[geordief (OP)]

Is there any way to show ,purely from the principle of relativity (thdt there is no privileged frame of reference) that   spacetime will always be curved?

That two objects will always diverge or converge eventually...

If so ,can this be shown in a simple ,intuitive way or does one have to go through the maths?

[Eternal Student]

LATE EDITING:  I'm human and make mistakes.  This doesn't change the spirit of what was said and is hidden under a spoiler in the appropriate place.

Hi.

Is there any way to show .....  that   spacetime will always be curved?

No, I don't think so.   If space is just empty (has no matter or energy in it at all) then according to General Relativity spacetime would not be curved, it would just be plain ordinary Minkowski space (or what we can call "flat").   That means the purely space part is just plain ordinary Euclidean space with exactly the simple geometry we all learnt in the early years of school.

Spoiler: show

   Actually, there are vaccum solutions of the Einstein Field Equations that aren't just flat Minkowski space.   For example, the Schwarzschild solution, or solutions with Gravitational Waves travelling through space.   This doesn't change the spirit of what was said because we only needed to know that we could get flat Minkowski space as a solution rather than that we will only get that.

I suspect you were interested in using just special relativity when you talk about "there being no privileged frame" rather than the bigger theory of general relativity.   Special relativity makes no use of spacetime curvature at all and it is just what you will obtain from general relativity where we insist that spacetime is perfectly flat (Minkowski space).   For example, if there is some region far away from any matter where the curvature tensor becomes almost zero valued, then that region of space can be well approximated by special relativity on its own.   So you certainly cannot prove that spacetime would be curved just from principles of special relativity - indeed it won't be and it can't be, it is a fundamental assumption of SR that spacetime is just described as flat Minkowski space.

You then went on to talk about two objects diverging.   Once you have some objects in space, for example matter with some mass, then you usually will get some curvature.   I'm tempted to say that you would always have some curvature but I can't be sure of that.    I can only tell you what I know.
    Any situation that I know about where there is some mass in the universe does produce a spacetime with some non-zero curvature tensor.       However, it might be possible to have some very special distribution of matter along with some special distribution of more exotic content that will contribute a negative enegy density (such as dark energy),  so that you manage to keep spacetime perfectly straight (zero curvature tensor) everywhere  -   I don't know for sure.  It's often thought that spacetime curvature is something described with just one number so that it's either poisitive, negative or zero valued but it is NOT like that.   Spacetime curvature is described by a curvature tensor, so this should be imagined as a matrix with many numerical values in it and all of these values must be 0 at the same time when we say the curvature is 0.   The content of the universe (matter, dark energy, stuff...) is similalry entered as a matrix of values and the equations involved are called the Einstein Field Equations (EFE).   Overall the EFE are notoriously difficult to find solutions for and we have not found all the solutions yet.   So that's why I've got to say that I can't be sure if there isn't some special distribution of exotic stuff spread through the universe that could still give you a zero valued curvature.

Finally you asked if this (convergance or divergance of objects) can be shown without complicated mathematics.   Well, yes we can do it without complicated mathematics - but what we will actually show is exactly the opposite thing (two objects do NOT have to converge or diverge).    Specifically, if we use only Special Relativity (SR) then we can just choose to have two objects with 0 velocity relative to each other.  SR doesn't involve any gravity or any reason why one object would attract or repel the other.   Furthermore, since spacetime is just flat Minkowski space, it doesn't expand, contract or change in any other way at all.   So the distance between the two objects never changes.   As such these two objects would NOT converge or diverege,  they would just stay the same distance apart for ever and ever.

I hope that's of some use to you.   Best Wishes.

[set fair]

Relativity isn't everything. It is claimed that empty space is negatively curved and the source of the increase in the rate at which the universe is expanding. If that is so, then by the scotch egg theorem aka the intermediate value theorem, it follows that if you set off from positive and travel to negaive curvature then you must pass through zero curveature. If you think of contour lines on a map then you could draw contours of curvature in sapce which would be 2 dimenioal contour membranes. Just like plateaus on land there might be extended volumes of no curvature.

[geordief (OP)]

Hi.

Is there any way to show .....  that   spacetime will always be curved?

No, I don't think so.   If space is just empty (has no matter or energy in it at all) then according to General Relativity spacetime would not be curved, it would just be plain ordinary Minkowski space (or what we can call "flat").   That means the purely space part is just plain ordinary Euclidean space with exactly the simple geometry we all learnt in the early years of school.

I suspect you were interested in using just special relativity when you talk about "there being no privileged frame" rather than the bigger theory of general relativity.   Special relativity makes no use of spacetime curvature at all and it is just what you will obtain from general relativity where we insist that spacetime is perfectly flat (Minkowski space).   For example, if there is some region far away from any matter where the curvature tensor becomes almost zero valued, then that region of space can be well approximated by special relativity on its own.   So you certainly cannot prove that spacetime would be curved just from principles of special relativity - indeed it won't be and it can't be, it is a fundamental assumption of SR that spacetime is just described as flat Minkowski space.

You then went on to talk about two objects diverging.   Once you have some objects in space, for example matter with some mass, then you usually will get some curvature.   I'm tempted to say that you would always have some curvature but I can't be sure of that.    I can only tell you what I know.
    Any situation that I know about where there is some mass in the universe does produce a spacetime with some non-zero curvature tensor.       However, it might be possible to have some very special distribution of matter along with some special distribution of more exotic content that will contribute a negative enegy density (such as dark energy),  so that you manage to keep spacetime perfectly straight (zero curvature tensor) everywhere  -   I don't know for sure.  It's often thought that spacetime curvature is something described with just one number so that it's either poisitive, negative or zero valued but it is NOT like that.   Spacetime curvature is described by a curvature tensor, so this should be imagined as a matrix with many numerical values in it and all of these values must be 0 at the same time when we say the curvature is 0.   The content of the universe (matter, dark energy, stuff...) is similalry entered as a matrix of values and the equations involved are called the Einstein Field Equations (EFE).   Overall the EFE are notoriously difficult to find solutions for and we have not found all the solutions yet.   So that's why I've got to say that I can't be sure if there isn't some special distribution of exotic stuff spread through the universe that could still give you a zero valued curvature.

Finally you asked if this (convergance or divergance of objects) can be shown without complicated mathematics.   Well, yes we can do it without complicated mathematics - but what we will actually show is exactly the opposite thing (two objects do NOT have to converge or diverge).    Specifically, if we use only Special Relativity (SR) then we can just choose to have two objects with 0 velocity relative to each other.  SR doesn't involve any gravity or any reason why one object would attract or repel the other.   Furthermore, since spacetime is just flat Minkowski space, it doesn't expand, contract or change in any other way at all.   So the distance between the two objects never changes.   As such these two objects would NOT converge or diverege,  they would just stay the same distance apart for ever and ever.

I hope that's of some use to you.   Best Wishes.

Thanks so much for your answer.I will study it and hope to be able to muster a considered  reply in  the next few days....

[Eternal Student]

Hi.

It is claimed that empty space is negatively curved....

Who exactly claimed that?

        If the energy?momentum tensor T_μν is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting T_μν = 0 in the trace-reversed field equations, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as     R_μν = 0 .   

[Reference and further explanation available at:   https://en.wikipedia.org/wiki/Einstein_field_equations#Vacuum_field_equations ]

    We're taking the modern approach of incorporating the cosmological constant into the stress-energy tensor, that is to say that it is treated as some content (such as the dark energy content) of / in  space.   So we have R_μν = 0 where there is no stress-energy content in the space,   or to say that another way .... the Ricci Curvature Tensor must be and can only be 0 in a region of space with no content.   

   I would be remiss if I didn't mention that a full description of spacetime curvature actually requires an even more complicated curvature tensor, called the Riemann Curvature Tensor.   The Ricci Tensor can be written as a matrix or of values with 4 rows and 4 columns   OR   treated as a 2-dimensional array of values with indices that range between 1 to 4.  So it has 16 individual numerical values associated with it.  The Ricci Tensor is a "medium" or "good enough" level of description for spacetime curvature.  The Riemann curvature tensor is an even bigger array of values with 4 indicies and each index can take values from 1 to 4.   So it has 256 individual numerical values associated with it.   This bigger and more complicated Tensor is what we need for a full description of spacetime curvature.

   As stated above, the Ricci Curvature Tensor is always 0 in any region of space with no content.  Note that this is a matrix or array of values, so when we say "it is 0" we mean that every individual entry in this matrix is 0.  I'm probably over-emphasising the point but it's important to recognise that we do have an agreed understanding of what 0 curvature means.  A matrix is said to be 0 if all of it's entries are 0.   However, we do not have any understanding of what a positive or negative curvature could mean - would that mean all of the entries in this curvature are positive, that most of them are, that the trace of the matrix is positive,... or something else?   Indeed we will not be able to talk about curvature as simply being a positive or negative number until we know a lot more about the space under consideration (see later, if we know we're working a FLWR universe then we could describe curvature this way).

    I've probably spent too long talking about how we can describe spacetime curvature - the most important thing to take away from this section is simply that we can't do it with a single number, we need some whole set or matrix of values.

....by the.... intermediate value theorem, it follows that if you set off from positive and travel to negaive curvature then you must pass through zero curveature.....

   There are several concepts described as "curvature" in GR or Cosmology.   As I have tried to emphasise above, an important one is the Ricci Curvature Tensor.   This is not a single numerical value, it's a matrix of many values.   As such the intermediate value theorem is not applicable.   A continuous single valued function that was negative and became positive must have passed through 0 somewhere along the way.  However, a matrix of values can start with all negative valued entries and end with all positive values and although each individual entry may have passed through 0, that doesn't have to happen to all the entires all at the same place, so the matrix may not have been 0 anywhere.

So when can we talk about curvature as if it is something that is just a single number?
    Another concept of curvature used in Cosmology is the notion of a purely spatial curvature.   So, that's a curvature of just a 3-dimensional thing we can call space,  it is not a curvature of the 4-dimensional thing like spacetime.   Furthermore, it's not as if we can describe any and all arbitrary 3-dimensional spaces with a single number for its curvature,  we can't.   We would still need some kind of matrix of values for this.   The most commonly used notion of curvature that really can be expressed as a single numerical value and typically given the greek letter  κ  (kappa),  only applies to the FLWR universe models (FLWR, FRW or some similar acronym is used because Friedmann, Lemaitre, Walker and Robertson are the scientists accredited to the development).   In those models space is assumed to have some important properties (e.g. space is isotropic - it's the same in every direction).   So the FLWR models involve maximally symmetric solutions and it is only because of this high-level of symmetry that we can fully describe the spatial curvature with just one single number, κ, instead of needing to use a whole set or collection of values to describe curvature.

    I don't know but I suspect you ( @setfair ) have blended notions of curvature from several PopSci articles and ended up imagining that curvature is always something we can describe with a single numerical value - but we can't.   I also don't think any of that is your fault -  PopSci articles want the reader to go away feeling good and believing they understand something.  So although they may be careful to restrict their attention to just FLWR models, they don't make any effort to point out that this doesn't apply to most things:  Spacetime curvature is never as simple as being a single number.

     Anyway, FLWR universe models do have enough symmetry to allow this simple description of curvature but even that is just going to be a description of the purely spatial curvature.     For example, a FLWR universe which is said to be flat, has curvature k=0, gives rise to spatial geometry that will be just perfectly ordinary Euclidean stuff,  is still NOT FLAT when regarded as a 4-dimensional spacetime manifold.

   See this reference for the equations describing the non-vanishing components of the Ricci tensor for a FLRW universe:   https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#Curvature
A moment of algebraic manipulation will show that
     A flat FLWR space will be Ricci flat                                  if and only if
     da/dt = 0   [where a = the scale factor of the universe]      if and only if
     The universe neither expands or contracts.

     We seem to be in a universe that is reasonably approximated as a FLWR universe which is spatially flat (has κ =0 ) but is expanding, so the 4-dimensional spacetime certainly will not have 0 curvature.  It's as bent as banana, despite the 3-dimensional spatial part being perfectly flat.

Best Wishes.

[geordief (OP)]

@Eternal Student,  you said
 " If space is just empty (has no matter or energy in it at all) then according to General Relativity spacetime would not be curved, it would just be plain ordinary Minkowski space (or what we can call "flat")."

The way I approach that part is that I don't separate "empty space" and the "contents" but I see them as part of the same entity -so that space is (as I see it for now) never empty but only filled with more or less matter.

That is why ,when I mentioned Relativity  in OP it was intended to refer to GR.

I also supposed that the lack of a privileged frame of reference applied also to GR and not  just SR.

So I wasn't really thinking about Special Relativity at all.

I was under the impression that all of SR's findings  would also be obtained in GR  since SR is described  as being a specific area of the general theory

But I could be wrong  since I have only a very broad /sketchy understanding of  either of them

So my question  is whether ,simply from  a requirement that all frames of reference are equally valid -and an understanding that the space is never empty -that the Euclidean notion of parallel lines that never meet  would be incompatible with those 2 basic requirements.

I am not suggesting  that gravity would have to  be the reason that coordinate lines would diverge or  converge but rather that  with no privileged  frame of reference  that perhaps  that in itself could be the readon that space is curved.

Not that I can see why it would have to be so (certainly not with  the weakness of my mathematical and  logical thinking) just that I wonder if it could possibly be the case.

[Eternal Student]

PART 1 of many....

Hi,

  I've spent a day trying to reply to your comments @geordief but they just get long and then I delete them.

It's probably best I keep it really short and worry less about accuracy.

The way I approach that part is that I don't separate "empty space" and the "contents" but I see them as part of the same entity -so that space is (as I see it for now) never empty but only filled with more or less matter.

OK.
I would look upon "space" as just being an abstract mathematical object.  All I need is some system, a co-ordinate system and I just consider everything between co-ordinates  x=0 and x=1  as being some region of space.
Sometimes there's a planet in the region between x=0 and x=1 and that's fine, I still consider that a region of space.  Sometimes there's nothing in the region between x=0 and x=1 and that's also fine.
Usually we can separate "space" from "the contents of space".

  I had a daughter who saw things differently.   She took the opinion that there was space around a planet but when you reached the edge of the planet then there would be only planet - but I still kept that daughter.


Best Wishes.

[Eternal Student]

Part 2 of many...

Hi,

I was under the impression that all of SR's findings  would also be obtained in GR  since SR is described  as being a specific area of the general theory

    So you're thinking of GR as something that grew out of SR...    Indeed historians say that is exactly the way it went.  I had a daughter who studied history (and I still kept that daughter).  In history you often do look upon development as just building more on the top of what you already had.
    However, it's a Friday, so we can choose to do whatever we want.   We can be less concerned about history, put more attention on mathematics and even take a bit of poetic licence.

All squares are rectangles.    The rectangle is a more generalised version of the square.   
So a rectangle is NOT required to have all the properties of a square (but it will probably just have some of them).
    SR is the square, while GR is a much more generalised version,  GR is the rectangle.   The rectangle isn't required to have all the properties of the square.

    We can still recover all the properties of the square from our theories about the more generalised rectangle if we try.   We just insist all the sides are the same length.   In a similar way, we will be able to recover everything about SR if we make certain stipulations in GR.

   I don't think Einstein built his new castle (GR) by keeping the first little castle (SR) and just building more on the top of it.   I think he started again with the new castle on some other plot of land and just glanced at the original little castle for inspiration.  Of course it wasn't quite as linear or simple as that ... his new castle fell down and was rebuilt a couple of times.  In its final form the big castle still looks like the little castle when viewed from one specific angle but only from that one specific angle.....  but I don't know, I wasn't there.

- - - - - - - - -
   

So my question  is whether ,simply from  a requirement that all frames of reference are equally valid -and an understanding that the space is never empty -that the Euclidean notion of parallel lines that never meet  would be incompatible with those 2 basic requirements.

   I don't know.    One of the first assumptions is suspect:   Laws of physics might be different in different inertial frames.   In SR that would be true, in GR ...ummm...  grey area...  but I'm biased and would say there are important changes in laws of physics between inertial frames.

Best Wishes.

[geordief (OP)]

Part 2 of many...

Hi,

I was under the impression that all of SR's findings  would also be obtained in GR  since SR is described  as being a specific area of the general theory

    So you're thinking of GR as something that grew out of SR...    Indeed historians say that is exactly the way it went.  I had a daughter who studied history (and I still kept that daughter).  In history you often do look upon development as just building more on the top of what you already had.
   

No I  think I see it the same way as you.
I feel he formed the two theories "back to front".
As if ,when he had finished GR he might have said to himself "what a waste of time all that SR was .If only I had started  with GR first, SR  would have followed as a consequence."


Isn't that why it is called (he called it?) GR  -to encapsulate that SR is just a small province of the  GR kingdom?

To your last point (I can't find the "quote selected" function that used to be there for me)

Don't you think there should be "one rule to rule them all"-from which all the sub rules  derive and which apertain to the allocated areas of relevance?

All the rules have to knit together  surely - they can't just be "laws unto themselves"?

   

[Eternal Student]

Hi.

(I can't find the "quote selected" function that used to be there for me)

   There are a number of features that don't work as well as they used to.
I've just lost the final part of the messages I was going to post because the  "save as draft" ability didn't work.   So there won't be any  "part 3 of many"  but that's not such a bad thing.

There are some work-arounds if you want quotes:

(i)   Use the "Insert Quote".    Much more than just the bit you wanted will be quoted but you can see this and should go in and delete all the surplus stuff by hand before hitting the  "Post" button.
(ii)  Just use the required tags in square brackets to start and end a quote yourself and construct a quote box.   If you use the "Insert Quote" feature a few times then you'll quickly work out and learn some more things you can add in these square brackets to make fancier things happen, such as adding the date and time of the quote  etc.

For example,  if you type this  (see below) 

Code: [Select]

[quote]  ES said:      The rain in Spain stays mainly on the plain.
[/quote]

... then the following will appear...

  ES said:      The rain in Spain stays mainly on the plain.

Best Wishes.

[Eternal Student]

Hi again,

   ..and yes,  GR does effectively supersede SR...

    However, enough people were relunctant to accept GR when it was published.   If there hadn't been a theory of SR published first which had already got some experimental evidence and support, then GR would probably have been thrown away and never accepted for any sort of publication.

Best Wishes.