If the effect gravity is considered to be due the curvature of space which keeps a satellite in orbit, what force causes the satellite to spiral "downwards" towards the earth if its velocity were to be reduced.

The curvature of space alone could not produce an attractive "downward" force, as in space no slope or incline could produce motion.

yes it would, just as a snooker ball would fall down a rug weighted in the middle, the force of gravity would attract the once orbiting object to its centre of mass :D

yes it would, just as a snooker ball would fall down a rug weighted in the middle, the force of gravity would attract the once orbiting object to its centre of mass :D

I'm glad you brought up the anology of the rug. The reason the ball falls down the rug is due to the force of gravity. The slope of the rug generates no force. if you were to perform the same experiment in space outside of the earths gravity, the snooker ball would go nowhere.

As I mentioned in my first post an incline or curvature could only produce motion in the presence of gravity.

why do you think it is said that space is curved?

arul...I think you are confusing concepts...I myself don't know much about this, but the way I understand it is like this...

space is not curved all by itself, the way I see it, space alone is not curved at all...it is the existence of gravity through space that makes space seemed to be curved, or at least we humans visualize it and model it as if it was curved.

so, when a satellite does not have enough of its own speed to keep itself from falling to earth, it does fall to earth BECAUSE of gravity.

I'm no authority on this, but when you say that the Earth's gravity causes the motion along the slope or incline, isn't it the same gravitational pull of the Earth that causes the fabric of space-time to curve around it? And as we all know, this curve decides how diff bodies interact gravitationally with the Earth. In this scenario, a ball spiralling down a rug is analogous to the satellite spiralling down. Both are caused by the gravitational force, the same curves exerted by the same body, albeit on a different scale!

So my guess is that it is the effect of the Earth's gravity that causes it to spiral down, just as it can't keep up it's circular motion due to its decrease in velocity.

Also, remember that we are used to visualizing space-time as a single fabric, whereas in reality it is a stack of such layers! :)

DrGreg
Gold Member
If the effect gravity is considered to be due the curvature of space which keeps a satellite in orbit, what force causes the satellite to spiral "downwards" towards the earth if its velocity were to be reduced.

The curvature of space alone could not produce an attractive "downward" force, as in space no slope or incline could produce motion.
You are right. The curvature of space alone does not explain gravity; it is the curvature of spacetime that does.

For the simplest of motion in one space dimension then "spacetime" is just a fancy name for a distance v. time graph. In the presence of gravity, instead of drawing the graph on a flat piece of paper, you have to draw it on a curved surface instead. On a curved surface, if you try to draw two "parallel" lines, each as straight as possible, you will fail. The lines start off parallel, but later the surface curvature forces the lines closer together (or further apart).

See Straight lines in a curved spacetime.

Yeah. curvature of space makes no sense to me in this context at-least! Thanks DrGreg :)

Thanks for all the replies. From what I understand there is no "Gravity" apart from a curved space-time.

Gravity = curved space time = Gravity. Space-time is not warped by gravity but by mass. according to GR there is no force of gravity apart from space-time warping.

You are right. The curvature of space alone does not explain gravity; it is the curvature of spacetime that does.

For the simplest of motion in one space dimension then "spacetime" is just a fancy name for a distance v. time graph. In the presence of gravity, instead of drawing the graph on a flat piece of paper, you have to draw it on a curved surface instead. On a curved surface, if you try to draw two "parallel" lines, each as straight as possible, you will fail. The lines start off parallel, but later the surface curvature forces the lines closer together (or further apart).

See Straight lines in a curved spacetime.

Thanks Dr. Greg for your reply and interesting link. How ever I still don't find an explanation as to why an orbiting satellite is pulled in towards the Earth. Assuming that space-time does not exert any force itself, a stationary object in curved space-time should experience no force at all.

Thanks for all the replies. From what I understand there is no "Gravity" apart from a curved space-time.

Gravity = curved space time = Gravity. Space-time is not warped by gravity but by mass. according to GR there is no force of gravity apart from space-time warping.

Exactly!

In short, you can imagine space-time as a plot of the gravitational field at every point. The lower the point gets on the fabric of space-time, the more more force of attraction it will experience, the more negative potential energy the body will posses (assuming PE at $\infty$ to be zero)! I made the previous statement up, is it valid?

I'm glad you brought up the anology of the rug. The reason the ball falls down the rug is due to the force of gravity. The slope of the rug generates no force. if you were to perform the same experiment in space outside of the earths gravity, the snooker ball would go nowhere.

As I mentioned in my first post an incline or curvature could only produce motion in the presence of gravity.
Because the space outside of our gravity has no curvature,if their was nothing else in the area using a gravitational force, the object would be in a state of inertia and not be attracted to anything

If the effect gravity is considered to be due the curvature of space which keeps a satellite in orbit, what force causes the satellite to spiral "downwards" towards the earth if its velocity were to be reduced.

The curvature of space alone could not produce an attractive "downward" force, as in space no slope or incline could produce motion.

The use of the word "downward" has no relevance here as we are talking about an overall 3D curvature, something that we can't imagine, it doesn't fall downwards it falls in all 3 dimensions :D

Exactly!

In short, you can imagine space-time as a plot of the gravitational field at every point. The lower the point gets on the fabric of space-time, the more more force of attraction it will experience, the more negative potential energy the body will posses (assuming PE at $\infty$ to be zero)! I made the previous statement up, is it valid?

When you say "force of attraction", are you refering to gravity as a independent force or the curvature of space-time?.

The use of the word "downward" has no relevance here as we are talking about an overall 3D curvature, something that we can't imagine, it doesn't fall downwards it falls in all 3 dimensions :D

Thats the reason I put the downward within inverted commas

When you say "force of attraction", are you refering to gravity as a independent force or the curvature of space-time?.

I believe that gravity as a force, is quite different from the others! I think of space-time as way of finding out how the force of gravity will actually act on a body in that particular region! A steep downward slope in the fabric of space-time implies the fat that the object will move in the direction of the downward slope due to gravitational attraction! I do not know if my ideas hold true, and the mathematical implications that what I have said can have!

Hope you get what I mean to say!

If the effect gravity is considered to be due the curvature of space which keeps a satellite in orbit, what force causes the satellite to spiral "downwards" towards the earth if its velocity were to be reduced.

The curvature of space alone could not produce an attractive "downward" force, as in space no slope or incline could produce motion.

Don't think of it as a "slope", as that infers that possibly a rolling, undulating almost hill like space-time makes the Multi-dimensional attractive force. A slope refers to a 2 dimensional curvature that always makes an object fall downwards. Space-time is warped by the presents of mass, which causes gravitational attraction.
An interesting point about gravity is that it could be a force carrying particle called the graviton, that if we were to include string theory would not be attached to a brane meaning that it could travel to all different brane-worlds in different dimensions, this is an idea of why gravity is so weak compared to the other forces, as it travels through all dimensions, unlike the other forces.

DrGreg
Gold Member
Thanks Dr. Greg for your reply and interesting link. How ever I still don't find an explanation as to why an orbiting satellite is pulled in towards the Earth. Assuming that space-time does not exert any force itself, a stationary object in curved space-time should experience no force at all.
In the analogy of the "trumpet shaped" spacetime diagram that I linked to, the graph of a "stationary" object would be a horizontal circle drawn round the vertical trumpet. But within the curved surface this isn't the straightest line possible. If you draw a straight-as-possible line it represents something rising and falling under gravity.

How ever I still don't find an explanation as to why an orbiting satellite is pulled in towards the Earth. Assuming that space-time does not exert any force itself, a stationary object in curved space-time should experience no force at all.
There is no such thing as a "stationary" (meaning its path through space-time consists of a single point, such a thing would be an event, with no duration) object in space-time. In the link to which DrGreg referred, an object that is initially motionless (i.e., the time derivative of its spatial trajectory is zero) has a non-zero "velocity" (four-velocity) vector (it would point along the blue curves (just as do the temporal components of the two vectors shown) and be of magnitude c, the speed of light). So, the motion of the initially motionless object will be represented by a curved path in space-time. The path is determined by the direction of the http://en.wikipedia.org/wiki/Four-velocity" [Broken] planes follow across the surface of the earth) which is why the two paths shown in the pictures look curved even though they are locally straight. An object that starts off with zero spatial velocity at some non-zero height would curve downward and to the left in the diagrams in the link, intersecting the horizontal axis at a larger (than initial) value of the vertical axis (time) coordinate. This represents the object falling as time progresses.

Similarly, the curvature of space-time would cause the path of a satellite to intersect that of the earth if it were initially motionless with respect to the earth. Since a satellite in orbit has a nonzero velocity with respect to the earth, it moves sufficiently far in the time it would take to fall to earth that it misses the earth and continues on its trajectory. http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates" [Broken] may be helpful for visualization. In these coordinates, the curvature manifests as "falling" spatial coordinates (the spatial coordinates contract toward concentrations of mass-energy terminating on its surface in finite time).

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Similarly said:
http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates"[/PLAIN] [Broken] may be helpful for visualization. In these coordinates, the curvature manifests as "falling" spatial coordinates (the spatial coordinates contract toward concentrations of mass-energy terminating on its surface in finite time).

When the satellite is at its orbital velocity it would follow a certain path in space-time around the earth. Now why can't it follow the same orbit when its velocity decreases? (assuming there is no force of gravity apart from space-time curvature). What is the connection between velocity and space-time curvature that forces it into a lower orbit?

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WannabeNewton
When the satellite is at its orbital velocity it would follow a certain path in space-time around the earth. Now why can't it follow the same orbit when its velocity decreases? (assuming there is no force of gravity apart from space-time curvature). What is the connection between velocity and space-time curvature that forces it into a lower orbit?
There would have to be some mechanism capable of decreasing the velocity. Since the satellite then would no longer be in free fall during the velocity decrease, the geodesic (the space - time path you were talking about) would start to diverge/converge as per $\frac{D^{2}\xi ^{\alpha }}{D\tau ^{2}} = R^{\alpha }_{\beta \mu \nu }V^{\beta }V^{\mu }\xi ^{\nu }$ where $\mathbf{V}$ would be the object's 4 - velocity and $\boldsymbol{\xi }$ would be a separation vector measuring the convergence with respect to some other nearby geodesic.

I'm totally making this up, and it's probably entirely wrong, but it's an interesting idea.

Perhaps the warping of spacetime causes this attraction due to a different phenomenon...

When I think about the warping of spacetime I note that the surface area of a warped region of space time is higher than the surface area of a nonwarped region. In other words, space is being "stretched" apart. This would in turn imply a lower density of the fabric of spacetime in the stretched area, which could in turn result in a force moving objects from the higher density area to the lower density area.

Looking at that again, I admit it's extremely far-fetched and kind of ridiculous. I'm not contending it isn't, I was just following a random mental tangent. Maybe someone else with more physics experience could see if it has any possibility of actually explaining the phenomenon - as a hypothesis; or they could note something which clearly disproves it.

-Mator

Now why can't it follow the same orbit when its velocity decreases? (assuming there is no force of gravity apart from space-time curvature). What is the connection between velocity and space-time curvature that forces it into a lower orbit?
If you change the initial slope of a straight line, it does not "follow" the same path as it would have (in this respect general relativity is just like high school geometry). A change in the initial velocity of the satellite would require a change in its four-vector, thus its path in space-time would be different. If the satellite's speed in the angular direction (assuming rain coordinates) is slightly less that necessary for a circular orbit, the straight path it follows will be more affected by the falling of the spatial coordinates toward the concentration of mass-energy (since its four-velocity vector will be smaller in magnitude and be pointed more toward the earth than originally). (Note that there are coordinate systems that do not fall that describe space-time just as well as rain coordinates, it is just a little less intuitive that objects would fall in those coordinate systems.)

If you are talking about the decay of a satellite's orbit, this is usually due to drag against the atmosphere (so the satellite really is being accelerated in this case and its path through space-time will not be straight).

This would in turn imply a lower density of the fabric of spacetime in the stretched area, which could in turn result in a force moving objects from the higher density area to the lower density area.
What does it mean for space-time to have a lower density? I think you may be thinking of the fact that time is dilated and lengths contracted (among other things) near concentrations of mass-energy relative to an observer far away. However, generally the "force" that causes objects to move from regions of high density to low is the result of statistics, objects on the boundary of the high density region more often collide with particles from that region and thus gain momentum in the direction opposite of that region. As far as I know, space-time is not composed of moving particles (I am not sure the idea would even make sense) so there is no similar tendency for objects to receive momentum. In any case, if space-time is stretched (assuming it is stretched in the right way and that objects follow straight paths) then there is already an explanation of gravity.