espen180 said:I suggest you take a look at the gravitomagnetic field equations, which are a first-order approximation to GR but good enough to give you an idea of what is going on. The effect is similar to the effect due to a magnetic field caused by a moving charge.
"Is space bent towards a mass?" is, I think, a strange question to ask. To properly examine the curvature, you have to take time into account as well. The result of the curvature is that straight lines in space-time appear to be curved toward masses in space.
jaketodd said:I appreciate your post. It sounds like I don't need to understand gravitomagnetic field equations. "...straight lines in space-time appear to be curved toward masses..." So the answer to my question seems to be a simple "yes." Right?
Thanks,
Jake
jaketodd said:Is it true that gravitating bodies actually warp the fabric of space towards them like in this picture? http://www.astronomynotes.com/evolutn/grwarp.gif
starthaus said:Like in http://www.wbabin.net/ntham/todd3.pdf "paper" you just "published" based on what you are learning here.
TCS said:If you think about space time as a baloon where the stretchiness of the balloon at a spot on its surface is determined by its mass/energy density, then the surface of the balloon will be dimpled. The rate of time and the spatial dimensions are all determined by the radius of the dimple. Motion across the surface of the baloon means that you will be moving through dimples in space time as well as causing a dimple to propagate over the surface.
espen180 said:Thank your for that idea. It sparked some of my own.
I guess it works as a 2D analogy of a closed universe, but it doesn't help jaketodd, since inhabitants on the baloon surface cannot experimentally determine the direction of the curvature (positive if on the outside, negative if on the inside, but this is impossible for the 2-dimensional inhabitants to determine).
Nevertheless, the baloon analogy is exellent for demonstrating that asking in what direction spacetime curves is nonsense. We can see that on the balloon, spacetime is embedded in 4 dimensional space (2 spatial dimensions, 1 temporal dimension and a fourth dimension into which spacetime also curves). By analogy we can see that we would need a 5-dimensional space in which to embed our 4-dimensional spacetime for us to be able to ask in which direction spacetime curves, and even then it would be a question of definiton.
espen180 said:We can see that on the balloon, spacetime is embedded in 4 dimensional space (2 spatial dimensions, 1 temporal dimension and a fourth dimension into which spacetime also curves). By analogy we can see that we would need a 5-dimensional space in which to embed our 4-dimensional spacetime for us to be able to ask in which direction spacetime curves, and even then it would be a question of definiton.
jaketodd said:You don't necessarily need a 5th dimension. Imagine, instead of a dimple, spacetime stretched toward a massive object without curving into a 5th dimension. However, the question remains: What force or tendency makes objects go into regions of stretched spacetime?
jaketodd said:What force or tendency makes objects go into regions of stretched spacetime?
A.T. said:It is the tendency to move on straight lines in spacetime:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
jaketodd said:but it doesn't explain why something starting from rest, relative to a massive object, starts falling toward the massive object.
It wouldn't be a free falling object then, because it's path through spacetime(worldline) wouldn't be a straight line anymore. In order to keep the object at the top of the house, you have to bend it's worldline by applying an upwards force on the object.jaketodd said:If the "falling object" mirrored the path of the proper time in the graphic, then it would stay at the top of the house.
jaketodd said:If the "falling object" mirrored the path of the proper time in the graphic, then it would stay at the top of the house.
TCS said:In the four dimensional model of space time, you are never stationary. In uncurved space, you are moving at a constant velocity in the direction of time. When space curves, some of your velocity is in the other three dimensions.
However, I think that five dimensional models provide a more intuitive picture of space time.
Because there is no real force acting on it (it is in free fall), it advances on a straight line trough spacetime.jaketodd said:In the graphic, why doesn't the object have to mirror the axis of proper time?
In GR you don't need a cause to advance straight in spacetime - it the default behavior of all objects. You need a cause (force) to deviate from that straight line.jaketodd said:What causes it to deviate from that path?
By moving locally straight you always tend towards the area of increasing distances (more stretched spacetime). This is dictated by geometry as shown in the pictures.jaketodd said:There still needs to be something that chooses which spatial direction to go in.
No, moving locally straight is enough.jaketodd said:And if you bring a 5th dimension into it, there needs to be a force that pulls things into a dimple of spacetime.
A.T. said:Because there is no real force acting on it (it is in free fall), it advances on a straight line trough spacetime.
No. The graphic shows how GR models gravitation, and in GR force free objects advance locally straight in spacetime. Maybe you are confusing GR with a different (your own?) theory.jaketodd said:Since the graphic defines the proper time as curved, then an object with no forces on it would mirror that curved path.
A.T. said:No. The graphic shows how GR models gravitation...
A.T. said:...gravitation...
A.T. said:gravitation
A.T. said:Because there is no real force acting on it (it is in free fall), it advances on a straight line trough spacetime.
In GR you don't need a cause to advance straight in spacetime - it the default behavior of all objects. You need a cause (force) to deviate from that straight line.
A.T. said:gravitation
Where did I say "force" ? "Gravitation" refers to the general phenomena, not a specific model.jaketodd said:Finally, a force...
No. In GR you don't need a force to make things advance in spacetime. All objects advance in spacetime by default.jaketodd said:...that makes things move in spacetime
No. In GR you don't need a force to advance locally straight in spacetime. It is the default behavior of force free objects.jaketodd said:and can lead to the straight line in the graphic.
Yes, in GR within inertial frames, free falling objects are force free.jaketodd said:As you can see, before you where claiming no force on the object.
A.T. said:Where did I say "force" ? "Gravitation" refers to the general phenomena, not a specific model.
No. In GR you don't need a force to make things advance in spacetime. All objects advance in spacetime by default.
No. In GR you don't need a force to advance locally straight in spacetime. It is the default behavior of force free objects.
Yes, in GR within inertial frames, free falling objects are force free.
No idea what you compare here, A dimension doesn't "take a path of a finite length",jaketodd said:Also, the graphic shows the object taking a path shorter than the curved, proper time dimension.
Time dilation in this diagram means the object advances less along the proper time dimension. This might help you to understand the diagram better:jaketodd said:This is incorrect because an object would take longer than the proper time since its motion causes time dilation. So its path should be longer than the proper time curve between the two end points of the object's path.
Yesjaketodd said:So objects fall according to what? The curvature of spacetime?
You could just as well ask "What bends a cricle?". It is simply a geometrical consequence of the mathematical model (geodesics on curved manifolds).jaketodd said:What pulls them into a dimple?
You can say a lot of things. But the things GR says also fit the observation quite well.jaketodd said:I guess you could say objects move by default to less dense areas of spacetime.
Actually, the curvature of the balloon surface is positive regardless of which "side" of the surface you are talking about. In fact, the concept of "inside" or "outside" the surface is only valid in the 3D embedding space and is meaningless within the 2D surface itself.espen180 said:I guess it works as a 2D analogy of a closed universe, but it doesn't help jaketodd, since inhabitants on the baloon surface cannot experimentally determine the direction of the curvature (positive if on the outside, negative if on the inside, but this is impossible for the 2-dimensional inhabitants to determine).
TCS said:If you think about space time as a baloon where the stretchiness of the balloon at a spot on its surface is determined by its mass/energy density, then the surface of the balloon will be dimpled. The rate of time and the spatial dimensions are all determined by the radius of the dimple. Motion across the surface of the baloon means that you will be moving through dimples in space time as well as causing a dimple to propagate over the surface.
I understand this is from a book by L Epstein. It clearly was never professionally reviewed as the concept of curved proper time is sheer nonsense.A.T. said:http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_time.gif
Passionflower said:http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_time.gif
I understand this is from a book by L Epstein. It clearly was never professionally reviewed
A single dimension cannot be intrinsically curved alone, that's why "curved time" is in quotes. But intrinsic curvature is not even necessary to have a "gravitational pull". The "gravitational pull" is related to the 1st derivates of the metric, while tidal forces are related to 2nd derivates (curvature). And the spacetime in the picture in fact doesn't have any intrinsic curvature. I agree that "curved time" is not a good title for the illustration.Passionflower said:as the concept of curved proper time is sheer nonsense.
Passionflower said:The reason a clock attached to a ceiling runs faster than a clock on the floor is that the clock on the floor has a greater proper acceleration than the clock near the ceiling.
Yes.Passionflower said:Curvature of spacetime simply causes objects to accelerate with respect to each other without any need for proper acceleration.
A clock in the center of the Earth accelerates away from all directions because all the mass surrounding it tries to attract it. Proper acceleration does not necessarily result in relative movement.A.T. said:- A clock resting in the Earths center experiences less proper acceleration but runs slower than a clock resting on the surface.
- You can have a two clocks resting (relative to each other) in an uniform gravitational field experiencing the same proper acceleration, but running at different rates.
That is just a very complicated way to say that the proper acceleration of a clock resting in the center is zero. This is less than the proper acceleration of a clock resting on the surface. Yet the center clock runs slower than the surface clock.Passionflower said:A clock in the center of the Earth accelerates away from all directions because all the mass surrounding it tries to attract it.
Passionflower said:The reason a clock attached to a ceiling runs faster than a clock on the floor is that the clock on the floor has a greater proper acceleration than the clock near the ceiling.
I don't quite see how this addresses my counter example to your claim that greater proper acceleration causes a slower clock rate.Passionflower said:An object can be both stationary and accelerating.
Start with the Schwarzschild solution in the weak field approximation:A.T. said:That is just a very complicated way to say that the proper acceleration of a clock resting in the center is zero. This is less than the proper acceleration of a clock resting on the surface. Yet the center clock runs slower than the surface clock.
starthaus said:Start with the Schwarzschild solution in the weak field approximation:
(cd\tau)^2=(1-2\frac{\Phi}{c^2})(cdt)^2+(1-2\frac{\Phi}{c^2})^{-1}(dr)^2+...
For the case dr=d\theta=d\phi=0 you get the well known relationship:
d\tau=\sqrt{1-2\frac{\Phi}{c^2}}dt
Writing the above for two different gravitational potentials \Phi_1 and \Phi_2 you obtain the well-known time dilation relationship:
\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-2\frac{\Phi_1}{c^2}}{1-2\frac{\Phi_2}{c^2}}}
At the Earth surface :
\Phi_1=-\frac{GM}{R}
At the Earth center:
\Phi_1=-3/2\frac{GM}{R}
Now, due to the fact that \frac{\Phi}{c^2}<<1 you can obtain the approximation:
\frac{d\tau_1}{d\tau_2}=1-\frac{\Phi_1-\Phi_2}{c^2}=1-1/2\frac{GM}{rc^2}<1
So, f_1>f_2
In addition, the time dilation depends on the gradient of the gravitational field \Phi_1-\Phi_2, i.e., it depends on acceleration.
TCS said:Shouldn't phi 1 be zero. I think that you are missing a little r in your calculation.
TCS said:This link shows that gravitational time dilation is proportion to small g, which should be zero at the center of the earth.
https://www.physicsforums.com/library.php?do=view_item&itemid=166
starthaus said:The formulas on the link you cited are valid for outside the Earth. The potentials inside the Earth are different.You need to be careful with what expressions you plug in into your calculations. The calculations I showed are correct.
Yes. E.g. as a photon goes up it gains potential energy, loses kinetic energy, and therefore becomes redshifted. This indicates that time is slower lower in the potential well.TCS said:Is the difference related to change in potential energy of the clock?
