Curved space

[Greg Boyles]

This might give a bit of insight, that Swansont mentioned in another thread, that's relevant here I think.

 

We can only ever investigate quantum mechanics and relativity etc through mathematical abstractions because the reality is beyond any detection and comprehension we can acheive through our senses, i.e. the real world.

[IM Egdall]

 

I would like to say, that;

 

the object takes a different path not because the space around the massive object is warped but instead because, it is forced to do so by the gravitational field of the massive body. May be the tangential component of the Gravitational force is forcing the object to change its path.

 

Like a magnetic needle gets deflected in a magnetic field.

 

Likewise, Gravity has the potential to deflect motion & duration of objects & events respectively.

 

what are the drawbacks or flaws in saying that?

 

Your description is incomplete. At the risk of sounding like a broken record, let me say again: According to general relativity, the gravitational field IS the warping of space and time or spacetime curvature.

 

A clock runs slower as it gets closer to the Earth's surface. The distance between the same two points stretches as the points approach the surface of the Earth (points perpendicular to the Earth's surface as seen from far away). In Einstein's construct, this warping of time and space is what tells objects how to move in a gravitational field. It is what is holding me down in my chair right now as I write this. So all the physical effects we attribute to the term "gravity" are produced by the warping of space (change in length or space interval) and warping of time (change in time interval). Thus the warping of space and time is gravity.

 

By the way, this change in the space and time intervals is represented mathematically in general relativity by the change in the spacetime interval. The spacetime interval is the square root of the difference between the square of the time interval and the square of the space interval (ignoring cross-terms).

 

I suggest you look up frame dragging and the latest Gravity Probe B results. Here the spinning of the Earth on its axis caused gyroscopes orbiting the Earth to change their spin orientation (very slightly). Why? How can the spinning of the Earth affect the orientation of gyroscopes which are themselves rotating in "empty" space? Because, per general relativity, the spinning of the Earth drags empty space around with it as it rotates.

 

The confirmation of this so-called frame dragging by Gravity Probe B (to about 19%) shows that space IS flexible, is warpable. No matter what we say about how crazy this is -- we now have emprical confirmation of this strangest of phenomena.

 

Link: http://einstein.stanford.edu/highlights/status1.html

[Anilkumar]

This might give a bit of insight, that Swansont mentioned in another thread, that's relevant here I think.

Thanks StringJunky,

 

That's one great article.

 

Tailor made for this thread.

 

I went through it once. But I need to go through it a couple of times more and give it a thorough thought to digest the core philosophy of the article.

 

That's a philosophy which can bring a significant change on the way we exert to perceive this world.

 

We have invented many concepts to explain things around us.

 

But there is a hazard that, those very concepts may push us into an illusory conceptual world.

 

 

-------------------------------*******************

 

Your description is incomplete. At the risk of sounding like a broken record, let me say again: According to general relativity, the gravitational field IS the warping of space and time or spacetime curvature.

 

A clock runs slower as it gets closer to the Earth's surface. The distance between the same two points stretches as the points approach the surface of the Earth (points perpendicular to the Earth's surface as seen from far away). In Einstein's construct, this warping of time and space is what tells objects how to move in a gravitational field. It is what is holding me down in my chair right now as I write this. So all the physical effects we attribute to the term "gravity" are produced by the warping of space (change in length or space interval) and warping of time (change in time interval). Thus the warping of space and time is gravity.

 

By the way, this change in the space and time intervals is represented mathematically in general relativity by the change in the spacetime interval. The spacetime interval is the square root of the difference between the square of the time interval and the square of the space interval (ignoring cross-terms).

 

I suggest you look up frame dragging and the latest Gravity Probe B results. Here the spinning of the Earth on its axis caused gyroscopes orbiting the Earth to change their spin orientation (very slightly). Why? How can the spinning of the Earth affect the orientation of gyroscopes which are themselves rotating in "empty" space? Because, per general relativity, the spinning of the Earth drags empty space around with it as it rotates.

 

The confirmation of this so-called frame dragging by Gravity Probe B (to about 19%) shows that space IS flexible, is warpable. No matter what we say about how crazy this is -- we now have emprical confirmation of this strangest of phenomena.

 

Link: http://einstein.stan...ts/status1.html

Hello IM Egdall,

 

just when the article suggested by StringJunky, had begun to give me an idea that spacetime is an abstract model, your post has given a blow to the idea by saying Space is flexible.

 

I definitely need time to come to terms with this.

 

Thank you for the conscientious appraisal.

Edited December 15, 2011 by Anilkumar

[StringJunky]

Thanks StringJunky,

 

That's one great article.

 

Tailor made for this thread.

 

I went through it once. But I need to go through it a couple of times more and give it a thorough thought to digest the core philosophy of the article.

 

That's a philosophy which can bring a significant change on the way we exert to perceive this world.

 

We have invented many concepts to explain things around us.

 

But there is a hazard that, those very concepts may push us into an illusory conceptual world.

 

Glad you found it useful Anikumar...I did too...kudos to seansont It's important to get a grip on what abstract constructs are because scientists, especially physicists, use them alot and spacetime is one of them. They are not intended to be an exact mirror of the world around us in visual terms but is a mental construct that conveys the information that like-minded people can have dialogue about and have mutually agreed what it means. If we amateurs wish to understand what they conceive the onus is on us to learn their language. As you will note in the article, even some qualified scientists fall into the trap of reifying abstract concepts as though they were real entities so it's clearly not an easy path to follow...take some comfort from that when you struggle to understand!

 

We can only ever investigate quantum mechanics and relativity etc through mathematical abstractions because the reality is beyond any detection and comprehension we can acheive through our senses, i.e. the real world.

 

Greg, that's how I understand the necessity of abstract models due to the limits of our perception.

Edited December 15, 2011 by StringJunky

[Anilkumar]

Glad you found it useful Anikumar...I did too...kudos to seansont It's important to get a grip on what abstract constructs are because scientists, especially physicists, use them alot and spacetime is one of them. They are not intended to be an exact mirror of the world around us in visual terms but is a mental construct that conveys the information that like-minded people can have dialogue about and have mutually agreed what it means. If we amateurs wish to understand what they conceive the onus is on us to learn their language. As you will note in the article, even some qualified scientists fall into the trap of reifying abstract concepts as though they were real entities so it's clearly not an easy path to follow...take some comfort from that when you struggle to understand!

Yes StringJunky,

it certainly shows the other face of our own endeavors.

And Swansont is a great guy.

 

 

 

------------------------------**********************

 

 

If you want to put it that way then I would say that the mathematical theory of relativity actually replaces 'space' with 'spacetime'.

 

'Space' is therefore relegated to the long list of discredited human ideas such as flat earth and the four elements or fire, water, earth and air.

Does GR say that no such thing as 'Space' - the empty/vacant room which accommodates all matter, exists?

Edited December 15, 2011 by Anilkumar

[swansont]

No 'Anilkumar' spacetime is not an mathematical delusion, spacetime is very real.

 

However we are 3 dimensional creatures trapped in 3 dimensions with senses that are restricted to 3 dimensions.

 

Spacetime involves a 4th physical dimensions that we can neither perceive nor ever experience through our senses.

 

But fortunately the tool of mathematics allows us to throw off these shackles and explore the universe beyond the limitations of our 3 dimensional senses.

 

Allow me to put this to you another way.

 

No doubt you would agree that mobile phones, computers and electron microscopes etc all work and are not the result of mathematical or psychological delusions.

 

All these things contain semi conductors and other eletronic components. And these designed to work a a specific and convenient way (for our electronic deivces) based on the fixed properties of atoms and sub atmoic particles.

 

There is no way that we can directly perceive atoms and sub atomic particles because they are simply to small. We can only rely on our mathematics to guide us in creating components that make our electronic devices work.

 

If we agree that mathematics works to describe and create our electronic devices, and that it is not all a mathematical delusion, then why can we not also agree that the same mathematics provides an accurate description of the true nature of spacetime/space even if we can't comprehend the answer in terms of sight and touch etc?

 

Abstraction and delusion don't mean the same thing. Mathematics tells us how things work, i.e. it tells us the results. It does not tell us why it works that way. Even the way we describe subatomic particles is an abstraction — we say that an electron acts like a wave and a particle, depending on what measurement you are doing, but that tells you how it behaves, not what it is. Wave and particle are abstractions that give you a fairly well-defined list of properties.

 

Delusion would mean we are imagining the whole thing. I don't think anyone is claiming that.

[Anilkumar]

Abstraction and delusion don't mean the same thing. Mathematics tells us how things work, i.e. it tells us the results. It does not tell us why it works that way. Even the way we describe subatomic particles is an abstraction — we say that an electron acts like a wave and a particle, depending on what measurement you are doing, but that tells you how it behaves, not what it is. Wave and particle are abstractions that give you a fairly well-defined list of properties.

 

Delusion would mean we are imagining the whole thing. I don't think anyone is claiming that.

I am happy.

[Greg Boyles]

Abstraction and delusion don't mean the same thing. Mathematics tells us how things work, i.e. it tells us the results. It does not tell us why it works that way. Even the way we describe subatomic particles is an abstraction — we say that an electron acts like a wave and a particle, depending on what measurement you are doing, but that tells you how it behaves, not what it is. Wave and particle are abstractions that give you a fairly well-defined list of properties.

 

Delusion would mean we are imagining the whole thing. I don't think anyone is claiming that.

 

 

I was getting the impression that that is how Anilkumar was thinking - that relativity is nothing more than a mathematical illusions that does not apply to reality as we experience it, i.e. his constant references to space not equalling spacetime etc.

 

My position is that relativity does indeed describe reality as we experience it but that we cannot directly perceive its full richness due our limited senses.

[Anilkumar]

I was getting the impression that that is how Anilkumar was thinking - that relativity is nothing more than a mathematical illusions that does not apply to reality as we experience it, i.e. his constant references to space not equalling spacetime etc.

 

My position is that relativity does indeed describe reality as we experience it but that we cannot directly perceive its full richness due our limited senses.

There are Mathematical models. There are Mathematical concepts. But no such thing called 'Mathematical illusion' exists. Mathematics is incapable of telling lie.

[Greg Boyles]

There are Mathematical models. There are Mathematical concepts. But no such thing called 'Mathematical illusion' exists. Mathematics is incapable of telling lie.

 

 

I am well aware of that Anilkumar. But you were giving me the impression that you believed that mathematics told lies as far warped spacetime that does not tally with your every day experience of the universe.

 

You are obviously not an entirely strong english speaker/writer so perhaps the point you have really been trying to make has been lost on me and we have been debating across purposes.

Edited December 16, 2011 by Greg Boyles

[Anilkumar]

I am well aware of that Anilkumar. But you were giving me the impression that you believed that mathematics told lies as far warped spacetime that does not tally with your every day experience of the universe.

 

You are obviously not an entirely strong english speaker/writer so perhaps the point you have really been trying to make has been lost on me and we have been debating across purposes.

I don't know how you would define 'Strong English'. But,

 

you are the only Pal who has got the impression that I believed, that mathematics told lies.

 

I think that speaks for both of us.

 

But petty things aside,

 

we have had a very good discussion & your contribution to it is invaluable. You have worked hard to convince me about what you believed is right and where you felt I was making mistakes. Your basic motive was to see that right is not suppressed but upheld and there are no wrong beliefs. That motive is worthy of salutation. However,

 

lets concentrate on what you, me & everybody loves the most.

 

Seeking knowledge.

Edited December 16, 2011 by Anilkumar

[Anilkumar]

Your description is incomplete. At the risk of sounding like a broken record, let me say again: According to general relativity, the gravitational field IS the warping of space and time or spacetime curvature.

 

A clock runs slower as it gets closer to the Earth's surface. The distance between the same two points stretches as the points approach the surface of the Earth (points perpendicular to the Earth's surface as seen from far away). In Einstein's construct, this warping of time and space is what tells objects how to move in a gravitational field. It is what is holding me down in my chair right now as I write this. So all the physical effects we attribute to the term "gravity" are produced by the warping of space (change in length or space interval) and warping of time (change in time interval). Thus the warping of space and time is gravity.

 

By the way, this change in the space and time intervals is represented mathematically in general relativity by the change in the spacetime interval. The spacetime interval is the square root of the difference between the square of the time interval and the square of the space interval (ignoring cross-terms).

 

I suggest you look up frame dragging and the latest Gravity Probe B results. Here the spinning of the Earth on its axis caused gyroscopes orbiting the Earth to change their spin orientation (very slightly). Why? How can the spinning of the Earth affect the orientation of gyroscopes which are themselves rotating in "empty" space? Because, per general relativity, the spinning of the Earth drags empty space around with it as it rotates.

 

The confirmation of this so-called frame dragging by Gravity Probe B (to about 19%) shows that space IS flexible, is warpable. No matter what we say about how crazy this is -- we now have emprical confirmation of this strangest of phenomena.

 

Link: http://einstein.stan...ts/status1.html

 

Hello IM Egdall & hello everybody,

 

You will not believe whatever I will say now.

 

I am pretty convinced now.

 

That Space warp does not happen.

 

I am not saying GR is wrong.

 

But a couple of the conclusions of GR are wrong.

 

And Space warp is one among them.

 

GR is the greatest theory that mankind has ever found.

 

If Albert Einstein hadn't done it, I don't think anyone could have done it, because it requires the highest level of perceptive capability that any human being can ever demonstrate. Einstein had that implausible intelligence. Yet he was human. The quality of humans, to err, was also in him.

 

Albert Einstein made a mistake and the Relativists have been carrying it forward.

 

Space does not warp.

 

And I will prove it here.

 

I won't be giving a mathematical proof.

 

It is just going to be suggestive, hint like. But logical & rational.

 

It is for the adequately placed logical thinkers & scientists to promote it, if they find any substance in what I say.

 

This thread will go down in history as the one which brought a correction to GR.

 

And we will be part of it.

 

I shouldn't be saying all this before proving it. But I am jubilant.

 

Jubilant that, at last I am able to backup, what I felt was right. My voice, though right in what it was saying, was till now feeble. Now I am relieved, at last, that I can say what I want to say, without hesitation, in a confident manner.

 

Enough of boasting; I know the more I blow my own horn, the more rotten tomatoes, I will have to face, if proven wrong. But there is no chance of that. Space can't be warped.

 

I will be back in a few hours. I am preparing the post as fast as I can.

 

Thank you.

Edited December 17, 2011 by Anilkumar

[Greg Boyles]

Hello IM Egdall & hello everybody,

 

You will not believe whatever I will say now.

 

I am pretty convinced now.

 

That Space warp does not happen.

 

I am not saying GR is wrong.

 

But a couple of the conclusions of GR are wrong.

 

And Space warp is one among them.

 

GR is the greatest theory that mankind has ever found.

 

If Albert Einstein hadn't done it, I don't think anyone could have done it, because it requires the highest level of perceptive capability that any human being can ever demonstrate. Einstein had that implausible intelligence. Yet he was human. The quality of humans, to err, was also in him.

 

Albert Einstein made a mistake and the Relativists have been carrying it forward.

 

Space does not warp.

 

And I will prove it here.

 

I won't be giving a mathematical proof.

 

It is just going to be suggestive, hint like. But logical & rational.

 

It is for the adequately placed logical thinkers & scientists to promote it, if they find any substance in what I say.

 

This thread will go down in history as the one which brought a correction to GR.

 

And we will be part of it.

 

I shouldn't be saying all this before proving it. But I am jubilant.

 

Jubilant that, at last I am able to backup, what I felt was right. My voice, though right in what it was saying, was till now feeble. Now I am relieved, at last, that I can say what I want to say, without hesitation, in a confident manner.

 

Enough of boasting; I know the more I blow my own horn, the more rotten tomatoes, I will have to face, if proven wrong. But there is no chance of that. Space can't be warped.

 

I will be back in a few hours. I am preparing the post as fast as I can.

 

Thank you.

 

How do you explain the experiments with sychronised atomic clocks, one of which is taken into space for a period of time and then both compared when brought back together. There is always a time difference which confirms the effect of gravity on time predicted by GR.

[Anilkumar]

How do you explain the experiments with sychronised atomic clocks, one of which is taken into space for a period of time and then both compared when brought back together. There is always a time difference which confirms the effect of gravity on time predicted by GR.

Hello dear Greg Boyles,

 

I have never said ‘Time does not warp’.

 

Throughout the thread I have been saying ‘Gravity affects Motion & Duration’.

 

I have said Mass can not affect Space other than occupying/filling it. Space can’t warp. As it does not have that capability.

[StringJunky]

 

I have said Mass can not affect Space other than occupying/filling it. Space can’t warp. As it does not have that capability.

 

Explain how Einstein Rings occur if space is not warped by mass. This is a Double Einstein Ring:

 

 

This is an image of gravitational lens system SDSSJ0946+1006 as photographed by Hubble Space Telescope's Advanced Camera for Surveys. The gravitational field of an elliptical galaxy warps the light of two galaxies exactly behind it. The massive foreground galaxy is almost perfectly aligned in the sky with two background galaxies at different distances. The foreground galaxy is 3 billion light-years away, the inner ring and outer ring are comprised of multiple images of two galaxies at a distance of 6 and approximately 11 billion light-years. The odds of seeing such a special alignment are estimated to be 1 in 10,000. The right panel is a zoom onto the lens showing two concentric partial ring-like structures after subtracting the glare of the central, foreground galaxy.

 

http://hubblesite.org/newscenter/archive/releases/exotic/gravitational%20lens/2008/04/image/a/

Edited December 17, 2011 by StringJunky

[michel123456]

Impatients. Why are you all attacking Anilkumar before he posts his hypothesis. He wrote

I will be back in a few hours. I am preparing the post as fast as I can.

Thank you.

Let's wait.

[Greg Boyles]

Impatients. Why are you all attacking Anilkumar before he posts his hypothesis. He wrote

 

Let's wait.

 

Are you seriously inending to single handedly replace general relativity and fundamentally alter our understanding of the universe?

 

You are either an undiscovered genius or just another Hutchinson.

[Mellinia]

Side note: Mass warps space time...at times :"only time is warped".

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved.

~http://en.wikipedia.org/wiki/General_relativity

 

just in case

[Anilkumar]

Explain how Einstein Rings occur if space is not warped by mass. This is a Double Einstein Ring:

 

 

This is an image of gravitational lens system SDSSJ0946+1006 as photographed by Hubble Space Telescope's Advanced Camera for Surveys. The gravitational field of an elliptical galaxy warps the light of two galaxies exactly behind it. The massive foreground galaxy is almost perfectly aligned in the sky with two background galaxies at different distances. The foreground galaxy is 3 billion light-years away, the inner ring and outer ring are comprised of multiple images of two galaxies at a distance of 6 and approximately 11 billion light-years. The odds of seeing such a special alignment are estimated to be 1 in 10,000. The right panel is a zoom onto the lens showing two concentric partial ring-like structures after subtracting the glare of the central, foreground galaxy.

 

http://hubblesite.or...008/04/image/a/

I am compiling it, dear fellow.

 

I am trying as much as I can to put my thoughts in precise, concise & simple words. So that my post doesn’t give any superfluous meanings, by which you could make me frequently edit it by adding or deleting words and let you have a proud smile.

 

Don’t be harsh on me fellows. But take it from me; I will face all the rage thrown at me but see that you won’t crumple that Space again. You can crumple me instead, if you like.

 

We all know truth is above all of us & everything in this universe, past present & future, put together.

 

By the way; yesterday night I had a dream. Someone was chasing me with a baseball bat in hand.

 

 

 

Impatients. Why are you all attacking Anilkumar before he posts his hypothesis. He wrote

 

Let's wait.

Thanks michel123456. I won't let you down.

[swansont]

!

Moderator Note

Anilkumar 's explanation has been split into a new thread http://www.scienceforums.net/topic/62342-ordinary-mans-attempt-to-explain-gr/

Any further discussion along the lines of "GR is wrong" should/will go there

[DrRocket]

Hello dear Greg Boyles,

 

I have never said 'Time does not warp'.

 

Throughout the thread I have been saying 'Gravity affects Motion & Duration'.

 

I have said Mass can not affect Space other than occupying/filling it. Space can't warp. As it does not have that capability.

 

In general relativity it is spacetime, not "space" that shows curvature. In fact in general relativity there is no such thing, on a global basis, as either time or space. Time and space are purely local notions, and are dependent on the arbitrary selection of a local coordinate system.

 

Mass and more precisely mass/energy not only warps spacetime, but determines the curvature of spacetime via the stress-energy tensor and the Einstein field equations.

 

The statement that "Gravity affects Motion & Duration" is meaningless babble. It is not even wronog, just nonsense words.

 

If you want to understand what general relativity really says, ralther than make a ridiculous criticism of a theory that you manifestly don't begin to understand, then you need to read a real book on the subject. Gravitation by Misner, Thorne and Wheeler would be a good start.

[Widdekind]

space-time "wants" to be flat ?

 

I understand, that, quantum mechanically,

 

"Energy is not characteristically a local phenomenon. The nature of energy is to expand, move, or propagate... Energy of any type disperses, from being localized to becoming spread out, if it is not constrained" (dickau)

Naively, if concentrated mass-energy curves space-time; but if concentrated mass-energy "seeks" to spread out through space-and-time; then, according to a crude combination of QM & GR, space-time "likes" being flat, and "hates" being curved ?? I.e. space-time "seeks" to "un-dent" & "un-dimple", a little like a self-re-forming car body, which "desire to un-dent" drives QM wave-functions to disperse ??

 

 

 

 

Just because you can't see or touch the fabric of space time does not amount to evidence that a fabric does not exist...

 

The fabric of space time is similar. When a massive object is introduced it warps, rather than displaces, the fabric of space-time.

spacetime is very real.

The "stage" (space-time) is as ontologically real, as the "actors" (fermions & bosons) ? Space-time is a "mesh", a fabric-like "weave", within which matter & energy is embedded, and into whose "weave" matter & energy is coupled, i.e. "hooked in" ? J.A.Wheeler wrote (Journey into Gravity & Spacetime) that matter "grips" space-time, giving a vague impression, of "rock-climbers" being "roped in" to a vast cargo-net-like "rigging", which rigging is distorted, as particles traverse it, cp. Gravity Probe B results, indicating that the "mesh" of space-time can be "twirled" by our spinning world ?

 

If our earth is 5 billion years old; then by how many degrees, has our earth "twirled" the local space-time fabric ? In animations, earth is depicted orbiting our sun. So, does the fabric of space-time "un-twirl", once earth "spins past" some particular place ??

Edited January 25, 2012 by Widdekind

[Santalum]

A straight line is only shortest distance between any two points. How can there be more than one straight line?

Thanks.

I believe that in Euclidian geometry (of flat spaces) then yes, the shortest distance between to points is a straight line.

 

But in non-Euclidian geometry (e.g. of a curved surface such as that of the earth) the shortest distance between to points is effectively an arc rather than a straight line.

 

So 4 dimensional space-time is defined by non-Euclidian geometry I presume, in which case your above assumption is false.

[Widdekind]

The geometry of space is inferred or extrapolated from the observed behaviour of bodies within it.

If space was 'nothingness' it could not bend!

...

The expanding universe says that the space is expanding; the question thus remains, where is the space expanding to

Intuitively, "reifying" space-time, as a real "thing", with real "substance", within which all matter-and-energy resides, like "insects encased in amber", seems plausible.

 

The presence of mass (and energy) causes space and time to warp...

 

Space warp

: Imagine two points in space where all objects are so far away that gravity is virtually zero. Now imagine placing the Earth near these two points. As seen from far away, the two points will now be a different distance apart. This in essence is space warp or space curvature.

 

Time warp

: Imagine a clock also in space where all objects are so far away that gravity is virtually zero. Now again imagine placing the Earth near this clock. The clock now runs slower. due to the Earth's presence. This is called time warp or curvature of time.

Together time warp and space warp are called spacetime curvature. And this spacetime curvature determines the path of the Moon around the Earth, and all the planets around the Sun etc., and holds us to the surface of the Earth. Spacetime curvature IS gravity.

Thank you very much for the clear-if-qualitative explanation. Also thank AJB very much for the Sean M. Carroll reference.

[Jiggerj]

Everytime I try to wrap my head around space-time illustrations someone drops a heavy ball on a stretched out blanket, and then rolls a smaller ball onto the curve in the blanket made by the big ball. This would be fine if all the universe were flat as a sheet with all the mass floating on top of it. But, isn't everything in the universe completely submerged in the universe? If so, how can the ball on the blanket serve as a valid example of space-time?