to know is how the curvature of time influences the slowly moving object, e.g. object with a trajectory carrying it close to Earth.

 

[PeterDonis]

it is specifically the curvature of time component

To the extent this is a valid description (see below), it is only valid in a particular frame, the rest frame of the mass. I have already pointed that out.

Even in that frame, "curvature of the time component" is not really accurate; the object's worldline has no curvature, and it doesn't even make sense to talk about the curvature of individual components of the object's 4-velocity. See further comments below.

Sean Carroll, whom you have cited, uses it

Note that he does not say "curvature of the time component". He says "curvature of time" (and "curvature of space"). Roughly speaking, in the planet's rest frame, the effect of the metric on the trajectory of an object can be split up into a "time" effect (due to the time component of the metric) and a "space" effect (due to the space components of the metric). But note that, as Carroll says, the effects are only equal for an object moving at or near the speed of light, such as a light ray. For an ordinary object moving very slowly compared to the speed of light, the "space" effect is negligible.

Even here the term "curvature" is something of a misnomer, because Carroll is not describing components of the Riemann curvature tensor, he is describing terms in the geodesic equation. Unfortunately this kind of misnomer is common in informal contexts (though you won't see it in actual textbooks or peer-reviewed papers, where things are required to be more rigorous).

Don't really need to generalize it to other situations, just wanted to understand it in this situation.

That doesn't change the fact that, to get the correct cause and effect, you have to consider the more general laws that the model for this particular situation is derived from.

the standard you are setting is even higher than someone like Sean Carroll sets for himself

Not really. See my comments on Carroll above. He is fairly careful with his phrasing (though he does have one misnomer, as I pointed out--but it's a very common one in informal contexts, as I said).

 

[A.T.]

https://www.physics.ucla.edu/demowe..._and_general_relativity/curved_spacetime.html

All I wanted to know is how the curvature of time influences the slowly moving object, e.g. object with a trajectory carrying it close to Earth.

The UCLA link you have posted shows the conical space-time model and the geodesic on it, which is just a straight line after you unroll the cone. That is all there is to understand geodesics on a cone.