Given is a finite, homogeneous, isotropic, three dimensional space with constant non zero curvature and a scale facter which is constant in time. Also given is an arbitrary geodesic in that space.
Does another geodesic exist, which at every point has the same nonzero minimal difference to the first geodesic?
If the anwer is no, do we expect consequences for photons with a non zero spatial extension travelling along a geodesic?
No. The reason is that, since the FLRW spacetime is curved, the distance between two spacetime points doesn't make sense by itself, since it isn't invariant under general coordinate transformations. Therefore the premise of the question doesn't make sense.
There aren't any consequences of the statement, since it's meaningless; how particles move in curved spacetimes is described by the geodesic equation (the mass of the particle enters into a constraint).
finite, isotropic, three dimensional space cannot be homogeneous, as finite implies edges. As far as I know all cosmological theories work with infinite space. Then, after adding another dimension, the system might be finite. But it is at least 4D.
It doesn't matter-there are boundary contributions, in any case and these ensure invariance under general coordinate transformations. It's this that implies that the notion of parallel geodesics is meaningless in curved spacetime.
Once more, it's not useful to mix up space and spacetime: spacetime geodesics aren't geodesics on space-like slices.
Hi Robert, Stam, Igor
What do you think about the following geometry:
At each point, every straight line (or geodesic line) is topologically equivalent to a surrounding bundle of tangent circles with a giant radius. Any infinitesimal small displacement in this geometry is accompanied with a random process selecting a specific one out of that 360 degree bundle of circles. The displacement then takes place on that specific circle.
It is not even clear that this kind of space is a Semi Riemannian manifold. Intuitively even a single geodesic has diverging properties. Something like parallel transport seems to be undefined.
The topology of that space is surely homogeneous, isotropic (and static) and the space is finite. It is imaginable that the diverging properties of that space allows explaining red shift phenomena.
Would you admit, that a profound mathematical analysis of this kind of geometry in which statistical and differential geometric concepts are merged, is urgently required.
Sincerely ,
Wolfgang
The statement doesn't make sense. A geodesic is the solution of the geodesic equation, along with boundary-or-initial conditions. It is possible to construct manifolds where the geodesic is a circle. In the Universe we live in this isn't true.
And it is, also, possible to study random manifolds in this way, too, where the metric depends on parameters, chosen from a distribution.
The mathematical analysis is known (e.g. through the work of Donaldson).
However the point is that the redshift phenomena relevant to astrophysics and cosmology are described by the FLRW metric.
Hi Stam,
Sorry, which statement does not make sense? Do you mean the definition of the geometry itsself with a combination of a geometrical and a random concept?
The given definition of the geometry only describes an infinitesimal displacement and not the geodesic. It is not even clear if a geodesic in the Riemannian sense with a defined parallel transport exists in the proposed geometry.
Hi Stam and others,
I did not believe in the red shift before because I thought it was a Doppler shift, and such are detectable only when the source mixes with, (/is close to) the observer.
And some painters schools instruct to use red colours when things should look distant. Painters also pictured galaxies originally.
After reading here on Reseach gate, I changed my mind so i think rctionedshift could be due to small distant change of curvatures in space. Otherwise it is violet, indigo or black for light traveling long distant. I can change my mind again, quite easily, since this is true only at Night. For light; there should be a description mod angle together with wavelength, to give that violet and indigo are close?
I wrote a book, suggesting that the stars are discrete solutions to electromagnetism on such spaces. Mathematicians call them Frechet spaces, and it is a present subject for them. The Southern Cross and parts of the Charles Vain may be such projections similar in their discrete shapes. In my opinion.
Hi again,
Concerning parallell geodesics. These star figures, visuable both on the Northern hemisphere and South could be that the eye makes such images, but I think it is objective and related to the Earth magnetic field. There is one more trace of evidence..
All these issues aren't the result of either beliefs or personal opinions-they're the result of impersonal calculations and measurements. it doesn't make sense to discuss random geometries, in the context of the FLRW metric: this is a family of metrics, that are particular solutions of Einstein's equations and have certain well defined consequences, results of calculations-and these results can lead to predictions for effects, that have been measured. To understand what these statements mean, it's necessary to learn general relativity.
Parallel geodesics only make sense in flat, Euclidian, space. The spacetime that describes our Universe is neither flat, nor Euclidian. Its geodesics are solutions of the geodesic equation for the FLRW metric.
I tried to adapt and understand the words in the question. What I meant was that there is a path of 3-4 stars towards the Pole star; Ursa Major. The path may be in a direction of some geometry. I thought of it perpendicular to the other in-plane motion giving the 4-stars.
Hi Stam and others,
the proposed geometry with a random selection of a specific circle out of a 360 degree bundle at every infinitesimal displacement, does not have an FLRW metric. Therefore there is no contradiction and no conflict with Einsteins theory, it is simply something else.
Due to the immense radius of the circles, the random selection at a specific displacement does not have any direct physical consequences. But on a larger scale, the displacement rule generates a closed space. A possible geodesic is not a specific circle because at any infinitesimal step another circle of the same radius is selected.
Nevertheless this space is homogeous, isotropic and static and has a constant curvature (if the curvature can be defined).
Because in that space light follows a curved path, this easily explains red shift and the observed time extension of supernova explosions:
The curved path is longer than a virtual Euclidian path. Delivering, as an example one day of real time information via this extended path, takes more than one day.
I had the pleasure of reading a paper on a model for a(t), recently. It was in the format dot(a)/a = f(a, and other cosmological properties). I compared with continuum mechanics, and my own discrete nonlinear equations to estimate a ratio. I don't know how much Einstein meant that it could vary. In continuum mechanics there are some connections between volumetric expansion and semi-rigid-body rotation
Hi Lena
Einstein did not try finding an appropriate model for the geometry of the universe. He only wanted to find and actually found a very precise model for phenomena in the universe.
There is also an ongoing misunderstanding in this discussion because it has in the headline "FLRW Metric", which is part of a solution of Einsteins equation.
The proposed geometry with a random component does no longer match to that headline. But it still matches to the symmetry properties.
It doesn't make sense to describe in words what is wished for, especially, if any calculation can contradict it-what matters is what can be calculated.
There are infinitely many metrics that are equivalent to the FLRW metric-the reason is invariance under general coordinate transformations. If one is interested in other metrics, these don't have anything to do with the question being discussed.
If the metric does have all the symmetry properties of the FLRW metric, it's equivalent with it-if not it isn't. So it doesn't matter how the coordinate system is defined. What matters are the properties that do not depend on the coordinate system. That's why describing the metric is pointless-the consequences only matter.
The Universe has a finite age. But it's expanding, as a spacetime, and the rate of this expansion is accelerating. These are the only things that matter. If the metric one is interested in doesn't have these properties, it doesn't describe the Universe we live in.
And it doesn't matter what Einstein or anyone thought about how the scale factor could vary-what matters are the equations that describe its variation and how they are solved.
Despite certain issues in common, general relativity is NOT equivalent to continuum mechanics and the differences matter and they're known. So it's much more useful to study a textbook or lecture notes on general relativity, at least the parts that refer to the FLRW metric, than trying to guess. This information exists, so it's pointless to pretend that it doesn't. Research in cosmology relies on mastery of this background.
Hi Stam
You wrote:
The Universe has a finite age. But it's expanding, as a spacetime, and the rate of this expansion is accelerating. These are the only things that matter. If the metric one is interested in doesn't have these properties, it doesn't describe the Universe we live in.
This sounds like a Dogma. But what would be if we simply analyse a Universe with a finite constant size?
This Universe can't be Euclidian and must be curved.
If on a large scale in this Universe light is forced using a longer curved path compared with an Euclidian path, this would lead to a stretched time scale along that path. This stretching, compared with the shorter Euclidian path, leads to a red shift and an extended duration of supernova explosions, very similar to that, what we observe in our Universe!
There's no dogma involved-these statements are the results of calculations and of measurements.
It's the statement that the Universe has a finite constant size that can be checked to be wrong-that's the work of Ryle and Hewish, for instance.
The statements that the Universe is or isn't described by Euclidian geometry can, also, be checked and the statement that its geometry is Euclidian can be found to be wrong and the correct statements can be found to be that it's expanding at an accelerating rate and it has Lorentzian, not Euclidian, signature. There's no meaning to the statement that ``the Universe can't be Euclidian''.
Once more, words just describe the results of calculations-and their meaning matters, not the formulation. There are many ways of saying the same thing, so it doesn't make sense focusing on the fact that the same thing is said another way.
If different formulations lead to the same results it's just a matter of personal taste, which to use-no one is better or worse than the other. The results matter more than how they're obtained. The vaguer the statements, the worse the method; the sharper the statements, the better the method. The method is only of interest to understand the bigger picture, beyond the immediate problem solved.
The Universe isn't dominated, as far as can be observed, by supernova explosions as such-so any description that relies on that fact has problems. And the quantitative description of its accelerated expansion, provided by general relativity, is sufficient for describing redshifts of galaxies quantitatively; so any other description is equivalent-only the description by general relativity is, already, known; so there's no point in studying a description that doesn't give anything more, a new prediction, all the wh!ile being consistent with general relativity.
Hi Stam
Thank you for your patience. Could yo provide a link to the according work of Ryle and Hewish?
But I still would like to know if you can follow the purely geometrical argument, that a finite size atomatically leads to a red shift, because a curved light path is longer than a straight path.
WK: leads to a red shift, because a curved light path is longer than a straight path.
Travelling a longer path doesn't imply a redshift, just a greater but constant time to reach us. "Redshift" describes a change of frequency between the value measured in the rest frame local to the source and that of the observer.
Hi George,
you are perfectly right in an Euclidian Geometry. But in a curved geometry the originally straight line is actually stretched like a gymnastics rope. And then the information dispersed on the stretched line becomes read with the original rate in units length per time. Because of the increased length, now more time is necessary to read the Information.
We also could use another view:
Compared with a straight geodesic a light pulse needs more time across a curved geodetic line. For the connection of two points A and B this means an effectively reduced speed of light compared with a straight path. The Information delivery rate of the connection from A to B is now proportional to this reduced speed of light. At point A this communication line accepts information with the standard rate, but delivers this information at point B it with an effectively reduced rate. We notice such a reduced information delivery rate as a red shift.
This view is only applicable to geodetic lines. An extended line length using mirrors is not geodetic.
That in any given spacetime, geodesics have greater proper length than other curves, connecting two spacetime points, doesn't imply that the light emitted from some source and detected at some receiver is redshifted, beyond proper motion. Redshift means that something's shifted-so in a universe of fixed size there isn't any redshift (or any blueshift), beyond that of proper motion-which isn't the issue here, since that's, already, been subtracted.
The redshift observed, that's independent of proper motion-for which Ryle and Hewish were awarded the Nobel Prize, https://www.nobelprize.org/nobel_prizes/physics/laureates/1974/
(the fundamental paper is http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1961MNRAS.122..349R&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf )
s due to the expansion of spacetime and can be quantitatively described this way, as the recent data by the Planck satellite, https://arxiv.org/abs/1502.01589 for instance, have, also, recovered.
Hi Stam,
Thank you for the informative answer and again for your patience.
There is still one point, I don't understand. If there is a communication line between two points and the rate with which this line accepts information is higher than the rate with which it delivers information, then isn't this property equivalent to a red shift?
No, because the statement about information and communication line doesn't make sense.
The statement that light detected anywhere from given sources is redshifted, beyond proper motion of the source, turns out to be equivalent to the statement that spacetime, as described by general relativity, is expanding in a certain way.
WK: There is still one point, I don't understand. If there is a communication line between two points and the rate with which this line accepts information is higher than the rate with which it delivers information, then isn't this property equivalent to a red shift?
It would be but a path of constant length will deliver the information at the same rate as it entered whether it is curved or not. A path of time-varying length will produce the data at a different rate, a lower rate if the length is increasing.
Think of the old trick of talking to someone using two tin cans and a piece of string. It doesn't matter if you hold it taught so the string is straight or let it sag into a catenary, if you put a 1kHz tone into one end, you hear 1kHz at the other. On the other hand, if the string is stretching due to the tension, the note heard would be lower.
The main difference between Euclidean space and curved is that the apparent size of distant objects is altered, or equivalently the apparent distance to objects of known size.
Hi George,
the critical point is, that compared with the straight path, the effective speed of light is reduced on the curved path. If you increase the path length in a curved space, then you decrease the transfer rate. This rate reduction doesnt occur in a string phone.
WK: the critical point is, that compared with the straight path, the effective speed of light is reduced on the curved path.
No it isn't, it remains the same at each point along the path and the longer path then just means it takes more time to get here. Even if what you said is true, the speed of light is reduced in glass but the frequency is unaffected.
WK: If you increase the path length in a curved space, then you decrease the transfer rate.
No, you just increase the travel time, there is no effect on the data rate.
WK: This rate reduction doesnt occur in a string phone.
Right, and it doesn't happen in space either.
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