I have a fundamental mis-understanding about gravity.

[Dr Ninja]

I have a layman understanding about how the curvature of space describes the motions of planets and other large celestial bodies. What I don't understand is how curved space makes say; an apple fall to the ground. Any help appreciated. Thanks.

 

[Pengwuino]

I have a layman understanding about how the curvature of space describes the motions of planets and other large celestial bodies. What I don't understand is how curved space makes say; an apple fall to the ground. Any help appreciated. Thanks.

The principle is the same. The apple is following the 'curvature' of spacetime that is caused by the Earth. All objects follow General Relativity, it's just that deviations from Newtonian gravity are only seen at the large celestial body scale.

 

[A.T.]

I have a layman understanding about how the curvature of space describes the motions of planets and other large celestial bodies. What I don't understand is how curved space makes say; an apple fall to the ground. Any help appreciated. Thanks.

It's not only curved space, but curved space-time. The time dimension is crucial to understand how objects at rest in space are affected at all. This links might help:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html
http://www.adamtoons.de/physics/gravitation.swf

Note that planets are also mainly affected by this "time curvature". It's only for very fast objects like photons, that the purely spatial curvature causes a significant fraction of the total path bending.

 

[Passionflower]

Note that planets are also mainly affected by this "time curvature". It's only for very fast objects like photons, that the purely spatial curvature causes a significant fraction of the total path bending.

Could you show the math to back this statement up?
Because I am not buying it.

 

[Mentz114]

Note that planets are also mainly affected by this "time curvature". It's only for very fast objects like photons, that the purely spatial curvature causes a significant fraction of the total path bending.

It's true. For example, you can see it from this metric
$$d\tau^2=g_{00}dt^2-\frac{1}{c^2}g_{ab}dx^adx^b,$$
where a,b,c are spatial indexes. The spatial contributions to the proper interval are all divided by c² so g₀₀ (time curvature?) dominates. The convention of setting c=1 hides this a bit, but doesn't change it.

 

[Passionflower]

It's true. For example, you can see it from this metric
$$d\tau^2=g_{00}dt^2-\frac{1}{c^2}g_{ab}dx^adx^b,$$
where a,b,c are spatial indexes. The spatial contributions to the proper interval are all divided by c² so g₀₀ (time curvature?) dominates. The convention of setting c=1 hides this a bit, but doesn't change it.

Why do you think that formula is relevant?

We are talking about why planets 'bend' inwards when they orbit a mass right?

In general you first have to define a map with a space-like and time-like component and then you have to take into consideration over which range this map is valid. Only then can you even begin to talk about how much curvature there is in the time-like and space-like parts. An observer's 'four legs' can point in any angle in the 'directions' of spacetime. Thus different observers can attribute different amounts of curvature in the spatial and temporal direction.

 

[Mentz114]

Why do you think that formula is relevant?

We are talking about why planets 'bend' inwards when they orbit a mass right?

In general you first have to define a map with a space-like and time-like component and then you have to take into consideration over which range this map is valid. Only then can you even begin to talk about how much curvature there is in the time-like and space-like parts. An observer's 'four legs' can point in any angle in the 'directions' of spacetime. Thus different observers can attribute different amounts of curvature in the spatial and temporal direction.

Of course it's relevant. The curvature is made up of first and second derivatives of the metric, including the 1/c² factors, so the contributions from the spatial curvature will be much less than that from the temporal components.

Since we're talking about orbits, I'll try to come up with something for the Hagihara frame that might convince you. The problem is that c is set to 1 in all my scripts ...

 

[Jonathan Scott]

Consider a plot of x against ct, starting at rest, so the line starts by rising parallel to the t axis. If there is a Newtonian gravitational field g along the x-axis, the line starts to curve towards the source with radius of curvature c²/g, which is a very large radius for normal gravitational accelerations because c is so large, but again because c is so large this also results in the velocity changing at rate g. This is effectively the "curvature of space with respect to time", that corresponds to the Newtonian acceleration. The linear curvature of space has a similar radius but only affects moving objects, proportionally to v²/c².

(In General Relativity the term "curvature" is often used to denote the intrinsic curvature of space-time created by mass-energy, which roughly corresponds to the sort of curvature on the surface of a ball that results in angles adding up differently from the way they do in flat space. In this case, I'm using the word "curvature" to describe the ordinary curvature of lines).

 

[A.T.]

It's true. For example, you can see it from this metric
$$d\tau^2=g_{00}dt^2-\frac{1}{c^2}g_{ab}dx^adx^b,$$
where a,b,c are spatial indexes. The spatial contributions to the proper interval are all divided by c² so g₀₀ (time curvature?) dominates. The convention of setting c=1 hides this a bit, but doesn't change it.

Why do you think that formula is relevant?

It is relevant because the word-line of a free-faller is maximizing the proper-time interval. And since for a planet those dx/c are relatively small compared to dt, the g₀₀ dominates. The two extremes are:
- An object at rest in space (dx=0) is not affected at all by spatial curvature.
- For photons (dx=c*dt) the contribution of spatial and temporal components is the same.

For math on those two cases see:
http://www.mathpages.com/home/kmath409/kmath409.htm
http://www.mathpages.com/rr/s6-03/6-03.htm

The planets are closer to rest than c.

Thus different observers can attribute different amounts of curvature in the spatial and temporal direction.

I was considering the rest frame of the gravitation source.

 

[Mentz114]

I think posts #8 and #9 wrap this up. But to answer this

An observer's 'four legs' can point in any angle in the 'directions' of spacetime. Thus different observers can attribute different amounts of curvature in the spatial and temporal direction.

For the stationary observer in the Schwarzschild vacuum the components of the 'fourleg' (cobasis) are
$$p_0= -c\sqrt{1-2m/r}\ dt,\ \ p_1=dr/\sqrt{1-2m/r},\ \ p_2= r\ d\theta,\ \ p_3=r\sin(\theta)\ d\phi$$
from which we can see cdt dominates for large r. For the orbiting frame the timelike cobasis vector is a mixture of the coordinate dt and dr directions
$$h_0 = -(1-2m/r)/(\sqrt{1-3m/r})\ dt-\sqrt{mr}/(c\sqrt{1-3m/r})\ dr$$
and again the spatial part is divided by c. The same pattern is shown by the new $\phi$ cobasis vector but I'm too tired to tex it in.

I think this shows that for these frames the new time direction takes only a small contribution from the coordinate spatial directions compared to the contributions from the coordinate time direction.

 

[pervect]

Going back to the original poster here, i.e. Dr Ninja, has the question been answered sufficiently.

Without getting into the technical details of how to specify curvature or some of the fine technical distinctions, the main point is that it's space-time that's curved, not just space.

One can consider a worldline as being a path through space-time, as a curve parameterized by proper time, tau. You can think of an object as "moving through" spatial coordinates and time coordinates as proper time advances.

Because space-time is curved, this "motion" through time (which is actually the change in coordinate time as proper time changes) causes relative accelerations relative to other worldlines of other objects, which is what we describe as gravity.

 

[Passionflower]

I think posts #8 and #9 wrap this up. But to answer thisFor the stationary observer in the Schwarzschild vacuum the components of the 'fourleg' (cobasis) are
$$p_0= -c\sqrt{1-2m/r}\ dt,\ \ p_1=dr/\sqrt{1-2m/r},\ \ p_2= r\ d\theta,\ \ p_3=r\sin(\theta)\ d\phi$$
from which we can see cdt dominates for large r. For the orbiting frame the timelike cobasis vector is a mixture of the coordinate dt and dr directions
$$h_0 = -(1-2m/r)/(\sqrt{1-3m/r})\ dt-\sqrt{mr}/(c\sqrt{1-3m/r})\ dr$$
and again the spatial part is divided by c. The same pattern is shown by the new $\phi$ cobasis vector but I'm too tired to tex it in.

I think this shows that for these frames the new time direction takes only a small contribution from the coordinate spatial directions compared to the contributions from the coordinate time direction.

Still what is your point?
A small contribution to the accumulation of proper time?
OK, I agree but so what?
What has that to do with spatial curvature?
What has the rate of the clock in a satellite to do witch spatial curvature?

Take one circumference of a satellite with a circular orbit of length r and compare this to a satellite going in a straight line. We can easily see how much the satellite was displaced by spatial curvature.

You are saying that it was displaced more by temporal curvature?
Show it!

 

[Mentz114]

Still what is your point?
A small contribution to the accumulation of proper time?
OK, I agree but so what?
What has that to do with spatial curvature?
What has the rate of the clock in a satellite to do witch spatial curvature?

Take one circumference of a satellite with a circular orbit of length r and compare this to a satellite going in a straight line. We can easily see how much the satellite was displaced by spatial curvature.

You are saying that it was displaced more by temporal curvature?
Show it!

The numbers I've shown have nothing to do with clock rates. They show the time cobasis vector of the orbiting observers frame. The time direction basis vector is only tilted by a small amount from the coordinate time direction. In other words the contribution of the spatial curvature is much smaller than the contribution of the time curvature. Read the equation.

The Newtonian orbit equations work very well, despite ignoring spatial curvature. Only tiny relativistic effects have been unexplained like the perihelion precession of Mercury.

We can easily see how much the satellite was displaced by spatial curvature.

Equations, please.

We are hijacking this thread. Start a new thread if you think it necessary.

 

[Passionflower]

In other words the contribution of the spatial curvature is much smaller than the contribution of the time curvature. Read the equation.

So you agree with AT that the curvature of spacetime depends on how fast one travels through it (whatever how fast might even mean)? If you go slow (again whatever that might mean) spacetime is curved more in the spatial direction and if you go fast (idem for fast) it curves more in the temporal direction?

Sorry but I think that is not even wrong.

Perhaps someone can point to a credible textbook that makes such statements.

 

[Mentz114]

So you agree with AT that the curvature of spacetime depends on how fast one travels through it (whatever how fast might even mean)? If you go slow (again whatever that might mean) spacetime is curved more in the spatial direction and if you go fast (idem for fast) it curves more in the temporal direction?

Sorry but I think that is not even wrong.

Perhaps someone can point to a credible textbook that makes such statements.

No, that is not what I'm saying. You raised the point the the axes of the local frame should be considered and I've answered that question for the orbiting observer. You obviously don't understand what I'm saying so let us just agree to disagree.

 

[Austin0]

They show the time cobasis vector of the orbiting observers frame. The time direction basis vector is only tilted by a small amount from the coordinate time direction. In other words the contribution of the spatial curvature is much smaller than the contribution of the time curvature. Read the equation.

The Newtonian orbit equations work very well, despite ignoring spatial curvature. Only tiny relativistic effects have been unexplained like the perihelion precession of Mercury.

If the planetary orbits are straight lines in curved spacetime doesn't their manifest ellipse shape in space result from and describe that curvature.

Is there any way you could conceptually describe how time effectuates this if the spatial curvature is negligible?
Thanks

 

[pervect]

Imagine drawing your space-time diagram on a curved surface. For a very simple example,, you might draw a space-time diagram on a sphere.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity/index.html#Geodesic

illustrates how initially parallel geodesics on a sphere converge.

So, "curved space-time" can be visualized as "drawing space-time diagrams on a curved surface".

As previously mentioned, in all of the space-time diagrams one can visulaize the static wordline as progressing through time by imagining a point on the worldline moving along it's length. Because time always "flows", the space-time diagram of an object is a world-line, not a world-point.

One other note - the underlying geometry is still Lorentzian, i.e. the standard flat space-time diagram in SR transforms via the Lorentz transform. So, if you really want to understand GR well, you need to understand SR first, you need to be familiar with the SR space-time diagrams and how they transform via the Lorentz transform.

 

[Austin0]

Imagine drawing your space-time diagram on a curved surface. For a very simple example,, you might draw a space-time diagram on a sphere.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity/index.html#Geodesic

illustrates how initially parallel geodesics on a sphere converge.

So, "curved space-time" can be visualized as "drawing space-time diagrams on a curved surface".

As previously mentioned, in all of the space-time diagrams one can visulaize the static wordline as progressing through time by imagining a point on the worldline moving along it's length. Because time always "flows", the space-time diagram of an object is a world-line, not a world-point.

One other note - the underlying geometry is still Lorentzian, i.e. the standard flat space-time diagram in SR transforms via the Lorentz transform. So, if you really want to understand GR well, you need to understand SR first, you need to be familiar with the SR space-time diagrams and how they transform via the Lorentz transform.

Thanks for the link .Very informative even if it didn't exactly resolve my question. I do have some conception of non-euclidean geometry but still am unclear on how time relates. It seemed to imply that particles on geodesics follow the path of greatest proper time but that seems more comprehensible as a result of geometry rather than a cause of it.
I am familiar with Minkowski diagrams but understood the geometry was only valid in a small local region in GR territory to an approximation?

It could be I won't be able to understand the answer to my query without a whole lot more basic GR study but thanks

 

[pervect]

Thanks for the link .Very informative even if it didn't exactly resolve my question. I do have some conception of non-euclidean geometry but still am unclear on how time relates. It seemed to imply that particles on geodesics follow the path of greatest proper time but that seems more comprehensible as a result of geometry rather than a cause of it.

A geodesic can be thought of as a curve that maximizes (more precisely, extrermizes) proper time, but it can also be thought of as a curve that is as is straight as possible.

An example of a geodesic on a sphere is a great circle.

In terms of parallel transport, a geodesic is a curve that parallel transports itself.

If you actually want to calculate a geodesic, you'll need the mathematical background to either derive the geodesic equation (extremizing proper time is one way, undesranding the ins and outs of parallel transport is another way). But if you consider 2-d surfaces embedded in a 3d geometry, it's not too hard to imagine what they look like without the math, I think.

If you consider a curved 2d surface embedded in a 3d space, (for example a sphere), at every point on the sphere there is a flat plane tangent to it.

This tangent plane is a convenient way of visualizing the more general "tangent space" that is flat and exists at any point on a general curved manifold of arbitrary dimension.

A geodesic is basically a curve that appears to be "straight" when projected onto its tangent plane, where you can use the usual definition of "straight" because the tangent plane is perfectly flat. (But I've skimped on describing the projection process.)

Basically, the space-time diagram maps the time dimension to a space dimension, and then you use your intuition about curved surfaces to translate the space-time curvature into spatial curvature, which you can visualize.

If you don't have an intuition about geodesics and curved surfaces, and if using spheres (for the curved surfaces) and great circles (for the geodesics) as a particular example isn't good enough for you, I guess you're out of luck (at least with this approach).

 

[A.T.]

Is there any way you could conceptually describe how time effectuates this if the spatial curvature is negligible?

Check the first picture:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

Initially, when at rest in space, the spatial geometry has no effect at all. But the object still advances at c trough time. When it moves at a small fraction of c in space, its advance in space-time is still mostly along the time-dimension, so the metric of that dimension has the most influence.

 

[A.T.]

Thanks for the link .Very informative even if it didn't exactly resolve my question. I do have some conception of non-euclidean geometry but still am unclear on how time relates. It seemed to imply that particles on geodesics follow the path of greatest proper time but that seems more comprehensible as a result of geometry rather than a cause of it.

Yes geodesics are extremizing the proper time interval between two points in space-time. But that doesn't directly help you if you know just the initial point and intial direction in space-time. It is better to imagine geodesics as paths going "straight ahead" on a curved surface. Like a toy car without steering:
http://www.relativitet.se/spacetime1.html

You can also take adhesive tape and stick it to a curved surface, without tearing or folding the edges. It also follows geodesics because the stripe is intrinsically straight (a long thin rectangle) and the sticking to the curved surface merely creates some extrinsic curvature.

It could be I won't be able to understand the answer to my query without a whole lot more basic GR study but thanks.

Understanding the concept is not difficult and should be possible form the picture in the links I posted.

 

[yuiop]

Yes geodesics are extremizing the proper time interval between two points in space-time. But that doesn't directly help you if you know just the initial point and intial direction in space-time.

I would like to understand this concept of extremizing the proper time a little better. For example imagine two points close together on the surface of a sphere. The proper time of a particle traveling the obvious short route between the particles is less than the proper time of a particle traveling at the same speed and going the long way round the great circle that the two points are on. Obviously there is some qualifier besides just the proper time. One qualifier that I know of is that the alternate paths should only deviate slightly from each other, but even in that case, if the particles all travel at the same speed then the particle traveling on the geodesic still has the shortest elapsed proper time. It seems that the qualifier is that the "straightest" path is the path that has the most elapsed proper time between A and B if all particles (travelling alternate paths) leave A at the same time and arrive at B at the same time.

It is better to imagine geodesics as paths going "straight ahead" on a curved surface. Like a toy car without steering:

You can also take adhesive tape and stick it to a curved surface, without tearing or folding the edges. It also follows geodesics because the stripe is intrinsically straight (a long thin rectangle) and the sticking to the curved surface merely creates some extrinsic curvature.

In slightly more geometrical terms, if we were to draw two short parallel lines either side of the initial velocity vector of the particle and start extending those parallel lines in such a manner that they always have the same length, then the geodesic is the line midway between those two parallel lines. Would that work? Parallel is here defined as remaining the distance apart as measured on the surface.

Is the curved surface in the drawing what is commonly referred to here as a manifold?

The curved surface depicts time, vertical distance and vertical velocity, but does not directly depict horizontal velocity. Am I curious if this is a limitation of the model? The light blue line illustrates an object thrown directly upwards with zero horizontal velocity and it eventually reaches an apogee and comes back down again. The purple line illustrates an orbiting object with zero vertical velocity and non-zero horizontal velocity. Now one aspect I am curious about, is that apogee the path of the radially moving object at the instant it at the apogee is indistinguishable from the path of the orbiting object and yet they diverge at a later time. How is this explained?

 

[Austin0]

Yes geodesics are extremizing the proper time interval between two points in space-time. But that doesn't directly help you if you know just the initial point and intial direction in space-time. It is better to imagine geodesics as paths going "straight ahead" on a curved surface. Like a toy car without steering:
http://www.relativitet.se/spacetime1.html

You can also take adhesive tape and stick it to a curved surface, without tearing or folding the edges. It also follows geodesics because the stripe is intrinsically straight (a long thin rectangle) and the sticking to the curved surface merely creates some extrinsic curvature.


Understanding the concept is not difficult and should be possible form the picture in the links I posted.

That was already the concept and understanding I was working with. In the Euclidean case of an inertial particle the shortest distance and time are simply consequences of the linear motion in flat geometry. Translated into curved geometry it is not hard to understand that the path would be deviated and the time altered as a consequence of that geometry.
What is hard for me to get is how the time factor could be a causative agent affecting the geometry and the perceived path. That was what seemed to be being proposed earlier in this thread. Equivalent to saying an inertial particle travels a straight path because it takes the least amount of time.
Thanks for the input. Perfect may be right and I may be out of luck until I get more background.

 

[A.T.]

I would like to understand this concept of extremizing the proper time a little better. For example imagine two points close together on the surface of a sphere. The proper time of a particle traveling the obvious short route between the particles is less than the proper time of a particle traveling at the same speed and going the long way round the great circle that the two points are on.

Yes, if the geometry is complex enough you will globally get multiple extrema. So that concept of extremizing proper time between two events (points in proper time) doesn't allow you to make an unique prediction, even if you know those two points. But usually you have one initial point and an initial direction - and that gives you a unique solution.

In slightly more geometrical terms, if we were to draw two short parallel lines either side of the initial velocity vector of the particle and start extending those parallel lines in such a manner that they always have the same length, then the geodesic is the line midway between those two parallel lines. Would that work? Parallel is here defined as remaining the distance apart as measured on the surface.

Yes, if you chose any two points on one line, and find the closest two points to them on the other line, then the two segments along the lines are same length.

Is the curved surface in the drawing what is commonly referred to here as a manifold?

Manifold is very general term. This is an embedding diagram, where a curved manifold is embedded in a higher dimensional flat manifold.

IThe curved surface depicts time, vertical distance and vertical velocity, but does not directly depict horizontal velocity. Am I curious if this is a limitation of the model?

It is a limitation of human spatial imagination, which ends at 3 dimensions. To embed a curved 2+1 space time, you would need at least a 4 dimensional embedding space. But you cannot visualize this.

The light blue line illustrates an object thrown directly upwards with zero horizontal velocity and it eventually reaches an apogee and comes back down again. The purple line illustrates an orbiting object with zero vertical velocity and non-zero horizontal velocity. Now one aspect I am curious about, is that apogee the path of the radially moving object at the instant it at the apogee is indistinguishable from the path of the orbiting object and yet they diverge at a later time. How is this explained?

The purple one is just an object held static at a constant height, by an upwards force. The world-lines of the objects diverge because the purple one has a force acting on it, so it's world-line is not a geodesic.

 

[A.T.]

So you agree with AT that the curvature of spacetime depends on how fast one travels through it

No, that is not what I'm saying.

Neither did I actually. The point was rather that the effect of different components of the metric on a free faller, varies with his direction in space time (or speed in space), when observed in the rest frame of the gravity source. Here another example of how GR-gravity effects depend on the free fallers velocity:

You shoot a laser and a rifle in flat spacetime between two distant points A & B and measure their travel times. Then you place a massive sphere midways between A & B which has a tunnel drilled along the line AB. You repeat the experiment. Will the travel times change compared to the first experiment? If yes, how?