geordief Posted July 7, 2016 Posted July 7, 2016 (edited) I should explicitly point out here that the "usual sense of the word" curvature is not the type of curvature that is used in GR at all. Should there be a different term to describe "intrinsic curvature" if the only similarity to what one might think of as normal curvature seems only to confuse? It seems to be related to the way distances (however they are defined ) change gradually in a smooth (I forget the precise meaning of "smooth" ) manifold. Would it be reasonable to say that any real object in nature that can be modeled by a manifold is in fact always intrinsically curved to some extent and only mathematical manifolds can be absolutely flat? I understand that the observable universe is supposed to be "flat" on the macro level... Edited July 7, 2016 by geordief
ajb Posted July 7, 2016 Posted July 7, 2016 Should there be a different term to describe "intrinsic curvature" if the only similarity to what one might think of as normal curvature seems only to confuse? We use extrinsic curvature to mean the 'curvature' associated with the embeding of the manifold into a higher dimensional Euclidiean space. For example, the cylinder has extrinsic curvature because of the way it is usually presented in R^3. However, the cylinder has zero intrinsic curvature - in is really flat! (Not that I have defined curvature here)
Markus Hanke Posted July 7, 2016 Posted July 7, 2016 Should there be a different term to describe "intrinsic curvature" if the only similarity to what one might think of as normal curvature seems only to confuse? The term has a mathematically precise definition, so I'm not sure what you are getting at. The word "curvature" is actually just an umbrella term for a number of related, but quite distinct mathematical concepts. It seems to be related to the way distances (however they are defined ) change gradually in a smooth (I forget the precise meaning of "smooth" ) manifold. Yes, pretty much. Would it be reasonable to say that any real object in nature that can be modeled by a manifold is in fact always intrinsically curved to some extent and only mathematical manifolds can be absolutely flat? I'm not sure I would agree with that. Even something as simple as a sheet of paper is intrinsically flat to a pretty high degree of accuracy.
geordief Posted July 7, 2016 Posted July 7, 2016 (edited) geordief, on 07 Jul 2016 - 10:59 AM, said: It seems to be related to the way distances (however they are defined ) change gradually in a smooth (I forget the precise meaning of "smooth" ) manifold. Yes, pretty much. Well I am glad I seem to be understanding the basics of manifolds in Relativity now. Edited July 7, 2016 by geordief
swansont Posted July 7, 2016 Posted July 7, 2016 . As with the electro magnetic field ,you guys helped me establish quite what was out there that the electro magnetic field could establishing itself in . Now , I wish your help to tease out , what quite is out there , that the gravitational field , which you indicate has something to do with space itself . But now we are talking of shapes , curves , measurable geodesic geometry . I need help to establish quite what the geometry is 'IN ' .or. 'ON' Now I know you want to say it's just there, in space . But I need a bit more than that . That is like saying I will draw a line in a vacuum . I need to 'pin you down ' a bit . With something I can identify with . Not just a mathematical operation ,something tangible . If you would not mind . Please . There , I have asked nicely . Asking nicely isn't the issue. You have asked a question AND placed restrictions on answers you deem acceptable. You have dictated that the answer must be tangible. What if there is no tangible answer?
Mike Smith Cosmos Posted July 7, 2016 Posted July 7, 2016 Asking nicely isn't the issue. You have asked a question AND placed restrictions on answers you deem acceptable. You have dictated that the answer must be tangible. What if there is no tangible answer? Then I am 'Sunk ' . Because I can only see things in my minds eye in ' Images' , however strange and convoluted, but images of what I construe to be REAL. Not mathematical constructs of something ' real ' . So a curve is space , I can imagine , but put in X squared /a + da de da , etc does not throw up any image of anything . You mathematicians , when you see or hear a formula , go into some form of ecstasy an no doubt get some form of meaning that you are happy with. If you describe space in a bunch of formulae , my brain goes 'numb ' . If you tell me gravity causes a bend in the fabric of space , then I start getting excited. What I do need to know in this instance is " what exactly ( in reality terms ) is Space or Space - Time that it can be bent or curved or what ever . Is it a field , if so what sort of field , grit field , rubber field , electromagnetic field . Atomic field , I know space is curved ....by gravity .... But what is IT Is it material , bits, fields ? MIKE
Strange Posted July 7, 2016 Posted July 7, 2016 That is like saying I will draw a line in a vacuum . I need to 'pin you down ' a bit . With something I can identify with . Not just a mathematical operation ,something tangible . If you would not mind . Please . You can conceive of a straight line between two points in space, can't you? Does that line have to be made of something? Can you only conceive of such a straight line if there is something there fore the line to pass through? If we had a perfect vacuum with two objects suspended in it, do you think it would be impossible think about the distance between them?
swansont Posted July 7, 2016 Posted July 7, 2016 Then I am 'Sunk ' . Because I can only see things in my minds eye in ' Images' , however strange and convoluted, but images of what I construe to be REAL. Not mathematical constructs of something ' real ' . So a curve is space , I can imagine , but put in X squared /a + da de da , etc does not throw up any image of anything . You mathematicians , when you see or hear a formula , go into some form of ecstasy an no doubt get some form of meaning that you are happy with. If you describe space in a bunch of formulae , my brain goes 'numb ' . If you tell me gravity causes a bend in the fabric of space , then I start getting excited. What I do need to know in this instance is " what exactly ( in reality terms ) is Space or Space - Time that it can be bent or curved or what ever . Is it a field , if so what sort of field , grit field , rubber field , electromagnetic field . Atomic field , I know space is curved ....by gravity .... But what is IT Is it material , bits, fields ? MIKE I'm sure you're familiar with the concept of a hole. Is a hole real? Is it made of a tangible substance? Would it be fair if I asked you (nicely) to bring me a hole, but just the hole and not the surrounding material, and also (for good measure) demand that it be tangible? (and does asking nicely change the ability for you to comply?)
Mike Smith Cosmos Posted July 7, 2016 Posted July 7, 2016 You can conceive of a straight line between two points in space, can't you? Does that line have to be made of something? Can you only conceive of such a straight line if there is something there fore the line to pass through? If we had a perfect vacuum with two objects suspended in it, do you think it would be impossible think about the distance between them? No not at all . But if you say Jupiter is over there and the earth is here . If we look half way between them gravity has distorted space -time there . What am I looking at ( there ) if I can't see it , o.k. .. But what is ' there ' that I can measure , test , probe . And see if it's bent . Where do I poke my ruler to measure , how do I make a time measurement to see if it's changed ? I am looking out at nothing , space , or do I spin a top, or drop something in a space ship , to see how it falls? Mike
Strange Posted July 7, 2016 Posted July 7, 2016 OK. So it turns out that what we think of as straight lines are not always straight. That's it. If you are still struggling, then maybe it is time to accept that your brain isn't suited for the concept and just move on.
Mike Smith Cosmos Posted July 7, 2016 Posted July 7, 2016 I am going out to look for a 'hole ' I will see what I come back with . Mike Ps. ( excuse me I have to go out on business for the afternoon . That is real , catch you later )
Markus Hanke Posted July 7, 2016 Posted July 7, 2016 (edited) I am going out to look for a 'hole ' I can think of a good few places where you might find those. Sorry, couldn't help myself. I'm too old now to pretend I still care about being politically correct Edited July 7, 2016 by Markus Hanke
swansont Posted July 7, 2016 Posted July 7, 2016 No not at all . But if you say Jupiter is over there and the earth is here . If we look half way between them gravity has distorted space -time there . What am I looking at ( there ) if I can't see it , o.k. .. But what is ' there ' that I can measure , test , probe . And see if it's bent . Where do I poke my ruler to measure , how do I make a time measurement to see if it's changed ? You could look at the path a photon takes as it passes by. You would notice that it doesn't follow what we would consider to be a straight line. The source of the light wouldn't be where the straight-line travel projection of the light would indicate.
geordief Posted July 7, 2016 Posted July 7, 2016 OK. So it turns out that what we think of as straight lines are not always straight. That's it. If you are still struggling, then maybe it is time to accept that your brain isn't suited for the concept and just move on. Here is a thought experiment . We have 3 bodies far removed from the gravitational influence of any other bodies. We join the centres of masses of each of the three with a line that is the shortest distance. Would those 3 lines define a flat plane or a curved one? My guess is it would be curved...
Strange Posted July 7, 2016 Posted July 7, 2016 Here is a thought experiment . We have 3 bodies far removed from the gravitational influence of any other bodies. We join the centres of masses of each of the three with a line that is the shortest distance. Would those 3 lines define a flat plane or a curved one? My guess is it would be curved... It depends on whether you are including the effect of the mass of the three bodies or not, I guess. If you treat them as "test particles" with no significant mass, then the lines will be straight (because the universe is, as near as we can tell, flat). Otherwise there will be some curvature introduced by the bodies.
swansont Posted July 7, 2016 Posted July 7, 2016 Here is a thought experiment . We have 3 bodies far removed from the gravitational influence of any other bodies. We join the centres of masses of each of the three with a line that is the shortest distance. Would those 3 lines define a flat plane or a curved one? My guess is it would be curved... Shortest distance according to whom?
geordief Posted July 7, 2016 Posted July 7, 2016 (edited) Shortest distance according to whom? I don't know. Can we have an unrealistic thought experiment? Can I (unrealistically) posit that one or all of the bodies are the observers? As Strange said the rest of the universe would define curvature on the 3 bodies ,I have tried to remove that context and ask if we can talk about the relationship between the 3 bodies per se . I suppose once we have 3 we could number up to any number and we end up with our own (imagined) universe. I thought in Relativity the observer didn't need to be an intelligent person ** necessarily ,just a point of reference. I hope some of this made sense **"intelligent observer" ,that is . I wasn't trying to be that flippant Edited July 7, 2016 by geordief
swansont Posted July 7, 2016 Posted July 7, 2016 I don't know. Can we have an unrealistic thought experiment? Can I (unrealistically) posit that one or all of the bodies are the observers? As Strange said the rest of the universe would define curvature on the 3 bodies ,I have tried to remove that context and ask if we can talk about the relationship between the 3 bodies per se . I suppose once we have 3 we could number up to any number and we end up with our own (imagined) universe. I thought in Relativity the observer didn't need to be an intelligent person ** necessarily ,just a point of reference. I hope some of this made sense **"intelligent observer" ,that is . I wasn't trying to be that flippant Yes, you can posit that. My point is that the distance generally depends on where you are when you make the measurement. "Shortest distance" might be preserved, but I'm not sure.
geordief Posted July 7, 2016 Posted July 7, 2016 (edited) It depends on whether you are including the effect of the mass of the three bodies or not, I guess. If you treat them as "test particles" with no significant mass, then the lines will be straight (because the universe is, as near as we can tell, flat). Otherwise there will be some curvature introduced by the bodies. Sorry , Disregard my earlier post (#291 and possibly the reference to you in post#293) I misread your reply. So the 3 bodies ,regardless of the environment (the universe as a whole) create their own local curvature simply because they have mass? And a "straight line" for the purposes of "communication" between these 3 massive bodies is not what is commonly accepted as straight but a geodesic even in this primitive set up ? Edited July 7, 2016 by geordief
Markus Hanke Posted July 7, 2016 Posted July 7, 2016 So the 3 bodies ,regardless of the environment (the universe as a whole) create their own local curvature simply because they have mass? Because they have energy-momentum. For ordinary bodies, their mass would be what makes up most of the contribution, but if they have a substantial amount of angular momentum or net electric charge, or if they carry a magnetic field, then that would play a role as well ( albeit a small one ). What's more, if the bodies have spatial extension ( i.e. if they can't be considered point-like ), then their shape and internal composition would also matter.
geordief Posted July 7, 2016 Posted July 7, 2016 (edited) Because they have energy-momentum. For ordinary bodies, their mass would be what makes up most of the contribution, but if they have a substantial amount of angular momentum or net electric charge, or if they carry a magnetic field, then that would play a role as well ( albeit a small one ). What's more, if the bodies have spatial extension ( i.e. if they can't be considered point-like ), then their shape and internal composition would also matter. So (repeating myself) we have 3 small massive bodies and ,if we factor out all the gravitational influences of the rest of the universe (impractical obviously - but in theory) any communication ** between them will take place in a space /spacetime that is curved ? Will that curvature be greater the closer they are to each other? **acceptable terminology? Edited July 7, 2016 by geordief
Mike Smith Cosmos Posted July 7, 2016 Posted July 7, 2016 (edited) . I have resurfaced ! I previously asked . ------------------------- Quote " X squared /a + da de da , etc does not throw up any image of anything . You mathematicians , when you see or hear a formula , go into some form of ecstasy an no doubt get some form of meaning that you are happy with. If you describe space in a bunch of formulae , my brain goes 'numb ' . If you tell me gravity causes a bend in the fabric of space , then I start getting excited. Edit ( there is something actually there getting bent ) ? What I do need to know in this instance is " what exactly ( in reality terms ) is Space or Space - Time that it can be bent or curved or what ever . Is it a field , if so what sort of field , grit field , rubber field , electromagnetic field . Atomic field , I know space is curved ....by gravity .... But what is IT Is it material , bits, fields ? " ----unquote ------------------ I have just read your comments since then . I went out and I am not sure you have given me a ' strait' answer . I do not mean that disrespectfully . I just mean , I still do not really know " what quite is out there in substantive form ? Namely . "Is there something tangible ' there ' . Not just a geometric idea , or geometric measurement . If there are two measurements , say ' of time and of distance " . What TANGIBLE thing have you measured ? Not just an arbitrary piece of space time but a " thing A FUZZY BIT OF SPACE TIME , but WHAT IS IT ? " at a particular time and a particular space . And do I need two probes , one to measure passage of , or , instantaneous time , and one to measure curvature , say ? WHAT am I measuring it ' on ' . What is it ? a field ? What is the field composed of ? What am I comparing the time to ? I am genuinely trying to separate arbitrary values ( say geometric points on a mathmatical graph , and imposed time duration ) I want to know that the ' things ,I am measuring on , are not imposed values , but measured values " but of What ? On what " What is this substance. " SPACE - TIME " and don't say it's a geodesic or I will shoot myself ! Mike Edited July 7, 2016 by Mike Smith Cosmos
disarray Posted July 7, 2016 Posted July 7, 2016 (edited) Because they have energy-momentum. For ordinary bodies, their mass would be what makes up most of the contribution, but if they have a substantial amount of angular momentum or net electric charge, or if they carry a magnetic field, then that would play a role as well ( albeit a small one ). What's more, if the bodies have spatial extension ( i.e. if they can't be considered point-like ), then their shape and internal composition would also matter. Amazing. So can I presume that mass, angular momentum, (angular velocity?), net electric charge, magnetic field, and spatial extension all have an effect on time as well as the 'curvature' of space? Isn't angular velocity, or just velocity, also a factor? To change topic a little, what seems odd to me is that everything in the spaceship, in effect, slows down, (though we say time dilates) in proportion to how fast the ship goes. This suggests to me that the faster all the molecules of the ship are traveling away from the twin on earth, the slower, in effect, that they are actually moving about as they go, e.g., the slower the DNA of an astronaut is replicating. Sounds like a contradiction. In any case, the twin on earth isn't really in some sort of inertial or stationary position as the earth is both turning at a rate of 1000 mph (at the equator), as well as around the sun at 67,500 miles an hour, as milling around the galaxy along with the sun at a rate of 43,000 mph towards the star Vega. I suppose that these are not considered a factor in the twin paradox because they are so small in comparison with the hypothetical speed at which the spaceship is said to go in the thought experiment. Edited July 7, 2016 by disarray
Strange Posted July 7, 2016 Posted July 7, 2016 "Is there something tangible ' there ' . Not just a geometric idea , or geometric measurement . If there are two measurements , say ' of time and of distance " . What tangible thing have you measured ? You have not measured anything tangible. Is 1 mile tangible? Is 1 second tangible? Space is just the distances between things. Why do you expect it to be "made of" something?
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