Curved space

[Anilkumar]

Hi everybody,

 

How and why does mass curve space?

 

Thank you.

[ajb]

You will need to "pull apart" the Einstein field equations to understand this in proper detail.

 

It is really the energy-momentum tensor of the matter that acts as a source of the gravitational field. In short you can paraphrase general relativity as

 

"Energy + momentum" = "local geometry of space-time".

[lemur]

Hi everybody,

 

How and why does mass curve space?

 

Thank you.

Hi Anilkumar, are you asking about how mass-space curvature is explained and represented mathematically or how mass actually causes space to curve and/or what it means for space to curve in the first place?

 

 

 

[Anilkumar]

Thanks ajb & lemur,

 

for your kind attention.

 

 

ajb said

It is really the energy-momentum tensor of the matter that acts as a source of the gravitational field.

 

 

Is there a theoretical and illustrated explaination for the TENSOR, ENERGY-MOMENTUM TENSOR ?

 

 

lemur said,

are you asking about how mass-space curvature is explained and represented mathematically or how mass actually causes space to curve and/or what it means for space to curve in the first place?

 

 

Yes, I am asking;

 

How mass-space curvature is explained and represented mathematically?

 

How mass actually causes space to curve?

 

What it means for space to curve?

 

Regards.

Edited June 4, 2011 by Anilkumar

[ajb]

Is there a theoretical and illustrated explaination for the TENSOR, ENERGY-MOMENTUM TENSOR ?

 

 

The wikipedia article I pointed to is a good place to start.

 

Yes, I am asking;

 

How mass-space curvature is explained and represented mathematically?

 

How mass actually causes space to curve?

 

What it means for space to curve?

 

You will need to know some differential geometry to understand that properly.

 

How much mathematics do you know already?

 

I have explained a simple example of curvature in 2-dimensions on this forum before. Have a search for it.

[lemur]

What it means for space to curve?

I think that I have a clear way of explaining what it means for space to curve, but it may be that someone corrects me: If an object moves from point A to B as a result of its own inertia/momentum, with no external force applied, it can be said to have moved in a straight line inertially. However, just because an object moved from A to B doesn't mean that the path it took was the only possible line connecting A and B. So, for example, the Earth may be at point A on January 1 and be at point B on December 1. The path between A and B is a straight inertial path within the space-curvature of the Sun but there can be other paths between the two points that are also straight inertial paths. The curvature may be different for different kinds of objects/particles moving at different velocities, I think, so light curves very little at the same distance from the sun that causes the Earth to remain in orbit at its velocity. I believe this basically explains space-curvature, but someone will probably correct some aspect of what I have said so don't take it as a perfect explanation. I just offer it as a simple basis for going further.

[Anilkumar]

The wikipedia article I pointed to is a good place to start.

 

You will need to know some differential geometry to understand that properly.

 

How much mathematics do you know already?

 

I have explained a simple example of curvature in 2-dimensions on this forum before. Have a search for it.

 

Thanks ajb,

 

I have read the wikipedia article. But didn't help me much.

 

Don't we have a theoretical explaination for all this without bringing much maths into it?

 

My mathematics is at UG level.

 

Thanks

 

I think that I have a clear way of explaining what it means for space to curve, but it may be that someone corrects me: If an object moves from point A to B as a result of its own inertia/momentum, with no external force applied, it can be said to have moved in a straight line inertially. However, just because an object moved from A to B doesn't mean that the path it took was the only possible line connecting A and B. So, for example, the Earth may be at point A on January 1 and be at point B on December 1. The path between A and B is a straight inertial path within the space-curvature of the Sun but there can be other paths between the two points that are also straight inertial paths. The curvature may be different for different kinds of objects/particles moving at different velocities, I think, so light curves very little at the same distance from the sun that causes the Earth to remain in orbit at its velocity. I believe this basically explains space-curvature, but someone will probably correct some aspect of what I have said so don't take it as a perfect explanation. I just offer it as a simple basis for going further.

 

Hi there,

 

Thanks for your painstaking explaination.

 

But I did not understand-

 

"The path between A and B is a straight inertial path within the space-curvature of the Sun but there can be other paths between the two points that are also straight inertial paths."

 

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

Thanks.

[ajb]

Don't we have a theoretical explaination for all this without bringing much maths into it?

 

That sounds like an oxymoron.

 

 

 

My mathematics is at UG level.

 

That should be enough. I will quote an earlier post of mine

 

It is not hard to understand curvature in 2d via parallel transport of vectors. You can think of vectors as little arrows and parallel transport along a curve just means "not changing the direction of the vector" , other than that due to following the curve. If you follow a close path, a loop and the vector is not the same as when you started, the space is curved.

 

Draw a closed path on a ball or balloon. Take a pencil to be a vector and parallel transport it around the closed path. Try not to introduce any change in direction other than that demanded by the loop. You will see the vector is not pointing in the same direction as you started. The ball (i.e. the sphere) is curved. (It is easier than I make it sound.)

 

Try it on (a piece of) the cylinder, i.e. a toilet roll tube.

 

The idea is the same in higher dimensions, but a lot more work is needed to understand vectors and parallel transport.

[Vilas Tamhane]

That sounds like an oxymoron.

 

 

 

 

 

That should be enough. I will quote an earlier post of mine

 

 

 

 

Ball is not space. Matter is not space. Space is something that matter occupies. I think what the original poster is asking is that, If space is nothing or atleast when it is not matter, how can it get curved?

[csmyth3025]

Ball is not space. Matter is not space. Space is something that matter occupies. I think what the original poster is asking is that, If space is nothing or atleast when it is not matter, how can it get curved?

This might just be a matter of semantics. It's the variations of the gravitational field in space that dictate the inertial path of objects. We can't see or touch this field any more than we can see or touch empty space. So, to avoid unnecessary complication, we just say that the inertial movement of objects follow the curvature of space (more precisely, spacetime).

 

As an approximation, if you're in a windowless space capsule orbiting the Earth you don't feel any force acting on you at all (ignoring miniscule tidal effects). For all you know you could be "coasting" along a straight line at a high or low velocity relative to some distant object or you could be sitting still relative to that object.

 

Even though your space capsule is essentially free-falling in a gravitational field (accelerating to an outside observer), you don't notice any such acceleration because free-falling in a uniform gravitational field is equivalent to inertial movement (traveling in a straight line at a uniform speed) when you're immensely far away from any gravitating body.

 

The equivalence principle proper was introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s² being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:

 

we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.Einstein, 1907

That is, being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines. From this principle, Einstein deduced that free-fall is actually inertial motion...

(ref. http://en.wikipedia....vitation_theory )

 

Chris

 

Edited to correct spelling error

Edited July 1, 2011 by csmyth3025

[Vilas Tamhane]

This might just be a matter of semantics.

 

 

 

Such misleading words create revulsion in the rational minds who wish to study physics. But then I think space is treated as an entity in relativity. After all, length contraction is space contraction in SR.

 

 

[swansont]

Ball is not space. Matter is not space. Space is something that matter occupies. I think what the original poster is asking is that, If space is nothing or atleast when it is not matter, how can it get curved?

Answered in ajb's first post in the topic: "Energy + momentum" = "local geometry of space-time"

 

i.e. it's the geometry

 

Such misleading words create revulsion in the rational minds who wish to study physics. But then I think space is treated as an entity in relativity. After all, length contraction is space contraction in SR.

I think it's clear the the OP is not a physicist. What are important details to a physicist may seem like semantics to an interested amateur, who does not understand the finer details and their importance.

[Vilas Tamhane]

Answered in ajb's first post in the topic: "Energy + momentum" = "local geometry of space-time"

 

i.e. it's the geometry

 

 

I think it's clear the the OP is not a physicist. What are important details to a physicist may seem like semantics to an interested amateur, who does not understand the finer details and their importance.

 

 

It is difficult to agree with you. I believe that if a theory in physics is correct then it should be possible to explain its basic verbose part to a layman of average intelligence. Mathematics is just a tool for making calculations and predictions. Therefore common notion that one can understand a theory only if one understands mathematics is wrong. Any way, SR also deals with space-time. Question is, ‘Does space contract’. Answer that this is only a measurement and only a perspective of an observer in a different inertial frame’ is not acceptable. Time is found to ‘really’ dilate. Therefore with or without experiment we can state that length contraction is also not apparent. You cannot say that it is just space-time geometry. Geometry is just visualization of mathematics. This geometry is applied to real world and when we do so we find that space contracts. This is not just semantics.

 

So the question ‘how space can contract’ remains.

 

 

[IM Egdall]

It is difficult to agree with you. I believe that if a theory in physics is correct then it should be possible to explain its basic verbose part to a layman of average intelligence. Mathematics is just a tool for making calculations and predictions. Therefore common notion that one can understand a theory only if one understands mathematics is wrong. Any way, SR also deals with space-time. Question is, 'Does space contract'. Answer that this is only a measurement and only a perspective of an observer in a different inertial frame' is not acceptable. Time is found to 'really' dilate. Therefore with or without experiment we can state that length contraction is also not apparent. You cannot say that it is just space-time geometry. Geometry is just visualization of mathematics. This geometry is applied to real world and when we do so we find that space contracts. This is not just semantics.

 

So the question 'how space can contract' remains.

 

 

 

Here is an excerpt from a book I am writing on relativity which discusses your question. Hope it helps:

For special relativity, Einstein came to embrace the notion that nothing exists beyond what we observe, what we can measure. He defined time as simply "what you read on a clock", and space as simply "the distance you measure between two points". The notion of time and space as anything beyond these "operational" definitions were, according to Einstein, simply the creation of the human mind.

In other words, time is relative; how fast it passes depends on the motion of the observer. This is measurable. However, the concept of time we hold in our minds is merely an abstraction. Einstein applied a similar view to "space".

 

But how can space contract? And how can the apparently same space contract differently for two (or more) observers in relative motion? Here Einstein is saying there is no "space" per se; only the distance between two points. The distance between the two points is measurable, albeit differently by the two observers. But "space" itself is again a mere abstraction.

 

In summary, we must "stop thinking about 'space and time' (as) something that is 'given to us', and must instead think about 'measuring positions and times', which is something we can do," writes Morton Tavel, professor of physics at Vassar College. "Only our measurements have real existence. We build up an intuition of something called space and time, which we believe exists beyond these measurements . . . (But) Einstein's first commandment was to pay attention only to your measurements and worry later about the properties of the more abstract notion of space and time." (My italics.)

 

Perhaps you find this approach to reality quite difficult to accept. So do I. Whether it is the years of thinking a certain way, or a particular bias of the human mind; I find it hard to accept that time and space are not real entities in their own right. But in special relativity, Einstein tells us that we must think of time and space as only the position on a clock and the markings on a ruler.

 

Einstein was to later develop a much broader view of space and time; first in considering the spacetime physics of his mathematics teacher, Hermann Minkowski, and most significantly, in his development of general relativity.

Ref: Victor J. Stenger, Quantum Gods, Creation, Chaos, and the Search for Cosmic Consciousness, p. 66.

J.A. Wheeler, A Journey into Gravity and Spacetime, p. 5

Morton Tavel, Contemporary Physics and the Limits of Knowledge (Rutgers, 2002), p. 58, as cited in Am. J. Phys. 74, 891 (2006).

Edited July 2, 2011 by I ME

[ajb]

I believe that if a theory in physics is correct then it should be possible to explain its basic verbose part to a layman of average intelligence.

 

I tend to agree with this, in principle anyway. The general idea of a theory should be able to be explainable to the interested layman.

 

Mathematics is just a tool for making calculations and predictions. Therefore common notion that one can understand a theory only if one understands mathematics is wrong.

 

Well, maybe a basic overall ethos of a theory could be understood without mathematics, but it would be very superficial. Using analogies is a good way of giving the ideas of a theory to people who are not experts, but analogies can also be misleading. The trampoline analogy springs to mind here.

 

Any way, SR also deals with space-time. Question is, ‘Does space contract’. Answer that this is only a measurement and only a perspective of an observer in a different inertial frame’ is not acceptable. Time is found to ‘really’ dilate. Therefore with or without experiment we can state that length contraction is also not apparent. You cannot say that it is just space-time geometry. Geometry is just visualization of mathematics. This geometry is applied to real world and when we do so we find that space contracts. This is not just semantics.

 

Right, so we use mathematics and in particular theories (or mathematical models) to describe nature, but this must not be confused with nature. Our best models suggest that the geometry of space-time is an important idea in gravity, and special relativity also. Of course the only "real things" are things that can be measured and the question is how do these measurements relate to our theories?

 

As an aside, I am not aware of any direct experimental evidence of length contraction. Not that I doubt it occurs, just that today the experimental accuracy is not there to observe this effect.

[DrRocket]

As an aside, I am not aware of any direct experimental evidence of length contraction. Not that I doubt it occurs, just that today the experimental accuracy is not there to observe this effect.

 

 

No such experiment has yet been performed. However, I am aware of one that is being planned, and may be carried out in the next two or three years. I have no idea how this experiment would be done.

 

I tend to agree with this, in principle anyway. The general idea of a theory should be able to be explainable to the interested layman.

 

 

 

Well, maybe a basic overall ethos of a theory could be understood without mathematics, but it would be very superficial. Using analogies is a good way of giving the ideas of a theory to people who are not experts, but analogies can also be misleading. The trampoline analogy springs to mind here.

 

Richard Feynman was as good as anyone at producing clear and elementary explanations of physics. Here is what he had to say on the subject:

 

"To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’. To those who do not know mathematics it is difficult to get across a real feeling as the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once." – Richard P. Feynman in The Character of Physical Law

 

 

[MigL]

Sure, you would bring up that specific quote since you are a mathematician. ( Just kidding )

[Vilas Tamhane]

Here is an excerpt from a book I am writing on relativity which discusses your question. Hope it helps:

For special relativity, Einstein came to embrace the notion that nothing exists beyond what we observe, what we can measure. He defined time as simply "what you read on a clock", and space as simply "the distance you measure between two points". The notion of time and space as anything beyond these "operational" definitions were, according to Einstein, simply the creation of the human mind.

In other words, time is relative; how fast it passes depends on the motion of the observer. This is measurable. However, the concept of time we hold in our minds is merely an abstraction. Einstein applied a similar view to "space".

 

But how can space contract? And how can the apparently same space contract differently for two (or more) observers in relative motion? Here Einstein is saying there is no "space" per se; only the distance between two points. The distance between the two points is measurable, albeit differently by the two observers. But "space" itself is again a mere abstraction.

 

In summary, we must "stop thinking about 'space and time' (as) something that is 'given to us', and must instead think about 'measuring positions and times', which is something we can do," writes Morton Tavel, professor of physics at Vassar College. "Only our measurements have real existence. We build up an intuition of something called space and time, which we believe exists beyond these measurements . . . (But) Einstein's first commandment was to pay attention only to your measurements and worry later about the properties of the more abstract notion of space and time." (My italics.)

 

Perhaps you find this approach to reality quite difficult to accept. So do I. Whether it is the years of thinking a certain way, or a particular bias of the human mind; I find it hard to accept that time and space are not real entities in their own right. But in special relativity, Einstein tells us that we must think of time and space as only the position on a clock and the markings on a ruler.

 

Einstein was to later develop a much broader view of space and time; first in considering the spacetime physics of his mathematics teacher, Hermann Minkowski, and most significantly, in his development of general relativity.

Ref: Victor J. Stenger, Quantum Gods, Creation, Chaos, and the Search for Cosmic Consciousness, p. 66.

J.A. Wheeler, A Journey into Gravity and Spacetime, p. 5

Morton Tavel, Contemporary Physics and the Limits of Knowledge (Rutgers, 2002), p. 58, as cited in Am. J. Phys. 74, 891 (2006).

 

I am afraid it doesn’t help. I call it motivated advocacy. You can always find such descriptions in the books on occult science. Things come first. Measurements later. Bertrand Russell would hate such a paragraphs. It conveys nothing meaningful.

[IM Egdall]

I am afraid it doesn't help. I call it motivated advocacy. You can always find such descriptions in the books on occult science. Things come first. Measurements later. Bertrand Russell would hate such a paragraphs. It conveys nothing meaningful.

 

 

 

 

I think you will also find this same focus on measurement in the so-called Copenhagen interpretation of quantum mechanics by Neils Bohr etc. The physics of relativity and quantum mechanics works superbly. That is their predictions agree with measurements of nature to extraordinary accuracy. So these theories must be telling us something about reality.

 

Perhaps "things" should come first, as you say, and the focus on measurement is a poor attempt to interpret what these theories are telling us about our world. It is, I think, a limit to our human perception.

 

Or perhaps "things" are merely a human abstraction, and Bohr and Einstein are on the right track. This is a matter of philosophy. We can argue about it till the cows come home, but, unfortunately, we cannot determine whose philisophical argument is more valid through experiment.

 

Or maybe we are like children who have discovered some remarkable models of the physical world. They work but we are unable to understand what they mean.

Edited July 3, 2011 by I ME

[Vilas Tamhane]

I think you will also find this same focus on measurement in the so-called Copenhagen interpretation of quantum mechanics by Neils Bohr etc. The physics of relativity and quantum mechanics works superbly. That is their predictions agree with measurements of nature to extraordinary accuracy. So these theories must be telling us something about reality.

 

Perhaps "things" should come first, as you say, and the focus on measurement is a poor attempt to interpret what these theories are telling us about our world. It is, I think, a limit to our human perception.

 

Or perhaps "things" are merely a human abstraction, and Bohr and Einstein are on the right track. This is a matter of philosophy. We can argue about it till the cows come home, but, unfortunately, we cannot determine whose philisophical argument is more valid through experiment.

 

Or maybe we are like children who have discovered some remarkable models of the physical world. They work but we are unable to understand what they mean.

 

This is true and for this reason a scientist should be an inquisitor and not a worshiper. I get feeling that something is wrong at the higher level. Not being a scientist myself, I am not aware if there is any rot at the top and extend of its spread. To give one single example, popular journal ‘American Journal of Physics’ clearly mentions that any paper that goes against established theories will not be accepted as this can be reviewed only by experts. This is wrong. A paper should be valued on its merit and in fact critical papers should be valued with greater respect, as it is far easier to add to knowledge which is already known.

 

 

[IM Egdall]

This is true and for this reason a scientist should be an inquisitor and not a worshiper. I get feeling that something is wrong at the higher level. Not being a scientist myself, I am not aware if there is any rot at the top and extend of its spread. To give one single example, popular journal 'American Journal of Physics' clearly mentions that any paper that goes against established theories will not be accepted as this can be reviewed only by experts. This is wrong. A paper should be valued on its merit and in fact critical papers should be valued with greater respect, as it is far easier to add to knowledge which is already known.

 

 

 

Unfortunately, there is always a bias towards accepted knowledge, whether conscious or not. Scientists are human. It is not impossible to get an unconventional theory published, it is just harder. And history shows that the theories we now accept initially also went through a period of doubt. When a measurement is made that agrees to good accuracy with a new theory (and not the established ones), then scientists take notice.

[Anilkumar]

I tend to agree with this, in principle anyway. The general idea of a theory should be able to be explainable to the interested layman.

 

 

 

Well, maybe a basic overall ethos of a theory could be understood without mathematics, but it would be very superficial. Using analogies is a good way of giving the ideas of a theory to people who are not experts, but analogies can also be misleading. The trampoline analogy springs to mind here.

 

 

 

Right, so we use mathematics and in particular theories (or mathematical models) to describe nature, but this must not be confused with nature. Our best models suggest that the geometry of space-time is an important idea in gravity, and special relativity also. Of course the only "real things" are things that can be measured and the question is how do these measurements relate to our theories?

 

As an aside, I am not aware of any direct experimental evidence of length contraction. Not that I doubt it occurs, just that today the experimental accuracy is not there to observe this effect.

 

Case 1: We can not see an object which is obstructed by another opaque body.

 

 

Case 2: We can see a star which is behind the Sun, even though the Sun is an opaque body.

 

 

The scientific explanation given for this mystery: The light coming from the star is passing near a massive body i.e. the Sun. And Sun's huge gravity distorts/bends the space surrounding it. The space in the vicinity of the Sun is curved. So the light, coming from the star, instead of moving in a straight line, follows the curved path provided by the curved space around the Sun. And thus we get to see the star situated behind the Sun.

 

My doubts : I can understand that the 'Length' is relative to space-time frames. It can be either short or long relative to, from which space-time frame we are measuring it. So to assume something as longer or shorter is an illusion. Because when someone says something is short, it is the perception generated by the conditions of that particular space-time frame. And when someone says something is long, it is the perception generated by the conditions of its own space-time frame. So the length is neither short nor long but is a perception/illusion generated by the conditions of their respective space-time frames.

 

But in Case 2,

 

We are not talking about RELATIVE PERCEPTION. We are actually seeing a star which we shouldn't see. And the reason given is that the light from the hidden star is brought to you by the bent space, like an optical fiber.

 

Space is nothingness. It is emptiness which gives space. The only thing that anybody can do to it is OCCUPY it. You just can not touch it or affect it in any other way except FILLING it, leave aside bending it.

 

My humble question is;

 

How can gravity, mathematics, you, me, or anybody bend 'NOTHING'?

 

 

I thank you for your interest in my doubts.

Edited November 25, 2011 by Anilkumar

[DrRocket]

How can gravity, mathematics, you, me, or anybody bend 'NOTHING'?

 

 

 

Differential geometry.

 

For the details see Gravitation by Charles Misneer, Kip Thorne, and John Archibald Wheeler.

[Anilkumar]

Differential geometry.

 

For the details see Gravitation by Charles Misneer, Kip Thorne, and John Archibald Wheeler.

 

Thank you for your interest.

 

OK, I will learn Differential geometry, and also read Gravitation. But till then to make things easier, isn't there a simplified theoretical explanation to it?

 

Thank you

[DrRocket]

Thank you for your interest.

 

OK, I will learn Differential geometry, and also read Gravitation. But till then to make things easier, isn't there a simplified theoretical explanation to it?

 

 

No.

 

Difficult questions have simple, easy-to-understand, wrong answers.

 

To understand any theory, and general relativity is no exception, you must invest intellectual capital, and that includes mastering the language in which it is formulated.