Does GTD Really Cause Gravity? | Brian Greene/Horn/e's 2011 The Hidden Reality

[Curious Cat]

TL;DR Summary

Well, I know that it doesn't but lately I've been seeing videos on
Youtube which claim that it does.

I didn't pay them much attention until I saw the PBS Space Time's one,
and I went "What!? Really!? Am I missing something, here, like a brain?
Have they all gone crazy"? And then I remembered where I first saw it:
In Brian Greene/Horn/e's 2011 The Hidden Reality book.
On P14 there is a footnote which says that the main cause of gravity for
a small body like the Earth/Sun is actually GTD!?
Read the whole footnote and You'll see what I mean.
At the time I just dismissed it, as nonsense, and I still do,
because even for the Sun the surface GTD is practically nothing.
Have they all gone mad, in the head!?
(Stop confusing the children).

 

[strangerep]

By "GTD", I'm guessing you mean "Gravitational Time Dilation". That certainly doesn't "cause" gravity, but is just one of the effects arising from a non-flat spacetime metric.

(I don't have Greene's book, so you might have to post the footnote here if you want a better answer. Maybe others can comment further.)

 

[Ibix]

I think the OP just asks "is gravity caused by gravitational time dilation".

Gravitational time dilation is directly related to gravitational potential, and in the weak field case you will find that if the difference in gravitational potential between two objects is $\Delta\phi$ then the time dilation factor between them is $\Delta\phi/c^2$. And also in the weak field limit the gravitational force is the gradient of the potential, so could be seen as the gradient of the time dilation factor. Itxs the factor of $c^2$ which makes time dilation effects many orders of magnitude smaller than the forces.

That said, in strong fields this is no longer a sufficient explanation. In fact, the anomalous precession of Mercury's orbit was one of the first tests of general relativity, and the reason it's anomalous is that you can't explain it in the weak field approximation - you need to take spatial curvature into account. So time dilation is not a complete explanation. And there are more general cases where time dilation can't be defined but gravitational effects still happen.

You didn't provide a link to any videos, so I don't know which ones you are talking about. There has been one posted here which has an animation of a squirrel whose feet are lower than its head, and its feet don't advance as far in time as its head due to time dilation, so its path curves down. There are a lot of problems with that video, notably that general relativity predicts that paths of point particles would curve downwards, and this argument can't explain that at all.

Finally, the truth is that gravity is spacetime curvature. Anything that isn't talking about that (and I mean spacetime curvature, not the rubber sheet diagram that only shows spatial curvature) is, at best, a partial explanation and probably misleading.

 

[Ibix]

This is a good video, by forum member @A.T.:

 

[A.T.]

On P14 there is a footnote which says that the main cause of gravity for
a small body like the Earth/Sun is actually GTD!?

The gradient in gravitational time dilation is directly related to the local gravitational acceleration. This is somewhat visible in the video in post #4.

for the Sun the surface GTD is practically nothing.

The natural scaling in space-time is :

1 second (short time for us) ~= 1 light-second (huge distance for us)

So a deviation in time that seems very small for us can result in a spatial effect that is noticeable for us.

 

[Motore]

This is the video the OP is referring to:



It's a pop sci video so I would caution the OP about making conclusions based on it.

 

[Ibix]

I had a quick look through the PBS video. It's a mix of true statements and misleading ones ("hurtling through time at the speed of light") and some completely untrue ones ("photons are fully rotated into the spatial direction").

It was the Science Asylum one, referenced in the middle of the PBS one, that had the squirrel that I was thinking of.

 

[A.T.]

I had a quick look through the PBS video. It's a mix of true statements and misleading ones ("hurtling through time at the speed of light") and some completely untrue ones ("photons are fully rotated into the spatial direction").

When you use space-propetime diagrams (instead of the more common space-coordiantetime diagrams), then the above makes more sense. Both are just different geometrical interpretations of the same theory.

 

[Ibix]

When you use space-propetime diagrams (instead of the more common space-coordiantetime diagrams), then the above makes more sense. Both are just different geometrical interpretations of the same theory.

How does one draw a spatial direction on a space-proper time diagram? I can't see how it can possibly be parallel to a null line.

 

[Sagittarius A-Star]

How does one draw a spatial direction on a space-proper time diagram? I can't see how it can possibly be parallel to a null line.

That are Epstein-diagrams:

The dogma reads: Everybody is always and everywhere moving at the speed of light c in 4D space-time.

Source:
https://www.relativity.li/en/epstein2/read/c0_en/c2_en

 

[Ibix]

That are Epstein-diagrams:

Source:
https://www.relativity.li/en/epstein2/read/c0_en/c2_en

I understand that. The problem with "everyone is traveling at $c$" is that it depends on a decision to normalise the four-velocity to length $c$, and you can choose other normalisations. What I don't understand is how you plot a spacelike displacement (which is how I read "fully rotated into the spatial direction") on Epstein's space-proper time diagrams.

Certainly light rays on an Epstein diagram are horizontal. I just disagree that horizontal is a spatial direction on an Epstein diagram.

 

[Sagittarius A-Star]

Certainly light rays on an Epstein diagram are horizontal. I just disagree that horizontal is a spatial direction on an Epstein diagram.

I would say that, in horizontal direction, the light world line and the spatial direction coincide, because the "time"-axis shows proper time of the moving object.

 

[Curious Cat]

By "GTD", I'm guessing you mean "Gravitational Time Dilation". That certainly doesn't "cause" gravity, but is just one of the effects arising from a non-flat spacetime metric.

(I don't have Greene's book, so you might have to post the footnote here if you want a better answer. Maybe others can comment further.)

Thank you, for your reply:
. Oh, yes, of course.
. And here it is:
It's easier to envision curved space than curved time, and that's why many popularizations of Einsteinian gravity focus solely on the former. However, for the gravity generated by familiar objects like the Earth and sun, it is actually the curvature of time - not space - that exerts the dominant impact. For an illustration, think of two clocks, one on the ground, the other on top of Empire State Building. Because the ground clock is closer to the Earth's center, it experiences slightly stronger gravity than the clock that's high above Manhattan. GR shows that because of this, the rate at which time passes on each will be slightly different: the ground clock will run a tiny bit slow (billionths of a second per year) compared to the elevated clock. The temporal mismatch is an example of what we mean by time being curved or warped. GR THEN ESTABLISHES THAT OBJECTS MOVE TOWARD REGIONS WHERE TIME ELAPSES MORE SLOWLY, IN A SENSE, ALL OBJECTS "WANT" TO AGE AS SLOWLY AS POSSIBLE. FROM AN EINSTEINIAN PERSPECTIVE, THAT EXPLAINS WHY AN OBJECT FALLS WHEN YOU LET IT GO.

What was that all about!? Did he have a senior moment, or something?

 

[Sagittarius A-Star]

Because the ground clock is closer to the Earth's center, it experiences slightly stronger gravity than the clock that's high above Manhattan. GR shows that because of this, the rate at which time passes on each will be slightly different

That's already wrong. The GTD comes from the difference in gravitational potential.

 

[Dale]

There is no way to claim that gravitational time dilation causes gravity. Causes always precede effects, and gravitational time dilation happens together with gravity.

I am assuming that “gravity” in this context refers specifically to the gravitational force and not to the general phenomena of gravity and all it various aspects.

In a spacetime where there is a gravitational potential then both gravitational time dilation and the gravitational force are determined by the gravitational potential. The difference in gravitational potential gives the gravitational time dilation and the gradient of the gravitational potential gives the gravity.

This relationship holds for any weak field metric that has a well defined gravitational potential, including ones like Rindler coordinates in flat spacetime. This should make you immediately question this line of thought since it is unrelated to curvature and is simply a coordinate-based statement.

 

[Ibix]

However, for the gravity generated by familiar objects like the Earth and sun, it is actually the curvature of time - not space - that exerts the dominant impact.

This isn't really correct or, perhaps better said, is so vague as to be meaningless. It is true that spatial curvature is (in all but the most extreme circumstances) a minor correction to Newtonian gravity. It is also true that if you adopt Schwarzschild coordinates that it is the $tt$ component of the Einstein equations that simplifies to Newtonian gravity. I wouldn't define that as curvature of time, though.

For an illustration, think of two clocks, one on the ground, the other on top of Empire State Building. Because the ground clock is closer to the Earth's center, it experiences slightly stronger gravity than the clock that's high above Manhattan. GR shows that because of this, the rate at which time passes on each will be slightly different: the ground clock will run a tiny bit slow (billionths of a second per year) compared to the elevated clock.

That argument is incorrect, although the bit about the clocks is true. A third clock at the center of the Earth experiences no gravity, yet ticks even slower than the ground clock. This is because time dilation depends on the gravitational potential difference between two heights while the gravitational strength (to the extent that makes sense in GR) depends on the gradient of the potential.

GR THEN ESTABLISHES THAT OBJECTS MOVE TOWARD REGIONS WHERE TIME ELAPSES MORE SLOWLY, IN A SENSE, ALL OBJECTS "WANT" TO AGE AS SLOWLY AS POSSIBLE. FROM AN EINSTEINIAN PERSPECTIVE, THAT EXPLAINS WHY AN OBJECT FALLS WHEN YOU LET IT GO.

That's definitely not true in general because it isn't generally possible to define "regions where time elapses more slowly". That can only make sense in static or stationary spacetimes, which are a very restrictive class. Read literally, it also denies the possibility of escaping trajectories, which clearly don't move towards lower gravitational potential.

It is true that geodesics, the paths that objects naturally follow, have a (local) maximum of proper time along them. I think that's been mixed into this.

What was that all about!? Did he have a senior moment, or something?

He wants to sell books to people who don't want to spend years studying the maths needed to follow a textbook. So he's saying things that appear to be understandable, and are kind of true in certain narrow circumstances, but presenting them as if they explained everything.

 

[A.T.]

I just disagree that horizontal is a spatial direction on an Epstein diagram.

How would you like to call that axis, and what is wrong with calling it spatial?

 

[Dale]

I don’t think that anything on a space-propertime diagram deserves to be called “spatial” or a “direction” or any other geometrical words.

 

[Ibix]

How would you like to call that axis, and what is wrong with calling it spatial?

The horizontal line gives me trouble. If it's a spatial axis then lightlike paths shouldn't lie along it, since those aren't spacelike. But (accepting a definition of proper time that includes $\Delta \tau^2=0$), clearly lightlike paths should have zero vertical extent. Essentially the half of a Minkowski diagram that lies outside the lightcone of the origin has to be degenerate along that line, but that doesn't mean that the lightlike path is rotated into the spacelike one, just that the Epstein diagram is a bad tool for drawing spacelike and lightlike paths. Or that's how it seems to me, anyway.

 

[robphy]

I just disagree that horizontal is a spatial direction on an Epstein diagram.

How would you like to call that axis, and what is wrong with calling it spatial?

I agree with @Ibix and @Dale .
Call it "horizontal axis" or "abscissa".

In relativity, "spatial" is already understood to suggest that a "spatial direction"
is perpendicular (via the spacetime metric) to a "timelike direction".

Despite its name, the Epstein diagram isn't equivalent to a Minkowski diagram.
(Minkowski diagram is probably a bad name...
position-vs-time graph or Minkowski-spacetime diagram is better.)
The Epstein diagram is not a map of the spacetime of special relativity.

In my opinion, the Epstein diagram is more an "infographic chart" or "nomogram" (useful for displaying a specific set of relationships), but not a "map" of spacetime (which may encode the same properties in Minkowskian-geometric way, but which may be unfamiliar or too complicated for the intended audience).

(I have similar opinions on Loedel diagrams and Brehme diagrams.)
They're all useful for a specific set of relationships, using Euclidean-like techniques (which only apply to specific configurations) because Minkowskian geometry is argued to be too hard...
but they [without the Minkowski spacetime diagram] won't be able take the audience to a fuller understanding of special relativity and general relativity.

These alternative diagrams are somewhat useful, but are limited.

In particular, where are the light cones along a worldline?
Without them, how does one discuss causality? (What is relativity without causality?)

My $0.02.

 

[Ibix]

I have similar opinions on Loedel diagrams and Brehme diagrams.

Brehme diagram is new to me, but isn't a Loedel diagram just a Minkowski diagram in a specific frame where two objects of interest have equal and opposite velocities?

 

[robphy]

Brehme diagram is new to me, but isn't a Loedel diagram just a Minkowski diagram in a specific frame where two objects of interest have equal and opposite velocities?

Yes, but its Euclidean method of calculation only seems to apply
to the triangle with sides along the given timelike axes.
(The method of calculation doesn't work for arbitrary triangles in that diagram.)

Thus, the Loedel method is limited.

 

[Ibix]

its Euclidean method of calculation only seems to apply
to the triangle with sides along the given timelike axes.

I hadn't realized there was a special trick with Loedel diagrams - I thought they were just a Minkowski diagram in the symmetric frame. If anyone else is similarly ignorant, Jorrie has the details.

 

[robphy]

I hadn't realized there was a special trick with Loedel diagrams - I thought they were just a Minkowski diagram in the symmetric frame. If anyone else is similarly ignorant, Jorrie has the details.

They are Minkowski diagrams in the symmetric frame.
In that frame, one can use Euclidean methods to relate the two frames.
But to handle, say, the clock effect (with unequal outgoing and incoming velocities),
it seems that one has to use a second Loedel diagram to do the Euclidean-type calculation for the return leg.
Using the diagram for the outgoing leg, you can't use the Euclidean-type method for the triangle describing the return leg... it appears that one has to boost into the symmetrical frame for the return leg.

With Minkowski spacetime geometry, the calculation can be done in any frame.
One doesn't have to boost each time to get things into "standard form"
(although that's a technique that is often used in teaching relativity... work in a frame where something is conveniently zero).

In Euclidean geometry, while we may take advantage of certain symmetries,
we generally don't have to rotate our diagram each time into "standard position"
to work on various legs of a triangle or polygon.

 

[DaveC426913]

Wow, the OP lasted just shy of 12 hours here...

 

[Dale]

These alternative diagrams are somewhat useful, but are limited.

In particular, where are the light cones along a worldline?

My main issue is that the points in a space proper time diagram are not events. So intersecting lines may not collide and colliding lines may not intersect.

That is why I don’t think anything in those diagrams should be given any geometrical names.

 

[Dale]

Wow, the OP lasted just shy of 12 hours here...

Given that we will close the thread. When the OP returns they can open a new thread.