Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: KennyC on 10/10/2013 12:55:56
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I'm a former physics major -- long long ago, no degree -- but do my best to read, understand and write about science. This one has been bugging me for a while.
According to Einsteins General Relativity the presence of mass curves space which we have observed during eclipses and through gravitational lensing. So my question is (and maybe the assumption is wrong but...)
If space is empty then what is being curved?
Seems that it must be something, not nothing if it is being curved.
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Einstein's big idea was to say that the basic laws of physics are the same for everyone. One of those laws says that things move in straight lines through space unless acted upon by forces. There are a couple ways of dealing with gravity, which pulls on things and deviates them from straight lines. Before Einstein, people modeled gravity as a pull on objects and deviated them from a straight line. Einstein realized that while this is true, that the curving cause by gravity could also be described by saying that space (and time) itself gets curved by gravity, which would naturally bend straight lines passing through that space. It turns out that Einstein's predictions actually work better than the old models based on predictions that leave space-time un-curved (flat), which is why his theory displaced the previous theories.
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I'm a former physics major -- long long ago, no degree -- but do my best to read, understand and write about science. This one has been bugging me for a while.
According to Einsteins General Relativity the presence of mass curves space which we have observed during eclipses and through gravitational lensing. So my question is (and maybe the assumption is wrong but...)
If space is empty then what is being curved?
Seems that it must be something, not nothing if it is being curved.
First off let’s be clear on one point. Einstein’s field equation of General Relativity relates mass to spacetime curvature, not merely spatial curvature.
Next – While spacetime curvature can cause gravitational lensing the presence of spacetime curvature is merely sufficient to cause lensing and is not necessary for it. As an example consider a straight cosmic string. Alaxander Vilenkin at Tufts University showed that the string alters the space around it by taking out a wedge of angle 8*pi*mu where mu is the linear mass density of the string. This represents the geometry of a conical space. If you’re familiar with geometry then you might also know that the surface of a cone is flat, i.e. it has no intrinsic curvature. The curvature that one speaks of in GR is intrinsic curvature. This geometry means that if two parallel rays of light move towards the string and each ray passes on opposites sides of the string that the rays will cross on the other side. In this case parallel lines cross even though the spatial geometry is Euclidean. How about that? :)
The emptiness of space has nothing to do with geometry other than there is nothing there to curve it. The terminology and visualizations used in GR are unfortunate. It gives the impression that what one means by curvature is that there is some sort of material which is being distorted. The worst and most misleading term in all of relativity is the fabric of spacetime.
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If you send an atomic clock and an uncompressible meter stick thru a strong gravity field, their size, speed and trajectory will depend on whether you define the speed of light as a constant, or whether you define the space-time grid as flat. You can't have it both ways; it just doesn't work.
If the space-time grid were flat by definition, the speed of light would have to change relative to the atomic clock and meter stick in a strong gravity field. The clock speed and the size of the meter stick could not be constant relative to the units that define the space-time grid.
But we have defined the meter and second to be the constant units of space and time; and we have defined the speed of light as a constant ratio of meters per second regardless of where you are; so we need a warped grid to make that so in a strong gravity field.
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Let's step a bit back and see how the curvature of space can be introduced.
The key point is the equivalence principle between gravitational field and the acceleration.
Let's consider an elevator that doesn't have any windows. The elevator moves along (say) 'z'-direction at an acceleration a=9.8m/s^2. This 'z' direction can be defined relative to an observer on Earth such that the elevator is moving perpendicular to Earth surface and away from Earth.
Next, let's image that one photon is passing through elevator along a direction perpendicular to our 'z' direction. Also, let's imagine we can trace the trajectory of the photon as it passes through elevator.
What will the observer on Earth see? Well, he will see that the distance between the bottom of the elevator and the 'location' of the photon will became smaller and smaller as the photon passes through elevator, because the elevator is moving up relative to him.
What the observer on the elevator will see? He must see the same thing: that the distance between the bottom of the elevator and the location of the photon became smaller and smaller as the elevator moves up and and photon passes through elevator.
In other words, the trajectory of the photon relative to the observer in elevator is not a straight line along a direction perpendicular to elevator but a curved one.
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Next important ingredient is the equivalence principle: specifically, there is no way for the observer in the elevator to distinguish between moving at the uniform acceleration a=9.8m/s^2 or sitting on the surface of a planet under a gravitational field g=9.8m/s^2 (ignoring the very small tidal effect or the very small changes of g over the height of elevator).
However, if there is no experiment to distinguish between being uniformly accelerated at a=9.8m/s^2 [in empty space] and sitting at the surface of a planet under a field g=9.8m/s^2, then for the observer on elevator the trajectory of a photon passing through elevator sitting on planet surface must be the same as the the trajectory of the photon passing through elevator moving at acceleration a=9.8m/s^2; and that is curved and not straight line.
If my understanding is correct, the fact that the photon follows a curved line (under gravitational field) can be reworded as the 'curvature of space due to gravity'. I must however admit that I am not so sure I have a good understanding of this concept and I am very interested in other opinions?
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Then is this 'curvature of space due to gravity' just a matter of pure geometry due to changing a system of coordinates [made of e.g. straight lines] with another system of coordinates [made of e.g. curved lines like those we can draw on a sphere]? Apparently yes, or so it seems to me.
Also, I am tempted to believe that relativity makes space looks more like a relationship between things rather than a reality on own.
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Let's step a bit back and see how the curvature of space can be introduced.
The key point is the equivalence principle between gravitational field and the acceleration.
That is incorrect. The key idea is tidal acceleration, not the equivalence principle
Let's consider an elevator that doesn't have any windows. The elevator moves along (say) 'z'-direction at an acceleration a=9.8m/s^2. This 'z' direction can be defined relative to an observer on Earth such that the elevator is moving perpendicular to Earth surface and away from Earth.
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In other words, the trajectory of the photon relative to the observer in elevator is not a straight line along a direction perpendicular to elevator but a curved one.
While this is true its totally unrelated to curvature. You’re referring to the fact that a straight spatial line in one coordinate system is a curved spatial line another coordinate system. Curvature refers to entirely different phenomena. Its unfortunate that the term “curvature” has two meanings in this context since it confuses a lot of people.
Next important ingredient is the equivalence principle: specifically, there is no way for the observer in the elevator to distinguish between moving at the uniform acceleration a=9.8m/s^2 or sitting on the surface of a planet under a gravitational field g=9.8m/s^2 (ignoring the very small tidal effect or the very small changes of g over the height of elevator).
This is also incorrect. An observer in such a field can tell that he’s not merely in an accelerating frame in flat spacetime by the presence of tidal acclerations
However, if there is no experiment to distinguish between being uniformly accelerated at a=9.8m/s^2 [in empty space] and sitting at the surface of a planet under a field
Also incorrect. Drop two apples. If they accelerate towards each other independent of their mass then you’re in the gravitational field which has tidal forces present
Then is this 'curvature of space due to gravity' just a matter of pure geometry due to changing a system of coordinates [made of e.g. straight lines] with another system of coordinates [made of e.g. curved lines like those we can draw on a sphere]? Apparently yes, or so it seems to me.
Sorry. But that’s all wrong. Fortunately there is a new text online which can explain all of this to you. See http://exploringblackholes.com
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That is incorrect. The key idea is tidal acceleration, not the equivalence principle
I am not sure how the tidal acceleration has anything to do with the equivalence principle.
Why would we need the tidal acceleration to understand the space curvature?
Curvature refers to entirely different phenomena. Its unfortunate that the term “curvature” has two meanings in this context since it confuses a lot of people.
These two different phenomena are equivalent because no physical measurement can distinguish between them.
Just out of curiosity: How would you explain the concept of space curvature in a more pedagogical/intuitive fashion?
This is also incorrect. An observer in such a field can tell that he’s not merely in an accelerating frame in flat spacetime by the presence of tidal acclerations
What tidal acceleration? In the first case the observer is assumed to be in empty space sufficiently far from any source of gravitational field and moving with an acceleration a=9.8m/s^2. In the second case the observer is at the surface of a planet under an uniform gravitational field g=9.8m/s^2. The gravitational field is assumed uniform (any tidal effect is ignored and it has nothing to do here).
My understanding of equivalence principle is that no physical measurement can distinguish between the above 2 situations.
If so, then the observer should see the same path of the photon in both situations (which is curved).
Sorry. But that’s all wrong. Fortunately there is a new text online which can explain all of this to you. See http://exploringblackholes.com
It may also be useful to show here within the context of the discussion why is a certain aspect incorrect.
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I am not sure how the tidal acceleration has anything to do with the equivalence principle.
The point is that it doesn’t.
Why would we need the tidal acceleration to understand the space curvature?
Spacetime curvature, not merely space curvature. GR is a theory of spacetime curvature.
Spacetime curvature and tidal accelerations is the same thing expressed in different languages. The former in terms of GR and differential geometry, the later in Newtonian mechanics terms.
I’m going to assume that you know what tidal acceleration/gradients are so I won’t define them.
Let’s talk about curvature for a moment. Picture the surface of the earth (i.e. the surface of a sphere). The surface of the sphere is an example of curved surface. Please don’t confuse the fact that the surface of a sphere also has what is called explicit curvature meaning that the curving occurs because the surface curves into a higher dimension. The surface of a cylinder has non-zero explicit curvature and zero intrinsic curvature.
Now go to the equator with a friend. Stand a distance apart along the equator while you each face North. Now both of you start walking north and remain walking in the “straightest possible line”. Eventually you’ll both reach the North Pole where your paths will meet. Thus two geodesics started off parallel but intersected.
From a physical standpoint the deviation of geodesics in spacetime in a gravitational field is what happens when two particles paths converge even though they started out parallel. This is caused by tidal accelerations. If you understand why dropping two apples at the surface of the earth will cause them to accelerate together then you’ll probably understand this argument.
Just out of curiosity: How would you explain the concept of space curvature in a more pedagogical/intuitive fashion?
I’d write a book and illustrate this in diagrams or create a web page with diagrams. It’s easier to see all of this if you visualize it.
What tidal acceleration? In the first case the observer is assumed to be in empty space sufficiently far from any source of gravitational field and moving with an acceleration a=9.8m/s^2.
While that’s true that is not what you described. You repeatedly spoke of this happening on the earth and the gravitational field of the earth is not uniform.
As far as why a certain aspect is correct that's why I provided a link to the book. It's too difficult to explain entirely and correctly in a thread.
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I think it's important to make the distinction between the curvature of a particles trajectory and the intrinsic curvature of a manifold, in this case space and spacetime.
The term curvature as it pertains to a property of a curve has to do with how the tangent of the curve changes as one moves along the curve. The term as it applies to a manifold has to do with the intrinsic properties of a surface. In this context we're speaking not of the curve inside the space but of a a property of the space itself (here I'm using the term "space" in the sense of a set of points).
Please see http://mathworld.wolfram.com/Curvature.html
In particular see http://mathworld.wolfram.com/IntrinsicCurvature.html
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I think it's important to make the distinction between the curvature of a particles trajectory and the intrinsic curvature of a manifold,to in this case space and spacetime.
Photons and particles in free fall follow the space curvature, right?
If one wants see the curvature of space, then just follow the trajectory of a free fall particle.
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Photons and particles in free fall follow the space curvature, right?
I'm sorry but I don't know what "follow the space curvature" means. I'm not even sure as to why you're speaking of space curvature. Why is that? Please recall reply number 2 of this thread where I wrote
First off let’s be clear on one point. Einstein’s field equation of General Relativity relates mass to spacetime curvature, not merely spatial curvature.
From herein I'm going to assume that when you use the (incorrect) term space curvature that you really refering to the (correct) term spacetime curvature.
As far as the meaning of "following the curvature" goes, it has none. Picture yourself walking on the surface of the earth. The surface of the earth is curved. The analogy of a man walking on the earth's surface to a free particle moving in spacetime is that the man walks along what is known as a "geodesic." On the surface of the earth great circles, or portions of them, are the only geodesics on the surface. So walking along a great circle is walking along a geodesic.
Back to spacetime - Can we determine if a spacetime is curved by observing only one worldine? No. We require other worldlines which we compare with each other or the original one. If two geodesics which start out parallel don't remain parallel then the spacetime is curved. This is the manifestation of tidal accelerations and is also the manifestation of the geometry of the spacetime when the spacetime is curved. I've drawn a diagram in a derivation for geodesic deviation here
http://home.comcast.net/~peter.m.brown/gr/geodesic_deviation.htm
If one wants see the curvature of space, then just follow the trajectory of a free fall particle.
I'm getting the feeling that you still believe that a twisting curve means spacetime curvature. Otherwise I can't understand why you'd believe such a thing.
Don't you recall my comments about regarding two apples? Only by comparing the relative motion of two or more particles can you whiff out whether the spacetime they're moving in is curved. Two apples accelerating towards each other are synonymous to two people starting to walk along parallel paths and each walks straight will eventually cross paths. In the case of the falling apples the relative acceleration of the two apples shows that their geodesics are deviating towards or away from each other.
I'm curious flr. Why do you think I brought up the relative acceleration of the two apples? This example is found in many books on gr and curved spacetime.
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I'm sorry but I don't know what "follow the space curvature" means.
I meant so say that photons and free fall particles follows a geodesic.
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The curvature is what we see when we map one kind of space-time grid onto a different kind of space-time grid. For analogy: Look what happens when you map the grid of log-log graph paper onto lin-lin paper, and vice versa.
Confusion results when we ascribe real properties to the grid, claiming that the grid IS the physical space in which things exist. In truth, the grid is merely a mathematical representation of reality. Reality can be represented by a variety of mathematical descriptions. More than one such description can be valid, though the one called Minkowski space-time appears to be the most useful for calculating trajectories at extremely high speeds, long distances and gravitational fields.
Perhaps a different mathematical space-time grid will be required to unify GR with quantum physics. If you map that grid onto Minkowski space-time, the new grid will appear warped in the vicinity of a particle.
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I'm sorry but I don't know what "follow the space curvature" means.
I meant so say that photons and free fall particles follows a geodesic.
100% correct. :)
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The curvature is what we see when we map one kind of space-time grid onto a different kind of space-time grid. For analogy: Look what happens when you map the grid of log-log graph paper onto lin-lin paper, and vice versa.
That is incorrect. Each piece of paper is flat having no curvature. Coordinate systems do not determine the presense of curvature.
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Can space the thought in some sense as 'inhomogeneous' due to its intrinsically curvature [near planets]?
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Can space the thought in some sense as 'inhomogeneous' due to its intrinsically curvature [near planets]?
Yes.
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According to Einsteins General Relativity the presence of mass curves space which we have observed during eclipses and through gravitational lensing. So my question is (and maybe the assumption is wrong but...) If space is empty then what is being curved? Seems that it must be something, not nothing if it is being curved.
To give an example of just space curvature you may like to read this from the book ‘Exploring Black Holes’. You can download the relevant chapter here…Pick the chapter ‘Curving 131004v’ http://www.exploringblackholes.com/ (http://www.exploringblackholes.com/)
From one of the authors Edwin Taylor, I have been told via email this is an example of just space curvature.
Two spherical latticework shells are built around a large mass/planet/star/ black hole. A plumb-bob is then lowered from higher shell to lower shell. The distance between shells is found to be greater than would be expected using Euclidean geometry.
Concerning next quote...
Here Reduced circumference is the radius you calculate from measuring around the spherical shell using Euclidean geometry.
Here the difference in r coordinates is the distance between shells using Euclidean geometry. The plumb-bob is assumed not to stretch.
From pages 3-13 and 3-14
Think of building two concentric shells, a lower shell of reduced circumference rL and a higher shell of reduced circumference rH, such that the difference in reduced circumference rH-- rL equals 100 meters. Stand on the higher shell and lower a plumb bob, and for the first time measure directly the radial distance perpendicularly from the higher shell to the lower one. Will we measure a 100-meter radial distance between our two shells? We would if space were flat. But outside a massive body space is not flat. The relation between global differential dr and measured radial di erential distance d comes from the spacelike version of the Schwarzschild metric (3.6) with dt = d = 0.
Why do we get a greater distance between shells than that expected with Euclidean geometry? It goes on to give this…
Objection Are you refusing to answer my question? What CAUSES the discrepancy, the fact that the directly-measured distance between spherical shells is greater than the difference in r coordinates between these shells? WHY this discrepancy?
ReplyA deep question! Fundamentally, this discrepancy shatters the notion of Euclidean space. We are faced with a weird measured result, which we can summarize with the statement, “Mass stretches space.” Your question “Why?” is not a scientific question, and science cannot answer it. We know only observed results and their derivation from general relativity. Does the following satisfy you? Space stretching causes the discrepancy! Section 391 3.8 exhibits one way to visualize this stretching.
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Thanks everyone for your posts and discussion. Interesting stuff.
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Thanks everyone for your posts and discussion. Interesting stuff.
You're welcome. And thanks for this comment. It's rare that we get feedback letting us know if we provided what the OP was looking for. Usually the OP will ask a question and sit back and do nothing but read the responses. It's better when we get feedback.
Call me needy but I like it when people tell us that we gave them what they sought like this. It's a good feeling to know you're being useful. :)
You may like to know that this sparked an idea to create a new addition to the general relativity section of my website at http://home.comcast.net/~peter.m.brown/gr/gr.htm to explain all this in a single web page. flr mentioned something to the effect of how would I describe this in some other context. So I'll use some spacetime diagrams as an aid to visualize what's going on. Would you like me to let you know when its finished?
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Would you like me to let you know when its finished?
Yes please!
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Would you like me to let you know when its finished?
Yes please!
Okey-dokey, Bill. I have to admit that my writing skills have a lot to be desired so this page will have a lot to be desired. Hopefully I'll get some input to help me write it better.
Here is what I have so far
The term spacetime curvature refers to the intrinsic curvature of a spacetime manifold as determined by the curvature tensor; also know as the Riemann tensor. It’s a term which is often misunderstood since in some cases the term curvature, when used unqualified, refers to the curvature of the worldline of a particle. Although Einstein used this term in his book Relativity: The Special and General Theory this referred to the curve and not spacetime. Spacetime curvature is a property of the spacetime manifold and not determined by trajectories of particles moving in said spacetime.
When a free particle is moving in an inertial frame of reference the spatial trajectory will be that of a straight line. The particles worldline (i.e. spacetime curve) will also be straight. If I now change to a frame of reference moving uniformly relative to the first then the trajectory will still be a straight line. If I now change my frame of reference to that of one, which is accelerating relative to the original inertial then the particle, will be deflected and the spatial trajectory will be curved. This kind of curvature represents the curvature of the trajectory/curve and is unrelated to any property of the spacetime manifold.
[tbd]
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If space is empty then what is being curved?
It seems to me that noone in this forum has answered this question. This is pretty consistent with what I have found in my travels in physics. It is possible that noone knows what spacetime curvature is all about but noone is gutsy enough to admit it! In his book "Simply Einstein Relativity Demistified", Wolfson states (p. 192) "We live, says Einstein, in a four-dimensional spacetime. The geometry of spacetime exhibits curvature and that spacetime curvature IS gravity". Easy peesy lemon squeezy n'est pas. I think not! Maybe all is needed is a great "explainer" to come along and elucidate the masses, or maybe the general understanding of the concepts involved has been lost in the mists of time. In the meantime most of us will just continue to scratch our heads and wonder.
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It seems to me that noone in this forum has answered this question.
If so then it's merely an oversight. The answer is very simple. First off, it's necessary to understand that the term curvature is a term used as an analogy to real and mathematical surfaces. When one is talking about a mathematical surface there is no real thing out there which actually "is" the surface itself. When we talk about curvature we're talking about what we deduce from measurements taken in the space of interest. If the distance measurements within the space are such that they describe a curved space then we say that the space is curved, even though there's no material body or material in the space.
In the text Exploring Black Holes - Second Edition the authors teach the reader all about black holes and black holes are described by Schwarzschild geometry. To see how they describe the curvature of the spacetime of Schwarzschild geometry please see:
http://www.eftaylor.com/exploringblackholes/Curving140829v2.pdf
Everything will be crystal clear as soon as we can visualize four-dimensional curved spacetime. But we do
not know anyone who can do this; we certainly cannot! So we compromise, we do our best to live with our limitations and develop intuition from the analogy to curved surfaces in space, such as the partial visualization of Schwarzschild geometry in the following sections.
This is pretty consistent with what I have found in my travels in physics.
Oh, really! And how long have you been traveling now and what is the extent of your travels? Have you ever even picked up a text about the philosophy of science to understand just what physics is and how it works? Have you taken courses in basic math, calculus, differential equations, vector and tensor analysis etc so you could understand the physics and solve homework problems to exercise the skills you're learning? Have you ever actually read a real physics textbook?
It is possible that noone knows what spacetime curvature is all about but noone is gutsy enough to admit it!
Why would you say something so rude about something you know nothing about? If you actually were to search for the places which teach the answers to the questions you're asking then you wouldn't make such rude assertions about people you don't know.
In his book "Relativity Demistified", Wolfson states "We live, says Einstein, in a four-dimensional spacetime. The geometry of spacetime exhibits curvature and that spacetime curvature IS gravity".
I have that text. What page is that on?
Easy peesy lemon squeezy n'est pas.
It's a very precise statement if one interprets gravity as modern gravitational physicist do. The use the fact that gravitational tidal forces cannot be transformed away and as such are what determines whether a gravitational field exists in the region of interest or not. The way to describe tidal gradients in terms of the geometry of spacetime is to say that the spacetime is curved since tidal forces are a manifestation of spacetime curvature.
I think not! Maybe all is needed is a great "explainer" to come along and elucidate the masses, or maybe the general understanding of the concepts involved has been lost in the mists of time.
Nearly every single general relativity textbook there is explains this.
In the meantime most of us will just continue to scratch our heads and wonder.
Not if you choose to read a GR text instead.
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When people first learn about relativity it is very unintuitive. Time dilation causes real problems for a lot of people until it clicks into place with a Eureka moment. The concept of the curvature of spacetime is just the same. It is best to read as many different explanations to get different ways of viewing it. It may take a while but without this effort no one can make reasonable judgements.
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It is best to read as many different explanations to get different ways of viewing it.
That's good advice, Jeff. My old prof gave me that advice. I.e. I asked him what I should do if I couldn't understand something that my text was trying to tell me. He said look in another text. I used that text all throughout my career. It's good advice. See http://bookzz.org/