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Author Topic: What is the meaning of "Spacetime Curvature"?  (Read 11901 times)

Offline Pmb

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What is the meaning of "Spacetime Curvature"?
« on: 04/07/2013 01:42:11 »
I'm curious as to what the average Joe thinks when he hears the term spacetime curvture. What did think when you heard of it the first time. By this I mean what did the concept bring to mind when you first learned about it? If you've studied it for some time now then what do you think it means now as compared to when you first started learning it?

My impression is that many people believe that spacetime curvature refers to the curving of a worldline in spacetime caused by a gravitational field. Am I close? I, of course, know what it means. I just want to know if people have the right idea about it. It seems to me that when people think of spacetime curvature and the phrase gravity is a curvature in spacetime that they think that the gravitational field we all know and love and have grown up in and live with all of our lives are the same thing.

What do you think? And than you for your thoughts in advance. :)
« Last Edit: 04/07/2013 22:50:37 by chris »


 

Offline galaxysim

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Re: The Meaning of "Spacetime Curvature"
« Reply #1 on: 04/07/2013 02:09:01 »
from my experience on other forums....and my oft quoted phrase 'cant count or wont count ? '  im pretty sure the 90% and perhaps 95%+ of the populace suffer brain implosion the moment things go beyond their tangible experience


You have to have what i call the maths gene AND an ability to carry out thought experiments AND a good instinct for self checking your progress.

Most lay peoples knowledge comes from opinion, rather than an internal matchstick pyramid of knowledge that they have painstakingly built from the ground up....this is why 'does not compute' is a common theme. Its very easy to get confused if you don't have the guardrails of science based education.

The mindset required to 'get laid and paid' is of a very different architecture to that of the scientific method. Thus if your new to science it can take some time to wrap your head around what might be a relatively straight forward scientific concept...this is of course frustrating for all parties in the discussion....people are impatient and the human mind tends to over simplify in when facing complexity.

A long winded reply to your OP

I think very few peoples understanding of Spacetime Curvature goes beyond ye olde cannon ball and rubber mat
http://en.wikipedia.org/wiki/Spacetime [nofollow]

 Joe Average meets the train in a tunnel paradox....paradox wins every time.



@Pmb says your a physicist ? curious what field ? casual or pro

....Mine is AI and general science Astronomy to zoology






« Last Edit: 04/07/2013 02:29:35 by galaxysim »
 

Offline Pmb

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Re: The Meaning of "Spacetime Curvature"
« Reply #2 on: 04/07/2013 20:57:32 »
Quote from: galaxysim
A long winded reply to your OP

I think very few peoples understanding of Spacetime Curvature goes beyond ye olde cannon ball and rubber mat
http://en.wikipedia.org/wiki/Spacetime
That doesnít tell me what they think it is though. For example; take a look at the uniform gravitational field inside a spherical cavity that was hollowed out from a spherical homogenous distribution of matter. The center of the cavity is offset from the center of the sphere.

http://home.comcast.net/~peter.m.brown/gr/grav_cavity.htm

Is the spacetime associated with the gravitational field inside the cavity curved? Hint: The field inside the cavity is uniform, i.e. the tidal force tensor is zero. If you donít know what that tensor is then please see

http://home.comcast.net/~peter.m.brown/mech/tidal_force_tensor.htm

Quote from: lean bean
Is this leading up to affine connections?
Not really but thanks for asking. Iím not sure whether the connection coefficients are geometric objects or not.

Quote from: lean bean
I was surprised to read Edwin Taylor say of John wheeler non-use of tensors.
Yeah. I know that about Edwin. I can never talk to him about anything to do with tensors. I think that after a while he developed a phobia about them. The same thing happened to me with the multiplication table. Lol!
 

Offline yor_on

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Re: The Meaning of "Spacetime Curvature"
« Reply #3 on: 04/07/2013 21:05:54 »
The way matter moves seems to be the best answer there, but it also connects to what you define as a universe, and its dimensions. As for a space without gravity? Can you get both? Defining it as the way matter moves, and also draw 'boundaries', as a ball shaped 'universe'. What defines that 'ball', a 'flat space'? The universe we have seem to 'stretch out', but if we use the 'ball' as our referee, the first thing to question should be. What defines that 'ball'? mass-'energy'? Matter? Gravity? Depending on choice, as for example me expecting gravity being the metric defining a 'shape', one then have to define what this stretched out universe is defined by, gravity too? If that is so, what is a 'flat space'?
 

Offline Pmb

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Re: The Meaning of "Spacetime Curvature"
« Reply #4 on: 04/07/2013 21:11:40 »
You didn't answer my question. There are only three possible responses to my question

Is the spacetime corresponding to the gravitationa field inside the cavity curved?

(1) Yes
(2) No
(3) I don't know
 

Offline yor_on

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Re: The Meaning of "Spacetime Curvature"
« Reply #5 on: 04/07/2013 21:23:43 »
How about a fifth?
5. it depends on the distribution of matter, and also from 'where' you define it, as well as what scale you use :)

You make hard questions Pete. You yourself has pointed out that you can transform away gravity, by 'free falling'. That's why I haven't found any better approximation for it than a 'preferred direction'. But 'free falling' and a 'flat space' is still questions unanswered to me, wondering what defines a universe. If gravity defines a shape, and have a infinite reach, then it shouldn't matter what 'shape' we define a universe to, should it? And whatever is meant by a 'flat space' it still must contain that metric, as I think for now, to exist.
 

Offline yor_on

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Re: The Meaning of "Spacetime Curvature"
« Reply #6 on: 04/07/2013 21:47:51 »
Ahh sorry, I was still stuck on your first there.  The Meaning of "Spacetime Curvature"  . As for the later, I would assume the space to be 'flat' and so 'non curved', but still containing a gravitational metric. And no, I don't think a space need to be curved as long as we have matter defining gravity. But it seems that some define a 'flat space' as something not using the metric of gravity? And that one is weird to me.
 

Offline Pmb

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Re: The Meaning of "Spacetime Curvature"
« Reply #7 on: 04/07/2013 22:33:09 »
Ahh sorry, I was still stuck on your first there.  The Meaning of "Spacetime Curvature"  . As for the later, I would assume the space to be 'flat' and so 'non curved', but still containing a gravitational metric. And no, I don't think a space need to be curved as long as we have matter defining gravity. But it seems that some define a 'flat space' as something not using the metric of gravity? And that one is weird to me.
Do you understand that spacetime curvature is merely a fancy name for tidal force?
 

Offline Bill S

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #8 on: 04/07/2013 23:16:23 »
Quote
Do you understand that spacetime curvature is merely a fancy name for tidal force?

No;  but I would like to!
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #9 on: 04/07/2013 23:30:55 »
Quote from: Bill S
No;  but I would like to!
Okay. It will help me if I knew where you are with math, Newtonian mechanics, gravity, tensors etc. I.e. can you follow these two pages
http://home.comcast.net/~peter.m.brown/mech/tidal_force_tensor.htm
http://home.comcast.net/~peter.m.brown/gr/geodesic_deviation.htm

See if you can find the subject in Taylor and Wheeler's new version of Exploring Black Holes at http://exploringblackholes.com/
« Last Edit: 04/07/2013 23:33:52 by Pmb »
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #10 on: 05/07/2013 00:04:08 »
Quote from: galaxysim
I think very few peoples understanding of Spacetime Curvature goes beyond ye olde cannon ball and rubber mat
http://en.wikipedia.org/wiki/Spacetime
Strangely enough those diagrams are really embedding diagrams and weren't supposed to represent spacetime curvature unfortunately it didn't work out that way.
 

Offline yor_on

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #11 on: 05/07/2013 11:00:23 »
Sure. everything defining a direction is 'gravity' Pete, and it's no 'force' either, as I get it. And all this about gravitomagnetic forces is just confusing to me, it splits gravity into two parts as I get it. One part of it being some 'component' making you anchored to earth (analogue to electricity) the other treating tidal forces (magnetic part), suggesting you can see it as some sort of electromagnetism. You can't have it as a force, as far as I see it's just a preferred direction. And we're all following geodesics. A force is what intrudes on a geodesic in this case, breaking it.
=

Who 'invented' gravitomagnetism btw? I just can't make me believe it was Einstein that created that slogan, someone clever did it, and it has caught on, but it don't fit Einsteins thoughts, to me that is.
« Last Edit: 05/07/2013 11:07:06 by yor_on »
 

Offline yor_on

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #12 on: 05/07/2013 11:25:36 »
Ahh, think I found him?

http://en.wikipedia.org/wiki/Gravitoelectromagnetism

"The analogy and equations differing only by some small factors were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law."

http://en.wikipedia.org/wiki/Oliver_Heaviside

On the other hand, this was before GR, so some must have reused it, banking into skulls using equations for electromagnetism, as presumed of a idea of all 'forces' coming from one origin, at some temperature for example (regime / symmetry breaks). If fits a Higg explanation, possibly? But I still don't see it fit Einsteins GR?
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #13 on: 05/07/2013 18:15:27 »
Sure. everything defining a direction is 'gravity' Pete, and it's no 'force' either, as I get it.
Not according to Einstein. GR states gravity is an inertial force rather than a 4-force like the Lorentz force.

For more on inertial forces and how some physicists percieve them as being "real" please see the web page I created on this subject at
http://home.comcast.net/~peter.m.brown/gr/inertial_force.htm0

For example please see the quote on that page from Introducing Einstein's Relativity, by Ray D'Inverno, Oxord/Clarendon Press, (1992) page 122
Quote
Notice that all inertial forces have the mass as a constant of proportionality in them. The status of inertial forces is again a controversial one. One school of thought describes them as apparent or fictitious which arise in non-inertial frames of reference (and which can be eliminated mathematically by putting the terms back on the right hand side). We shall adopt the attitude that if you judge them by their effects then they are very real forces. [Author gives examples]
Other quotes are like that.

Here's what Einstein said about this point

From an article in the February 17, 1921 issue of Nature by Albert Einstein
Quote
Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" gravitational field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia from the point of view of a system not taking part of the rotation can be interpreted as gravitation when considered with respect to a system that shares this rotation. According to Newton, this interpretation is impossible, because in Newton's theory there is no "real" field of the "Coriolis-field" type. But perhaps Newton's law of field could be replaced by another that fits in with the field which holds with respect to a "rotating" system of co-ordinates? My conviction of the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in the correctness of this interpretation.


 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #14 on: 05/07/2013 18:17:17 »
Sure. everything defining a direction is 'gravity' Pete, and it's no 'force' either, as I get it.
Everything? The electric field defines a direction. Are you saying that the electric field is a gravitational field?

Anyway, not according to Einstein (and me) it's not. GR states gravity is an inertial force rather than a 4-force like the Lorentz force.

For more on inertial forces and how some physicists percieve them as being "real" please see the web page I created on this subject at
http://home.comcast.net/~peter.m.brown/gr/inertial_force.htm

For example please see the quote on that page from Introducing Einstein's Relativity, by Ray D'Inverno, Oxord/Clarendon Press, (1992) page 122
Quote
Notice that all inertial forces have the mass as a constant of proportionality in them. The status of inertial forces is again a controversial one. One school of thought describes them as apparent or fictitious which arise in non-inertial frames of reference (and which can be eliminated mathematically by putting the terms back on the right hand side). We shall adopt the attitude that if you judge them by their effects then they are very real forces. [Author gives examples]
Other quotes are like that.

Here's what Einstein said about this point

From an article in the February 17, 1921 issue of Nature by Albert Einstein
Quote
Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" gravitational field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia from the point of view of a system not taking part of the rotation can be interpreted as gravitation when considered with respect to a system that shares this rotation. According to Newton, this interpretation is impossible, because in Newton's theory there is no "real" field of the "Coriolis-field" type. But perhaps Newton's law of field could be replaced by another that fits in with the field which holds with respect to a "rotating" system of co-ordinates? My conviction of the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in the correctness of this interpretation.


« Last Edit: 05/07/2013 19:35:56 by Pmb »
 

Offline niebieskieucho

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #15 on: 06/07/2013 00:18:14 »
I'm curious as to what the average Joe thinks when he hears the term spacetime curvture. What did think when you heard of it the first time. By this I mean what did the concept bring to mind when you first learned about it? If you've studied it for some time now then what do you think it means now as compared to when you first started learning it?
It's a double rubbish. Spacetime is a sick idea. It's just space that has nothing to do with time (of what BTW?), as time spontaneously does not exist. As to the second rubbish, i.e. alleged curvature of space, it's simply impossible. Space does not undergo deflection, is indestructible, does not expand, is of the same volume as it was before origin of matter.
Quote
My impression is that many people believe that spacetime curvature refers to the curving of a worldline in spacetime caused by a gravitational field. Am I close? I, of course, know what it means. I just want to know if people have the right idea about it. It seems to me that when people think of spacetime curvature and the phrase gravity is a curvature in spacetime that they think that the gravitational field we all know and love and have grown up in and live with all of our lives are the same thing.

What do you think? And than you for your thoughts in advance. :)
The only curvature of space I can accept is due to spherical shape of the (finite) universe. But this is its natural shape.
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #16 on: 06/07/2013 00:54:51 »
Quote from: niebieskieucho
It's a double rubbish. Spacetime is a sick idea. It's just space that has nothing to do with time (of what BTW?), as time spontaneously does not exist.
From these comments its clear that you donít know what spacetime is. Please learn about these things before you make another attempt at commenting on them. I.e. please study what spacetime is. I created the following for this purpose
http://home.comcast.net/~peter.m.brown/sr/spacetime.htm

Whether spacetime is curved or not is not up for debate. Itís an observable fact. Itís a measurable phenomena. But until you learn what it really is as opposed to what you think it is (which youíve proven that what you think it is is wrong) you wonít be able to understand that very simple fact. If you had chosen to first learn about the subject that youíre criticizing then you wouldnít have made these mistakes. I showed you where to read about it online. Please do so before you make another attempt to argue that itís wrong.

Quote from: niebieskieucho
As to the second rubbish, i.e. alleged curvature of space, it's simply impossible. Space does not undergo deflection, is indestructible, does not expand, is of the same volume as it was before origin of matter.
These comments also tell me that you donít know what space curvature is. You incorrectly assumed that it meant that space is deflected. It doesnít. It has to do with the measured distance between various points in space. The amount of deflection of star light by the sun is a measure by how much space is altered by the sunís gravitational field.


Quote from: niebieskieucho
My impression is that many people believe that spacetime curvature refers to the curving of a worldline in spacetime caused by a gravitational field. Am I close?
No. Youíre way off. A charged particle moving in flat spacetime in an electric field will have a worldline that curves. That in no way shape or form means that the spacetime is curved.

If the particleís worldline is a geodesic (i.e. the 4-force on it is zero Ė only inertial forces are acting on it) and it curves then all that tells you is that youíre observing the motion of the particle from a non-inertial frame of reference. Spacetime curvature pertains to the divergence of two geodesics which start out parallel and deviate. See
http://home.comcast.net/~peter.m.brown/gr/geodesic_deviation.htm

Quote from: niebieskieucho
The only curvature of space I can accept is due to spherical shape of the (finite) universe.
Then why not choose to learn about it and learn what it really means as opposed to what you think it means.

If you really want to learn what spacetime curvature is then read Exploring Black Holes at http://exploringblackholes.com/

Or read
http://home.comcast.net/~peter.m.brown/sr/sr.htm
http://home.comcast.net/~peter.m.brown/gr/gr.htm

I know itís a lot but nobody said that learning about spacetime curvature could be easy to learn.

Unless you really donít want to learn it?


I'm sorry if I come across as being rude to you. I don't mean to be. I simply get irritated when people claim that things are wrong when it's also clear that they've never learned what it is in the first place.
« Last Edit: 06/07/2013 00:58:27 by Pmb »
 

Offline yor_on

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #17 on: 06/07/2013 01:10:54 »
"The electric field defines a direction. Are you saying that the electric field is a gravitational field?"

No, I was talking about what you discussed, 'gravity', but there are several descriptions to it. gluing a name to it, as gravitomagnetism, doesn't catch it for me though. I think I prefer it as a direction so far :)
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #18 on: 06/07/2013 01:51:48 »
"The electric field defines a direction. Are you saying that the electric field is a gravitational field?"

No, I was talking about what you discussed, 'gravity', but there are several descriptions to it. gluing a name to it, as gravitomagnetism, doesn't catch it for me though. I think I prefer it as a direction so far :)
Gravitomagnetism refers to a specific gravitational effect. It's about a velocity dependant gravitational force just like the magnetic force is velocity dependant. The direction of the gravitational force depends on the direction of the velocity of the particle. So what direction are you talking about?
 

Offline yor_on

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #19 on: 06/07/2013 01:55:21 »
Fictitious force is a very nice description of it Pete. As you can transform it away by a free fall. If I assume that all matter in the universe has a preferred direction (geodesic) in where no forces act on it, then I suppose this should be 'gravity'. And that should be a preferred direction to me, or 'SpaceTime curvature'. Even though uniform motion is a relative motion, we still can use fixed stars, CBR, etc to define a direction for it. But if I go strictly by what 'relative motion' state, that you can redefine any (uniform) motion by just changing a reference frame? Then I'm not sure how to describe the 'direction' of matter, because you need a 'motion' to have a direction. And that one is intuitively weird to me, and in some way more correct, although I can prove motion just by introducing more uniformly moving objects in a 'system' and so be able to define different speeds relative those objects. If I can find that, then motion should exist, as I think for now :)
 

Offline yor_on

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #20 on: 06/07/2013 02:11:16 »
Maybe one could turn it around and state that as I feel a gravity, standing on earth, then that too must be a proof for a preferred direction, or geodesic, existing. And if matter tells space how to 'bend', and space tells matter how to 'move' then we all are in a geodesic, having a preferred direction, defined by all matter in a universe, or locally defined (earth) if one want, defined by the matter close to me. But if gravity has a infinite reach the first statement should be more correct I think?
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #21 on: 06/07/2013 02:46:14 »
Fictitious force is a very nice description of it Pete. As you can transform it away by a free fall.
Here's the problem with that. It has given the false impression that such a field cannot be caused by matter, which is false. People think that if it's fictitious then it can't be caused by a gravitational source. That's what happens when you label things as fictitious. You can call it what you like. I prefer to call it real because it is real. You can't tell the difference. Calling it fictitious because you can transform it away is like saying that an electric field is fake because you can transform that away too.
 

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #22 on: 06/07/2013 09:45:27 »
Fictitious force is a very nice description of it Pete. As you can transform it away by a free fall.

Here's the problem with that. It has given the false impression that such a field  cannot be caused by matter, which is false. People think that if it's fictitious then it can't be caused by a gravitational source. That's what happens when you label things as fictitious. You can call it what you like. I prefer to call it real because it is real. You can't tell the difference.

Pete
 What do you mean by field here "It has given the false impression that such a field cannot be caused by matter, which is false.''

Are you saying...the region of altered spacetime in the vicinity of mass is real, but locally it can be transferred away?
 To make it clearer for me, and if it helps others, is this field you refer to Newton's gravitational force of attraction or a region of altered spacetime? Evidence of this altered region of spacetime being accelerations and the converging together of test particles (originally on parallel paths) as they move towards the centre of mass of a planet?
« Last Edit: 06/07/2013 16:55:45 by lean bean »
 

Offline Pmb

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #23 on: 06/07/2013 17:12:00 »
Quote from: lean bean
Pete
 What do you mean by field here "It has given the false impression that such a field cannot be caused by matter, which is false.''
Iíve been discussing general relativity with people for over a decade now. It seems that people think of spacetime curvature as being what we normally think of as a gravitational field (i.e. things fall when we drop them at a rate which is independent of their mass). However that is not the case. But people have come to expect that when there is a body falling from the gravitational acceleration caused by a planet it is due to spacetime curvature (Riemman tensor) when actually itís due to whatís called the affine connection. Since people have the wrong idea about it Iíve found that in almost all circumstances people have expected spacetime curvature where there shouldnít be any. The example I always use is the uniform gravitational field. Such a field is actually defined as having no spacetime curvature yet since people are always associating spacetime curvature with gravity they also expect there to be spacetime curvature in a uniform gravitational field. And Iím not talking about the layman. Iím talking about experts. This occurred in the American Journal of Physics as well as in Gravitation by Misner, Thorne and Wheeler (MTW). In the case of MTW they wrongly assert that gravitational redshift implies spacetime curvature, which it doesnít. In fact the very first calculation of gravitational redshift, by Einstein of course, used flat spacetime!

Quote from: lean bean
Are you saying...the region of altered spacetime in the vicinity of mass is real, but locally it can be transferred away?
Not just locally but in a finite region of spacetime. The term local means a small region. The region need not be small when the field is a uniform gravitational field. However the equivalence principle states that the gravitational field (the affine connection) can always be transformed away whereas spacetime curvature cannot.

Quote from: lean bean
To make it clearer for me, and if it helps others, is this field you refer to Newton's gravitational force of attraction or a region of altered spacetime?
You seem to be indicating that there is a difference between Newton and GR in this area and there isnít. The area of physics where Newton differs from Einstein is not the gravitational force, because that also exists in GR (see http://home.comcast.net/~peter.m.brown/gr/grav_force.htm). Its in the area of the field e equations and the fact that Newtonís gravitational field acts instantaneously with no delay in propagation time between a change in the distribution of matter to a change in the gravitational field.

Quote from: lean bean
Evidence of this altered region of spacetime being accelerations and the converging together of test particles (originally on parallel paths) as they move towards the centre of mass of a planet?
That exact same thing happens in Newtonian physics too you know?

Quote from: lean bean
Am I wasting my time asking direct questions of you? :)
I'm sorry but I don't understand this question. Why would it be a waste of time? Donít you think I'm capable of answer your questions?
 

Offline galaxysim

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Re: What is the meaning of "Spacetime Curvature"?
« Reply #24 on: 06/07/2013 20:08:54 »
ref What do you think? And than you for your thoughts in advance. :)

I think you should make the effort to make a series of video shorts, or at the very lest build up a link of video shorts you can send people off to watch. Assuming your mission is education of the many ?

pages of text and diagrams is not only dry but a poor communication medium...it might suit some people in some circumstances but people tend not to posses the mindset of a low grade bank clerk...which is what you would have to have to enjoy wading through reams of dry text and line drawings

Far too many intellectuals have poor multi media and communication skills
Thus much of that knowledge remains buried in books or the halls of academia

The mind that can understand deeply is often hampered by the inability to communicate widely

We are much like ants, and the intrinsic complexity of the universe and our finite brain capacity means overspecialization has its downsides.

Sagan, Feynman & Neil deGrasse Tyson the exception rather than the rule....i think thats a great shame for all of us

Dr. Neil DeGrasse Tyson: A fascinatingly disturbing thought

Dawkins & Tyson on Intelligent Aliens


communication the other half of the 'lack of wisdom problem'



« Last Edit: 06/07/2013 20:34:32 by galaxysim »
 

The Naked Scientists Forum

Re: What is the meaning of "Spacetime Curvature"?
« Reply #24 on: 06/07/2013 20:08:54 »

 

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