Source of Gravity of heavy mass as per GTR

[eaglechief]

TL;DR Summary

(wiki) In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

Regarding the equivalence principle of GTR, inertial mass and heavy mass are comparable, and are causing identically effects on mass objects.

The pseudo force of inertial mass could be experienced in an accelerated frame of reference. In that point the "Force" appers, while energy is accelerating an object and causes a change of velocity. An observer could see that or even experience that on his own, i.e. when he is inside an accelerating car. This effect is intuitive.

But how does gravity in heavy mass work ? As far as i understood, in an example where an apple is falling from a tree, the apple could be considered as a freefalling frame of reference, and the ground is accelerating towards the apple (because the mass of the Earth is deforming spacetime).

But is that "accelerating of the ground towards the apple" only a mathematical effect, a kind of way the GTR is solving this problem, and/or why is mass curving spacetime at all ?

Thx in advance
Carlo

 

[Ibix]

Pedantic point: the source of gravity in GR is the stress-energy tensor, which is a more general entity than just mass.

You seem to be rather confused about the equivalence principle. The point is that, in Newtonian physics, it is a fact that the $m$ in $F=ma$ is the same as the $m$ in $F=GMm/r^2$, but there is no explanation for this. It is merely an observed fact that this is so. GR provides a reason - gravity is just geometry, so cannot depend on the characteristics of test particles such as what they are made of. The only thing that can matter is where they are and how fast they are going in what direction.

The point about free falling objects is that they are moving inertially. Thus, something that is at rest on the surface of the Earth (not in free fall) is not moving inertially. This is an inversion of the Newtonian model, where a falling body is accelerating due to a gravitational force while an object at rest on the surface is inertial with no net force. Realising that he could turn things round this way was a key part of Einstein's road to relativity.

And yes, it means that the floor is accelerating upwards. However the geometry of spacetime is not Euclidean, so this does not mean that it is moving outwards.

 

[PeterDonis]

in an example where an apple is falling from a tree, the apple could be considered as a freefalling frame of reference, and the ground is accelerating towards the apple

That's true, but the more important thing is that you, standing on the ground, feel a force--weight--whereas the falling apple doesn't. But if you limit yourself to just local observations (say you're standing inside a room with no windows and no other way of sensing the world outside and you drop the apple), you can't tell whether you feel that force, and see the dropped apple accelerate downwards relative to you, because you're standing at rest on Earth or because you're standing inside a rocket accelerating at 1 g in empty space. That's the basic content of the equivalence principle.

is that "accelerating of the ground towards the apple" only a mathematical effect

No, it's not, any more than it would be if you were standing inside a rocket accelerating at 1 g in empty space and released an apple--the rocket, and you with it, would be accelerating towards the apple, which would be in free fall.

why is mass curving spacetime at all ?

Spacetime curvature doesn't even come into play in the local experiments discussed above. It only comes into play globally, when you have to fit together all the different local patches into a single global solution. Globally, it's easy to distinguish the Earth and its gravity from a rocket accelerating at 1 g in empty space: the Earth's gravity points in different directions in different parts of the Earth, but the "pseudo-gravitational field" inside the accelerating rocket does not. But that's going beyond the equivalence principle.

 

[Dale]

Summary:: (wiki) In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

But is that "accelerating of the ground towards the apple" only a mathematical effect

It is not only a mathematical effect. It is directly measured by an accelerometer. An accelerometer placed on the ground directly measures an upward acceleration of 1 g.

 

[vanhees71]

The equivalence principle is more complicated than one thinks. I think the only really correct version is the pretty abstract postulate that at any spacetime point there exists a local inertial reference frame (geometrical approach) or that the laws of nature are described by a theory where Lorentz invariance is a local gauge symmetry (physical approach, extending GR to Einstein-Cartan theory).

The reason why the more heuristic equivalence principle used by Einstein to "derive" GR and which is also used in most textbooks to derive it, is not completely correct, because a true gravitational field can never be completely described by a reference frame that is accelerated relative to a local inertial frame, because if there is a true gravitational field (e.g., the gravitational field of the Sun or the Earth) the curvature tensor is non-zero, and that is an invariant feature, i.e., it holds in all reference frames.

A local inertial reference frame can be constructed by Fermi-Walker transporting a tetrad along a geodesic of space time, i.e., by a freely falling test-particle like observer. In a sufficiently small neighborhood the gravitational field only appears as "tidal forces".

 

[eaglechief]

It is not only a mathematical effect. It is directly measured by an accelerometer. An accelerometer placed on the ground directly measures an upward acceleration of 1 g.

going back to the point of dale: beneath the accelerated rocket we can observe the movement of the rocket and understand instantly, how the gravitational effects inside the rocket work. So "where" is the movement i.e. the acceleration measured at the ground of the Earth ? Would a 5d observer "see" any movement esp. changing of position of the ground towards the free-falling apple ?

 

[Ibix]

So "where" is the movement i.e. the acceleration measured at the ground of the Earth ?

It's seen by an inertial observer - one in free fall. Just the same as an inertial observer outside a rocket sees it accelerate, while one attached to it sees it motionless but feels a force.

Obviously if you look on a larger scale there are differences. But that's why we say the equivalence principle is only true locally.

Would a 5d observer "see" any movement

All of GR is formulated in terms of a 4d spacetime that need not be embedded in any higher dimensional space. Mathematically you are free to consider spacetime embedded in such a space, but there is no unique way to do such an embedding. So the answer to this is honestly that it's an unhelpful line of thinking, not related to the way GR works.

 

[Dale]

So "where" is the movement i.e. the acceleration measured at the ground of the Earth ? Would a 5d observer "see" any movement esp. changing of position of the ground towards the free-falling apple ?

The movement is in the frame of the apple. In that frame the apple is at rest and the ground is indeed accelerating upward.

What is interesting is not that. That is rather clear. What is interesting is how the USA and Australia can be constantly accelerating away from each other without getting any further away from each other. That is not possible in flat spacetime, but it is possible in curved spacetime.

 

[eaglechief]

Hi Dale,

is there any picture available visualizing that topic in curved spacetime ? Thats my point, i can imagine and visualize the acceleration of rocket and its interiors. But i am not able to do that with USA and Australia ...

Thx
Carlo

 

[eaglechief]

All of GR is formulated in terms of a 4d spacetime that need not be embedded in any higher dimensional space. Mathematically you are free to consider spacetime embedded in such a space, but there is no unique way to do such an embedding. So the answer to this is honestly that it's an unhelpful line of thinking, not related to the way GR works.

Yes i do agree. But given the reality is not only containing 4d but maybe more, than a 4d+ description could even do better than GR because describing for instance a 5d process in a 4d reference frame is limited in its expressivness.

 

[Dale]

is there any picture available visualizing that topic in curved spacetime ?

There are visualizations, but they require a bit of background. Please read this:

https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/

 

[A.T.]

Thats my point, i can imagine and visualize the acceleration of rocket and its interiors. But i am not able to do that with USA and Australia ...

Try this post:
https://www.physicsforums.com/threa...lly-at-rest-begin-to-fall.995946/post-6416452

The key is: Proper acceleration is different than coordinate acceleration.

Proper acceleration is deviation from geodesic free fall worldlines. In curved spacetime geodesic free fall worldlines can converge. Staying at constant distance on opposite sides of the Earth, means opposing free fall towards each other and thus deviating from geodesics (proper acceleration away from each other)

So proper acceleration away from each other doesn't imply moving away from each other, if spacetime is curved. It is needed to keep a constant separation too.

 

[Profevil]

Summary:: (wiki) In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

Regarding the equivalence principle of GTR, inertial mass and heavy mass are comparable, and are causing identically effects on mass objects.

The pseudo force of inertial mass could be experienced in an accelerated frame of reference. In that point the "Force" appers, while energy is accelerating an object and causes a change of velocity. An observer could see that or even experience that on his own, i.e. when he is inside an accelerating car. This effect is intuitive.

But how does gravity in heavy mass work ? As far as i understood, in an example where an apple is falling from a tree, the apple could be considered as a freefalling frame of reference, and the ground is accelerating towards the apple (because the mass of the Earth is deforming spacetime).

But is that "accelerating of the ground towards the apple" only a mathematical effect, a kind of way the GTR is solving this problem, and/or why is mass curving spacetime at all ?

Thx in advance
Carlo

If mass is nothing more than densely packed partials of mater that when increased begin to create pressure and heat; what causes gravity? If you exclude the pressure and heat as the cause, then you are left with the interaction of densely packed particles.

In my quest to try and figure out how the interruption of spacetime by mass creates the effect of gravity I am imagining theoretical experiments that may inspire thought. I am sure all of these have been discussed at one time or another, so I apologize if they are old shoes.

Knowing that the mass of a celestial object determines the amount of measured gravity that effects other objects near by thus causing them to move towards the center of that mass; what would happen if the physical properties of the mass were different than what the physical laws have provided us to examine?

What are the possible outcomes if we were to release a bowling ball over the center of a "theoretical" 20' wide hole drilled straight through the moon?

What if that hole were 10 miles wide?

What do the gravitational measurements look like if the hole were increased to 1/3 the moons diameter, or the theoretical moon were shaped like a doughnut?

What would the gravitational measurements look like around a theoretical large mass shaped like a cube?

For myself, answering these questions may help me or someone else understand how the interruption of spacetime by crushed partials, create the known laws of gravity.

 

[Dale]

For myself, answering these questions may help me or someone else understand

It would be better to post that as your own thread rather than hijack someone else’s thread. And it is better to focus on a single well-motivated and clearly-described simple question rather than a rapid barrage of poorly-described complicated questions.

 

[PeroK]

Hi Dale,

is there any picture available visualizing that topic in curved spacetime ? Thats my point, i can imagine and visualize the acceleration of rocket and its interiors. But i am not able to do that with USA and Australia ...

Thx
Carlo

As an analogy, you could have two objects at opposite sides of rod spinning about its centre. Both objects have a real, centripetal acceleration, hence are continuously accelerating towards each other; yet, they are not getting closer together. Acceleration in a given direction does not imply motion in that direction. In this case the constant rotation is the key factor.

Two objects on opposite sides of the Earth are, likewise, continuously accelerating in opposite directions but not moving apart. In this case, curved spacetime is the key factor.