A different way of looking at the trampoline analogy

[geordief]

One of the main arguments against the trampoline analogy for spacetime  curvature  is that the heavy object distorting the fabric of the trampoline/spacetime fabric itself relies on gravity to act ...

Suppose we just have two bodies ,one of which is a source of mass/energy and they are connected by a geometric line  that is marked off in equal lengths of spacetime distance.

There is  a tension between the two objects (and so along the geometric line?) for the "simple" reason * *that the two objects have  in principle a common source at the beginning of the universe ,whatever that looked like.

Going back to the two objects,as time progresses the massive/energetic object "pulls on " the geometric line representing spacetime  and "reels it in" like a measuring tape that springs back into the body of the tool.

But the line is held at the location of the second body  and ,moreover is elastic .

And so the spacetime intervals written on the  geometric line are stretched on the side of the massive body and this stretching is continues all the way along the geometric line  as far as the second body,with the distortion from the ideal even spacing greatest  in the immediate vicinity of the massive body ,but still existing (propagating?) as far as the second body ,or indeed to infinity if that  is where the second body is.

This is different from the trampoline analogy in the it just deals with one spatial  dimension (at a time).

The idea occured to me as I have been forced to wear  compression stockings for now  and when you pull them up the leg it looks a bit like stretching  a trampoline or spacetime fabric in action

 

Also ,in the What is Gravity thread  a few of you are talking about whether spacetime curvature can cause mass or vice versa.

In "my" analogy  the mass can "ingest" spacetime  and so it "causes" it.

And the stretchy geometric  line (spacetime) cause the second object to move relative to the massive object.

 

**well,I think both these objects have a prior relationship  and so a "historic" connection in that sense

Edited March 17 by geordief

[Bufofrog]

It's just an analogy.  No analogies will completely describe the actual phenomena 

[geordief]

19 minutes ago, Bufofrog said:

It's just an analogy.  No analogies will completely describe the actual phenomena 

Yes ,it is intended to be an analogy. 

But analogies can perhaps be useful as a way of getting a clearer  understanding of the actual phenomena or just encouraging further inquiry.

So, can some analogies be better than others?

The trampoline analogy has the  fault ,if taken literally**of mistaking an effect if gravity for gravity itself.

If that is disregarded  it is a lot "catchier" than mine -maybe too catchy for it's own good.

 

**of course students should be repeatedly warned that it is just a flawed analogy (as all analogies  intrinsically are)

[Genady]

The main problem with the trampoline analogy is that it has nothing to do with the gravity related spacetime curvature. A curious student should ask after watching it, 

"How can the tracks of a ball and of a bullet be curved so differently if that curvature arises from the geometry of space?" (MTW, Gravitation (p. 32).)

[KJW]

2 hours ago, geordief said:

One of the main arguments against the trampoline analogy for spacetime curvature is that the heavy object distorting the fabric of the trampoline/spacetime fabric itself relies on gravity to act ...

That's a standard argument, but my argument is that the familiar gravity we recognise as Newtonian gravity is caused by time dilation, whereas the trampoline analogy indicates spatial curvature as the cause of gravity. Thus, the trampoline analogy is misleading about what causes gravity. A correct analogy that I discovered recently is two wheels of unequal radius joined by an axle. As this rolls along a flat road, the trajectory will curve towards the smaller wheel, and the larger the difference between the radius of the two wheels, representing time dilation, the larger the curvature of the trajectory, representing the acceleration we feel as gravity.

 

[Genady]

2 minutes ago, KJW said:

That's a standard argument, but my argument is that the familiar gravity we recognise as Newtonian gravity is caused by time dilation, whereas the trampoline analogy indicates spatial curvature as the cause of gravity. Thus, the trampoline analogy is misleading about what causes gravity. A correct analogy that I discovered recently is two wheels of unequal radius joined by an axle. As this rolls along a flat road, the trajectory will curve towards the smaller wheel, and the larger the difference between the radius of the two wheels, representing time dilation, the larger the curvature of the trajectory, representing the acceleration we feel as gravity.

 

You are spoiling my question in the post right above yours ... 😉

[md65536]

1 hour ago, KJW said:

A correct analogy that I discovered recently is two wheels of unequal radius joined by an axle. As this rolls along a flat road, the trajectory will curve towards the smaller wheel, and the larger the difference between the radius of the two wheels, representing time dilation, the larger the curvature of the trajectory, representing the acceleration we feel as gravity.

Or use wheels that are the same size and curve the road intrinsically... like with a trampoline.

How is your analogy "correct"? What do the wheels represent and are you saying that spacetime (the road) is not really curved???

 

This also shows that gravity is not needed to show curvature in the trampoline analogy. Pin a rubber sheet flat against a wall in zero-g. Stick a large ball representing a gravitational mass under the sheet, stretching it (or even a long pipe sticking out from the wall, to imagine it more extremely). Roll an axle with 2 wheels of the same size along it, and the path will curve, analogous to null geodesics.

[KJW]

2 hours ago, md65536 said:

Or use wheels that are the same size and curve the road intrinsically... like with a trampoline.

No, that's not how gravity operates.

 

 

2 hours ago, md65536 said:

How is your analogy "correct"? What do the wheels represent and are you saying that spacetime (the road) is not really curved???

The two wheels represent two clocks at different locations in space. The two wheels rotate at the same angular rate, representing simultaneity. But each wheel rolls different distances on the road, representing different proper times. The different proper times at different locations in space is the time dilation that causes the path to curve. The road represents a two-dimensional spacetime where width is space and length is time, and this spacetime need not be curved, for example in the case of an accelerated frame of reference in flat spacetime. In other words, the analogy doesn't only describe true gravity caused by matter, but also artificial gravity caused by an accelerated frame of reference. The equivalence principle ensures that the same analogy applies to both.

 

 

2 hours ago, md65536 said:

This also shows that gravity is not needed to show curvature in the trampoline analogy. Pin a rubber sheet flat against a wall in zero-g. Stick a large ball representing a gravitational mass under the sheet, stretching it (or even a long pipe sticking out from the wall, to imagine it more extremely). Roll an axle with 2 wheels of the same size along it, and the path will curve, analogous to null geodesics.

The rubber sheet analogy is wrong because the curvature of the rubber sheet is not a correct representation of what causes gravity.

 

 

Edited March 17 by KJW

[geordief]

5 hours ago, Genady said:

The main problem with the trampoline analogy is that it has nothing to do with the gravity related spacetime curvature. A curious student should ask after watching it, 

"How can the tracks of a ball and of a bullet be curved so differently if that curvature arises from the geometry of space?" (MTW, Gravitation (p. 32).)

Because it is spacetime that is curved and not space?

[Genady]

14 minutes ago, geordief said:

Because it is spacetime that is curved and not space?

Exactly.

In the spacetime, they both have curvature of about 1 lightyear radius:

[geordief]

13 minutes ago, Genady said:

Exactly.

In the spacetime, they both have curvature of about 1 lightyear radius:

Thanks.I did track down that passage in the  book.

For those of us who find it hard to visualise a curved 4d (3d+time) object  is the way to do this to break the 4d down into  all its 2d surfaces?

Would every surface need to be curved for the 4d object to be classed as such?

Or could in theory only one of eg spatial surfaces be curved and all the other surfaces be "straight" and the 4d construct would also be curved?(not that I expect such a thing to exist in nature)

[KJW]

Curvature is a two-dimensional notion. It requires a surface to be curved in both directions. So, a sphere is curved but a cylinder is not. In higher dimensions, curvature can also be considered in terms of sectional planes. However, in two dimensions, there is only one type of curvature; in three dimensions, there are two types of curvature; and in four or more dimensions, there are three types of curvature.

 

[Genady]

27 minutes ago, geordief said:

Thanks.I did track down that passage in the  book.

For those of us who find it hard to visualise a curved 4d (3d+time) object  is the way to do this to break the 4d down into  all its 2d surfaces?

Would every surface need to be curved for the 4d object to be classed as such?

Or could in theory only one of eg spatial surfaces be curved and all the other surfaces be "straight" and the 4d construct would also be curved?(not that I expect such a thing to exist in nature)

It depends on the symmetries of a situation. For example, a spherically symmetric case, e.g., Schwarzschild black hole, is usually visualized in 2D, i.e., temporal and radial.

[Markus Hanke]

7 hours ago, KJW said:

However, in two dimensions, there is only one type of curvature; in three dimensions, there are two types of curvature; and in four or more dimensions, there are three types of curvature.

I’m probably forgetting something obvious but…what are you referring to by “types”? Do you mean the Weyl tensor, Ricci tensor, and Ricci scalar?

[KJW]

12 hours ago, Markus Hanke said:

I’m probably forgetting something obvious but…what are you referring to by “types”? Do you mean the Weyl tensor, Ricci tensor, and Ricci scalar?

Strictly speaking, I mean the Weyl tensor, traceless Ricci tensor, and Ricci scalar. One can combine these in a multitude of ways, but these three are the true algebraically distinct curvature tensors.

 

[geordief]

On 3/17/2024 at 4:07 PM, KJW said:

That's a standard argument, but my argument is that the familiar gravity we recognise as Newtonian gravity is caused by time dilation, whereas the trampoline analogy indicates spatial curvature as the cause of gravity. Thus, the trampoline analogy is misleading about what causes gravity. A correct analogy that I discovered recently is two wheels of unequal radius joined by an axle. As this rolls along a flat road, the trajectory will curve towards the smaller wheel, and the larger the difference between the radius of the two wheels, representing time dilation, the larger the curvature of the trajectory, representing the acceleration we feel as gravity.

 

Can you explain (in simple terms for me and perhaps in greater detail for others) how time dilation causes gravity?

I thought time dilation was caused by relative motion  and that relative motion does not necessarily entail ,or cause  gravity.

So you can have time dilation  where there is no curved spacetime. 

[MigL]

Before KJW and Markus make this discussion too mathematical, and the rest of us can only look on, let me put in my two cents.

The rubber sheet ( trampoline ) analogy fails because it is 2D, gravity pulls the trampoline down, and we cannot show 4D curvature ( yes I know, KJW and Markus, except mathematically ).
You would need to imagine trampolines above the mass, on all sides of it, and even extending into past and future.
And all of those are not pulled down by gravity.

Can, or is, space-time really curved ?
All we know is our model does a very good job of describing how test masses act in space-time, by using the curved geodesics that ascribe curvature to our model.

On 3/17/2024 at 12:07 PM, KJW said:

As this rolls along a flat road, the trajectory will curve towards the smaller wheel, and the larger the difference between the radius of the two wheels

And that's why I never engaged full-time 4 wheel drive with my Jeep Grand Cherokee on firm, dry surfaced roads.

Edited March 18 by MigL

[geordief]

1 hour ago, MigL said:

Can, or is, space-time really curved ?

Ouch ,did you mean to say "really"?.

 

I think it is just a mathematical construct to model real effects.

[KJW]

On 3/19/2024 at 8:03 AM, geordief said:

Can you explain (in simple terms for me and perhaps in greater detail for others) how time dilation causes gravity?

It appears that my analogy has failed. I shall abandon it in favour of an even simpler analogy:

Imagine walking along a road that is constant in width. The journey along the road represents the journey through time. It does take time to travel along the road in the analogy, so the analogy is somewhat realistic in this way. At regular intervals along the road, there are lines perpendicularly across the road from kerb to kerb. Each of these lines represent a spatial slice through spacetime such that each spatial slice is at least approximately the same as every other spatial slice, and different points on any given line represent different locations in space. As you walk parallel to the kerb anywhere on the road, you are at the same location in space at different times. The line is one-dimensional and represents the dimension along which the time dilation occurs. In the case of the earth and its gravity, a line on the road represents a vertical line, and the constant width of the road represents a constant height over time.

While the road is straight, the distance between the lines along the left and right kerbs is the same. But as the road curves to the left, the distance between the lines along the left kerb will be shorter than the distance between the lines along the right kerb. And as the road curves to the right, the distance between the lines along the left kerb will be longer than the distance between the lines along the right kerb. That is, the road always curves toward the shorter distance along the kerb and away from the longer distance along the kerb. There is a mathematical relationship between the curvature of the road and the relative difference between the lengths along the left and right kerbs. The relative difference between the lengths along the left and right kerbs represents the gravitational time dilation between the two different heights. And the curvature of the road represents the upward acceleration we perceive as we stand on the ground.

Although we can see ahead along the road, seeing the curve of the road, this would represent seeing into the future, which we can't do. So, we don't actually see a curved trajectory in spacetime. We perceive it as gravity. It should be noted that there is also time dilation for an accelerated frame of reference in flat spacetime, and that there is the same mathematical connection between accelerational time dilation and acceleration as between gravitational time dilation and gravity, as required by the equivalence principle.

However, there is an unavoidable complication between the analogy and the physics it represents that is due to the peculiarity of the geometry of spacetime compared to the familiar Euclidean geometry of the analogy. Thus, the acceleration of a trajectory in spacetime points in the opposite direction to the curvature of the road. That is, on earth we accelerate upward and away from the shorter distances in time, not towards it as in the analogy. The analogy is still correct, the mathematics describing it is the same as the mathematics describing gravity. In familiar Euclidean geometry, the two-dimensional shape that is a constant distance from a given point is a circle. In the case of spacetime geometry, the corresponding shape is a hyperbola, the central point that is equidistant from every point of the hyperbola is the intersection between the two asymptotes. Note that a hyperbola curves relative to the central equidistant point in the opposite direction to a circle. This is the difference between the analogy and the physics it represents.

Thus, unlike the trampoline analogy, which is a "lie-to-children" that does not correctly describe gravity, the road analogy actually does explain the relation between time dilation and gravity.

 

 

Edited March 22 by KJW

[Mordred]

While analogies can be useful, I've always found they tend to mislead.

  In this instance I have always found the method of describing spacetime curvature using geodesics paths of two parallel light beams far more useful.  If the two beams remain parallel then you have flat spacetime. If the beams converge the positive curvature. If they diverge then you have negative curvature. It's a simple descriptive provided you include the details that mass is resistance to inertia change as well as ensuring that the reader further understands that the (ct) interval is what provides time with dimensionality of length.

[geordief]

42 minutes ago, Mordred said:

While analogies can be useful, I've always found they tend to mislead.

  In this instance I have always found the method of describing spacetime curvature using geodesics paths of two parallel light beams far more useful.  If the two beams remain parallel then you have flat spacetime. If the beams converge the positive curvature. If they diverge then you have negative curvature. It's a simple descriptive provided you include the details that mass is resistance to inertia change as well as ensuring that the reader further understands that the (ct) interval is what provides time with dimensionality of length.

Is the region between,say  two BHs a region of negative curvature?(the two beams could diverge if the one moved towards the first CoG and the other beam towards the other)

I think I heard(from Markus ,perhaps if I remember rightly) that there are no  physical regions of negative  curvature and ,so it would just be a mathematical concept.

 

[Mordred]

That example doesn't particularly apply. The reason being that two falling objects as they approach either CoM of either BH would converge. They would also converge as they approached the barycenter. The most commonly known example is Anti-Desitter spacetime.  This is used in the FLRW metric as one of the viable solutions historically it applied to an open as opposed to a closed universe. Which is another topic lol as explaining the two can get tricky. 

https://en.wikipedia.org/wiki/Anti-de_Sitter_space

[md65536]

31 minutes ago, Mordred said:

The reason being that two falling objects as they approach either CoM of either BH would converge.

Two objects falling directly towards a BH can diverge (as with spaghettification). Two objects falling indirectly and parallel can diverge, eg. if only one of them has escape velocity due to different distance from the BH. I think the analogy needs more details.

On the other hand, if you have two side-by-side geodesics both directed toward a single point (like a CoM or barycenter), they shouldn't be parallel at any finite distance, in general?

[Mordred]

Probably easiest to simply examine what each means under the metric. One of the better ways has been how Barbera Ryden teaches each in her Introductory to cosmology. A key feature I haven't mentioned is how Pythagorus theorem applies in each case. Without trying to go too  far astray one of my webpages uses her method and the paths descriptive.

http://cosmology101.wikidot.com/universe-geometry

this page just describes the critical density relations the next page has the needed images

http://cosmology101.wikidot.com/geometry-flrw-metric/

note the following section

 Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices α
β,γ obey the relation α+β+γ=π−AR2.

 

a key detail to recall or understand is a negative curvature is a hyperboloid example the saddle image in the second link

 

Edited March 23 by Mordred