Details are in the text of the Hobby Project "A small thought experiment about a finite size universe".
Redshift and non-Euclidian space structure. It might be useful to study general relativity first, before embarking on possible claimed alternatives.
Well a non-Euclidian space structure doesn't mean anything, first of all. There are many non-Euclidian spaces and not all of them are spacetimes at all; and all spacetimes described by a Lorentzian metric are non-Euclidian-but only a subset of those that solve Einstein's equations describe an expanding Universe-that, also, happens to be the one we observe.
Incidentally: not all cosmological solutions of Einstein's equations describe an expanding Universe-i.e. cosmological redshift.
The question is well put: non Euclidean spacetime. This is a non-Euclidean manifold in general or an abstract geometrical object; it is spacetime. The main description (and the first in the history of physics) was given by Einstein's general relativity (gravitation), where if a strong gravitational source emits (or receives) photons it is easy to calculate that they reduce ( or increase).
The standard form is to assume a Schwarzschild solution of Einstein's equations and to compare the Schwarzschild radius with the distance of the center of mass of the gravitation body with the place where the photon is emitted. Thus if you have non Euclidean spacetime you are not going to obtain the red-shift or the blue-shift that the relativistic theories of gravitation predicts (Einstein's is only one of them).
A spacetime is a solution of Einstein's equations-by definition. The Schwarzschild solution is static-so it doesn't have anything to do with cosmology.
Distances in a curved spacetime aren't physical, because they aren't invariant under general coordinate transformations-they only make sense as building blocks of invariant quantities.
Similarly, the definition of a redshift requires considerable care, in curved spacetime.
Dear Adrian.
Thank you for your obvious remark, but my question was trying to go a little bit further,
If all of the metric components are independent of the coordinate t, Birkhoff's theorem states that the Schwarzschild solution is the unique spherically symmetric solution to Einstein's equations in vacuum, i.e. any spherically symmetric vacuum metric possesses a timelike Killing vector.
But this is not a static metric, but a metric which possesses a timelike Killing vector which is stationary. The metric would be static only if it has a timelike Killing vector which is orthogonal to a family of hypersurfaces. Broadly speaking, a static metric is one in which nothing is moving, while a stationary metric allows things to move but only in a symmetric way.
Is anything against the redshift that you have asked and I have used the Schwarzschild metric to do it? Is anything to avoid that Schwarzschild metric could forbid in a cosmological model to enter this metric in its vacuum or in a very heavy body?
Frankly I cannot understand all the answers that Stam Nicolis has given in his posts to this question.
Let us ask more precisely:
Can a stationary non Euclidian space structure cause a red shift? (Key words are Cosmologie, General Relativity)?
or
Does "crossing a curved space region" have an influence on the wavelength of an electromagnetic wave?
Yes, I agree, it is an excellent answer to the original question, but the geodesic equation cited in the answer contains time dependency.
Thank you for the clear answer Adrian.
As a consequence we may ask: Is it possible that we live in a finite universe with a static homogenous and isotropic geometry in which the red shift is caused by space dilatation?
The validated concept of the Hubble constant is at least an indication for isotropy.
A non static, homogeneous and isotropic solution of Einsteins field equations is of type Friedmann-Lemaitre-Robertson-Walker.
But why should the Einstein field equations describe the geometrical structure we are talking about?
Primarily we are only looking for a static geometry which is finite, isotropic and homogeneous. If we have a mathematical (or numerical) definition of this kind of geometry it should be possible evaluating the impact on wave propagation.
But still, the FLRW metric is not static and is related to the speed of light.
As an example, a homogeneous, isotropic and finite geometry could be achieved by an appropriate mapping of a three dimensional cartesian product of circles. Shall we expect a large scale impact on wave propagation whithin such a geometry?
Did somebody analyse wave propagation within such kind of geometry?
Hi Wolfgang!
Applying wave theory to these problems is problematic, because it tends to lead to the idea that cosmological shifts, gravitational shifts and motion shifts all ought to obey the same set of rules and a common set of shift equations ... and that idea is problematic because it tends to wreck the basis of special relativity.
Under an SR-based general theory, gravitational horizons are strictly one-way surfaces, where events behind the horizon have no effect on the outside universe, by definition. They're assigned dates in the infinitely-far future, and therefore cannot affect the rest of the universe "now" without reverse causality.
However, cosmological horizons cheerfully ignore this argument, and happily fluctuate, radiate indirectly and leak in response to events happening behind them, regardless. They don't really care about extreme mathematical tidiness, and instead obey the more messy rules of an acoustic metric, and follow a corresponding set of non-SR shift equations. Their resulting classical indirect-radiation effects seem to map directly to Hawking radiation effects under QM.
If we now invoke wave theory to map between cosmological and gravitational curvature effects, and accept that the cosmological horizon behaviour is unavoidable, then gravitational horizons will also have to fluctuate, radiate indirectly, and give off Hawking radiation ... so we'd seem to solve the black hole information paradox, and reconcile GR with QM, and finally get ourselves a theory of quantum gravity.
However, the cost of achieving this is that changing the gravitational equations to match cosmological behaviour has the knock-on effect that the same non-SR cosmological relationships then have to apply to cases of simple relative motion, too ... so although we'd gain a theory of quantum gravity, we'd lose special relativity.
A lot of physicists would consider this price to be too high, as it would mean that one of the surest facts in gravitational physics – that curved-spacetime models must physically reduce to the SR equations over smallish regions – would be wrong. So I'd be surprised if you can find a full, proper, rigorous, comparative analysis of wave propagation applied to cosmology-and-gravitation in the literature.
If a researcher does this work and does it badly (or incompletely), it's not worth publishing, and if they do it properly and fully, the results still won't be publishable because they'll inevitably end up contradicting SR.
Hi Eric,
you see severe consequences if we analyse wave propagation in a nearly cartesian geometry in which x,y,z is mapped to circles with a radius of about 7 billion light years?
Let's consider that as a challenge for physicists and mathematicians who like differential geometry. I am sure that this will not have any influence on the reputation of Einsteins work. And I am convinced, that the methods used in Einsteins work will help to find the tensor expressions for the differential properties of that geometry.
One static spacetime is a contradiction by itself, because statics means independent of the time or even as if the time didn't exist. Schwarzschild metric produce a stationary spacetime which is a very different thing.
Hi Wolfgang!
The problem is that SR-based theory, and "acoustic metric" or "gravitomagnetically-based" theory have fundamentally different and incompatible definitions of how causality is supposed to work. Under SR's flat Minkowski spacetime, we assume that all events are visible and there's no such thing as a horizon, so reality is defined purely by what is directly visible, and the concept of indirect observation doesn't apply.
However, with a cosmological horizon, and/or with quantum mechanics, the ideas of indirect observation and indirect radiation reappear, and we have to use a more sophisticated concept of causality in which there's no real safe way to assign nominal dates and times to remote events, because those assigned dates can change when the properties of the signal path alter.
Einstein touched on this when he discussed the definitional basis of quantum mechanics versus that of SR with Werner Heisenberg – with SR, direct observation defines what is allowed to be said to exist, whereas with QM, the physics of what is thought to exist controls what we are permitted to be able to see. With acoustic metrics and QM we can suggest a deeper underlying reality that explains what we see, while with SR-based theory, reality is defined as being what we directly measure (with a certain amount of procedural interpretation). This is why there's a fundamental incompatibility between SR-based GR and QM-and-cosmology, and this is why cosmological shifts have to use a non-SR shift equation, to allow leaky horizons.
Heisenberg:
'' ... And since it is rational to introduce into a theory only such quantities as can be directly observed ...
To my astonishment, Einstein was not at all satisfied with this argument. He thought that every theory in fact contains unobservable quantities. The principle of employing only observable quantities simply cannot be carried out. And when I objected that in this I had merely been applying the type of philosophy that he, too, had made the basis of the special theory of relativity, he answered simply:
"Perhaps I did use such philosophy earlier, and also wrote it, but it is nonsense all the same." ''
" you see severe consequences if we analyse wave propagation in a nearly cartesian geometry in which x,y,z is mapped to circles with a radius of about 7 billion light years? "
Wave theory, applied to curved-spacetime problems, has a habit of generating arguments that seem to invalidate SR-based theory.
Example 1: For a moving high-gravity star, additional gravitomagnetic effects superimposed on top of SR would seem to require the star's shift relationship to then depart from the SR predictions. But wave theory plus the principle of relativity requires all distant objects to follow the same velocity-shift law regardless of their surface gravity, so if the velocity/shift relationship for a moving black hole diverges from SR, then so should a moving tennisball or a moving atom.
https://www.researchgate.net/publication/316981511_When_a_black_hole_moves_The_incompatibility_between_gravitational_theory_and_special_relativity
Example 2: If our universe contains cosmological horizons, then signals have to be able to propagate across a patch of spacetime intersected by a cosmological horizon, in both directions. This means that the horizon has to be an "effective" horizon that fluctuates and allows signals and objects from behind the horizon to suddenly appear in front of it, apparently acausally to a distant observer, although the local physics is entirely causal (see: cosmological Hawking radiation). The physics in the horizon-intersected patch seems to have to obey the more non-linear laws of an acoustic metric rather than those of Minkowski spacetime, and the associated cosmological redshift has to obey a redder first-order shift law rather than the SR version.
But if the internal physics of the patch is "acoustic", then since every other patch of spacetime in the universe (including ours!) can be considered as straddling a cosmological horizon for some distant future observer, this suggests that the entire universe operates according to acoustic metric physics rather than Minkowski-spacetime-based logic.
That's two propagation-based arguments that seem to lead to the downfall of SR-based logic, I'm sure that there will be others.
Working Paper When a black hole moves: The incompatibility between gravita...
Hi Daniel
"One static spacetime is a contradiction by itself, because statics means independent of the time or even as if the time didn't exist."
This statement is not applicable to a mere geometry in which time is not even mentioned. The basic question only considers the possible influence of a curved geometry on wave propagation.
A possible geometry is three dimensional cartesian with the coordinates mapped to giant circles. The proposed metric in this geometry is using the secant instead of the arc length. Within distances of a few light years, this geometry only has tiny differences to the Euclidian metric.
Similarities to other geometries (keyword Schwartzschild metric) are highly welcome because within those geometries wave propagation already has been investigated.
Let me add the last paper on this metrics ( Schwarzschild or Kerr as its generalization) for seeing that they are called properly stationary and no static, which is obviously wrong in its basic meaning. If you want we can enter in more details for seeing the vector fields associated to their symmetries (Killing vectors) where the time plays an important role.
Another more for your collection ...Would you like to have more? Can you discuss something so obvious by the definition of static and stationary? I know that somepeople, over all in cosmololly use this term in a wrong form, but this doesn't mean that we need to accept it.
Hi Daniel,
with Schwartzschild or Kerr you are talking about a distortion of metrics caused by the influence of matter and energy on spcaetime. But the question here is about the possible influence of an intrinsically distorted space on physical laws, without taking into account matter, energy, static or stationary.
Dear Wolfgang,
I think that it would be convenient that you read the last posts for seeing what was the discussion, at least with respect to me.
Let me to repeat your last post:
Hi Daniel
"One static spacetime is a contradiction by itself, because statics means independent of the time or even as if the time didn't exist."
This statement is not applicable to a mere geometry in which time is not even mentioned. The basic question only considers the possible influence of a curved geometry on wave propagation.
A possible geometry is three dimensional cartesian with the coordinates mapped to giant circles. The proposed metric in this geometry is using the secant instead of the arc length. Within distances of a few light years, this geometry only has tiny differences to the Euclidian metric.
Similarities to other geometries (keyword Schwartzschild metric) are highly welcome because within those geometries wave propagation already has been investigated.
Dear Wolfgang,
I also remember you that your question was:
Can a non Euclidian space structure cause a red shift? (Key words are Cosmologie, General Relativity).
And Stam Nicolis (SN)said that it was not well put. My answer (initial answery) was:
The question is well put: non Euclidean spacetime. This is a non-Euclidean manifold in general or an abstract geometrical object; it is spacetime. The main description (and the first in the history of physics) was given by Einstein's general relativity (gravitation), where if a strong gravitational source emits (or receives) photons it is easy to calculate that they reduce ( or increase).
The standard form is to assume a Schwarzschild solution of Einstein's equations and to compare the Schwarzschild radius with the distance of the center of mass of the gravitation body with the place where the photon is emitted. Thus if you have non Euclidean spacetime you are not going to obtain the red-shift or the blue-shift that the relativistic theories of gravitation predicts (Einstein's is only one of them).
And SN replied in general:
A spacetime is a solution of Einstein's equations-by definition. The Schwarzschild solution is static-so it doesn't have anything to do with cosmology.
Distances in a curved spacetime aren't physical, because they aren't invariant under general coordinate transformations-they only make sense as building blocks of invariant quantities.
Similarly, the definition of a redshift requires considerable care, in curved spacetime.
Do you remember now what was the discussion?
Dear Wolfgang,
By the way, Einstein's equation don't work with distortions of a metric but with a given metric solution of an energy-momentum tensor. Nothing more.
Adrian Sfarti "Eric, You should stop peddling your fringe theories."
Adrian, if you have a scientific argument supporting your point of view, then by all means, feel free to share it with the group. Otherwise you're lowering the signal-to-noise ratio. Right, now back to the science ...
Adrian ... which is, what, exactly? Do you have some reason to disapprove of research into how we might reconcile general relativity with quantum mechanics?
Oh dear. "Work" in inverted commas. Adrian, once again, please, if you have an actual counter-argument to what I posted, please do tell us all what it is. Point out the flaw. I'd be genuinely interested to read it.
If you're asserting that the "acoustic metric"/"gravitomagnetic" approach can't reconcile QM with GR, then please explain why you think this is true. Give us some arguments based on science or logic or a least some form of reasoned argument.
Remember, though, that the quantum gravity community have already produced quite a significant body of work concluding that we can replicate some of the most difficult QM behaviours in a classical model using acoustic metrics, specifically the Hawking radiation effect that's at the heart of the black hole information paradox and is considered to be the main faultline dividing QM and GR.
The problem with the approach seems to be not that the approach doesn't work, it's that the approach isn't compatible with SR, so to get that solution, we have to rewrite some of the foundations of GR to eliminate the SR layer and make it a curved-spacetime theory "all the way down". The result is, I think a structurally superior general theory to what we have now.
But if you believe that you know better, please explain your line of reasoning.
Adrian: "Rubbish, SR is the limit of GR in the absence of gravitating bodies. This is basic physics that you are totally ignorant of."
Really, that's all you've got? That SR has to be the limit of a general theory? That's old and obsolete.
The "geometrical reduction" argument was well-known, but not watertight. For a counter-argument, go back to the C19th and read William Kingdon Clifford's concept of physics-as-space-curvature. Clifford was the guy who invented Clifford algebra, and his "take" on curved-space theories was that curvature could (eventually) be used to explain and generate all known properties of matter.
Now, you don't have to believe that Clifford was correct about this, but a Cliffordian universe is a logical counter-example to the rule that gravitational theories have to reduce to flat-spacetime physics over small regions. Because in a Cliffordian universe, there's no such thing as" flat-spacetime physics". It's all curvature-based. Certainly, we can zoom in on a section of a CU and obtain a region that's arbitrarily flat ... but only if it contains no particles with relative motion with which we can do physics – it's a purely mathematical limit, but derivations made at that limit don't then correctly describe the way that real inertial physics operates in a CU. In a CU, the flat spacetime limit isn't the limit at which a gravitational theory reduces to SR, it's the limit at which meaningful physics has already ceased to exist.
So in a Cliffordian universe, special relativity would be a null solution, with no theoretical domain of applicability.
You are not forced to like the idea of a Cliffordian universe, or to believe that it is a correct description of our universe, but you are forced to accept it as a logical possibility, so until someone can demonstrate that a CU is unworkable, or that we don't happen to inhabit one, the hoary old "geometrical reduction" argument simply isn't a valid proof of the correctness of special relativity.
What does physics in a Cliffordian universe look like? Well, suspiciously like ours. If we associate every particle with a little dent in spacetime, the relative motion of those dents gives a gravitomagnetic derivation of the equations of motion, and the characteristics of the resulting physics are those of a relativistic acoustic metric. So the Cliffordian approach seems to reconcile classical and quantum physics, it generates a classical field model that seems to quantise cleanly to produce QM statistics.
Logically, we seem to have two different possible approaches to relativity theory, one in which GR reduces to flat Minkowski spacetime, and one in which it reduces to a relativistic acoustic metric, instead. The first one has the advantage that Minkowski spacetime is an easy system for people who find spacetime curvature scary, the second has the advantage that it seems to mesh with QM and has a much stronger internal structure.
If we now revisit your little statement,
"Rubbish, SR is the limit of GR in the absence of gravitating bodies. This is basic physics that you are totally ignorant of."
, then it would seem that one of us might well be a little bit ignorant about this subject. But that person probably isn't me, because I have the distinction of being the guy who actually did the study on this subject and identified the counter-argument.
I've enjoyed out little chat, Adrian, but you must realise by now that you are really not qualified to be participating in this conversation.
Dear Robert,
Let me start that static means that time doesn't play any role in the physics of such solution. One simple example is electrostatics where the electric charges are assumed to be fixed. The same could happen with masses fixed in a Newtonian interaction. The metric obviously in both cases must be static and time a constant, which naturally could be chosen as zero.
But what happens if we have not a classical interaction but a relativistic one? In such a case we could try to find a simple spherciall symmetric solution as Schwarzschild solution. Suddenly non classical effects arise and time plays a fundamental role even in such simple situation. For instance, if you concentrate all the mass of the Earth in a sphere of radius ( Schwarzschild's radius) around one centimer or 3 km for the Sun, the spacetime is very different appearing imaginary surfaces associated to the "event horizon"; i..e surfaces which can be only crossed in one direction. Mathematically speaking the event horizon is characterized by having light like vectors or null at all points.
Thus, the event horizon is a null hypersurface which has what is known as stationary limit. Taking into account the spherical symmetry it is convenient to calculate the Killing vectors of the Schwarzschild metric for being more precise. Outside of the stationery limit the Killing vectors are timilike, on it they are light vectors and inside spacelike Killing vectors.
Going to the question of this thread, the relativistic proper times are related by
τ (xA)= (goo(xA)/gtt(xB))1/2 τ (xB)
which produces an infinite redshift when gtt -> 0.
Do you think that theses issues associated to a Schwarzschild metric must be considered as a static spacetime?
Hi Daniel,
Let me try to comment your answer directed to Robert because it seems to me that there is again a misunderstanding concerning static.
The considered space in the original question has a Riemannian metric. It is like an Euclidian metric, but the coordinates are cyclically mapped to a giant circle and the distance is defined by secants instead of arc differences.
A Lorentz-Metric also can be defined in that space but I do not know exactly how, because of the cyclic propreties of the mapping. If we restrict the consideration to the mathematical concept of a Pseudo-Riemannian space based on the Lorentz metric, we surely can avoid misunderstandings related to terms static and stationary.
In my understanding a static space geometry may contain non static objects. The fact that objects have an influence on the geometry is not necessarily subject of the consideration.
We may consider a single photon in that virtual space, which surely will not have any effect on the geometry.
Adrian, the pdf has one comment so far (which you posted last night), citing that mainstream "reduction to SR" argument. If you look at this discussion, the post directly above yours addresses your concern, and quotes your comment in its entirety, twice, in convenient italics and quotation marks.
Rather than having "nothing", I have supporting logic, reason, mathematics, geometry, history, peer-reviewed results from other researchers, mechanisms, physical phenomenology, superior design, and a more efficient and more robust framework with fewer special-case rules and exceptions, a wider range of applicability, and improved compatibility with QM, statistical mechanics, thermodynamics and information theory. That's obviously not all going to be in that single four-page essay, but is in other online sources and documents.
I also have experimentally testable predictions, and a long list of verifiable mistakes made by the community which prevented them from seeing the alternative solution.
When you call it a "so-called ''paper'' ", it was actually written as an essay for the Gravity Research Foundation's annual competition (whose format explains the minimal number of references) ... but that information is clearly given on the title page and the first line of the RG description. RG's list of categories for classifying uploaded documents is somewhat restricted.
So far you've referenced one standard counterargument, for which I put the logical invalidation into print probably about ten years ago, and you've relied a lot on innuendo, "scare quotes" and hyperbolic exaggerration ( "so-called", "totally ___")
You've also so far posted at least two counterfactual and possibly libellous statements, one about my being "totally ignorant" of a principle for which (ironically) I'm actually one of the few researchers who's done any work on it, and a statement that "Your ''theory''makes no falsifiable prediction", when in fact, it makes some very specific testable predictions.
For a scientist, there is no excuse for this sort of misbehaviour, and while slandering your opponents may well succeed in damaging them, you have to understand that this behaviour also damages your own reputation and people's willingness to be associated with you, and also damages science.
Dear Robert,
I think that we are discussing very basic concepts that doesn't deserve to be explained too much. For me Schwarzschild metric is in general stationary as Kerr metric (both have metric no dependence of time) and never static because that is in contradiction with the relativistic concept of spacetime ( time= 0 is meaningless). And less if this argument is the basic to say that it cannot be used in cosmology by this reason.
But let me to give two more reasons:
1. How do you can employ the usual coordinates within the Schawarzschild spacetime metric as can be Eddington-Finkenstein (where time is explicit) or Kruskal-Szekeres (introduced through a velocity).
2. The Penrose-Kruskal diagrams of the Schwarzchild spacetime are also impossible to follow in a "static" metric.
3. In the reference
H.Quevedo, Multipole moments in general relativity-static and stationary solutions, Fortschr. Phys./ Progr. Phys. 38/ 733-840 (1990)
you can find quite well what is a physical separation between these two kind of solutions.
Dear Adrian,
Thank you. My resoning is a little bit more complex but at the end you have summarized it quite well. Static is R3 Euclidean metric where time is not a coordinate but a parameter as it happens in Newton's gravitation or in electrostatic. Schwarzschild, Kerr, Reissner-Nordstrum are stationary.
Speaking very simply spacetime cannot be static, if static means no existance of time. That is a contradiction. I know that in gravitation some people use this terminology for Schwarzschild metric but this is meaningless from my humble point of view.
Robert, I have said also all that I could about this issue.
Dear Adrian and Robert,
I think that our personal positions are clear on what is the meaning of static and stationary. For me it is a contradiction to speak about static spacetime, that is absurd!
Robert, please, could you in a simple and possible form to see how the presence of an irrotacional timelike Killing vector is always associated to a Schwarzschild metric? What would happen in the case of Kerr metric?
Thank you.
Dear Daniel,
Yes, I agree with you there is a huge difference in Physics between static and stationary.
An orbit is stationary but cannot be considered static.
Actually the Schw metric of a massive object is considered static, it is considered time independent. It is the same for every t, unless the massive object interacts with some other big massive object. In such case the metric has to be considered time dependent.
Dear Robert,
Just for answering your "fantastic" paragraph I attach a paper where you can find a stationary Schwarzschild spacetime. I have more but I think that this is enough. By the way they do not use Birkhoff's theorem, but they obviously have it in mind.
Dear Stefano,
Everybody who has received an initial course of gravitation knows the physical conditions of the Schwarzschild metric. We are speaking of one spheric symmetric solution of Einstein's equations where all the mass is assumed to be concentrated in the center of the sphere. This is the main difficulty to traslate this metric to cosmology and no to be "static" as some people has said here in this forum.
It is true that in some cases and for certains zones of the space time associated to this solution, time can be considered a parameter constant if a hypersurface is chosen. This is not general and it is also know that Schwarzschild metric is the ideal for studying black holes where the structure of spacetime contains singularities and the time never can be fixed and less be made constant.
In any case, spacetime is always in contradiction with static configurations. This is not a convention but only an instant of thought.
Dear Robert,
You start:
"As always, I stress that Schwarzschild means exterior, so only the region outside the event horizon".
That is not right. The Schwarzschild metric contains more parts than that. If this were true then your interpretation would be right and our discussion would be a waste of time. Please don't try to make cheats.
" in terms of the standard coordinates, t is a time coordinate (since the vector T=d/dt is timelike), so that the Lie derivative of the metric wrt T is just the coordinate derivative".
This is also true for Kerr metric that you say that it is stationary.
In any case, you never has employed the property of being irrotational the Killing vectors. Frankly, Robert I think that your arguments can be read for many people and I don't want to put you in evidence. If you want we can finish just with the discussion of static and stationary because I doubt of your knowledge ( I hope that this is only a lack of space for explaining you enough well).
Dear Robert,
Just in the abstract the authors start saying that they are going to find a stationary,
Dear Robert,
(Sorry )
Just in the abstract the authors start saying that they are going to find a stationary, spherically- symmetric vacuum metric.
Birkhoff's theorem tell us that every metric with this conditions is equivalent to Schwarzschild's one. Thus.....
I know that in GR most of the people call this metric static without thinking too much in the physical consequences. If you only use it as a pure name there are no problems, What is a problem is if you think that this name is correct and even is in agreement with the physics behind it.
Dear Robert,
I apology for telling these words, but I don't know how to explain me in more clear form. Please, don't consider it as a personal attack and let me try to justify it.
1. For me it is a cheat that you enter at this stage of the discussion that you only consider a part of the Schwarzschild metric. The timelike one avoiding the important one that I have told you many times.
2. I attached you many references that you read, from my humble point of view, without understanding them properly and just repeating as a mantra that Schwarzschild spacetime is static.
3. You gave a definition of static using the irrotational property of the Killing vector bases. But instead of applying your definition, which was my question to you, you just refers to a choice for the Lie derivative in general without taking into account the Schwarzschild symmetries. Frankly, I hope that this is not offensive, but I have a serious doubt of your knowledge. This is only going to the short answer that you have given me and the considerations made on the references that I have shown you that I have attached you.
In any case, if you want, forget the discussion that I hope that everybody can get his own oppinion, provided it can have enough interest.
Dear Stephen,
First of all I am a theretician that I have been working in many different fields of physicas and mathematics. I have a book written many years ago on classical fields (electromagnetism and gravitation as gauge theories) on the language of differential forms (within principal fiber bundles as a geometrical background). Thus I think that I am asking or arguing in a reasonable form. Science is not for definitions and less physics, if you write
Yes, the Scwarzschild metric is static
That means that I have lost my time and what is worse, the time who is following us.
Dear Robert,
I would repeat my apologies if I have offended and also give you thanks because I have made one idea of what could be understood by static Schwarzschild metric. I have to say that always thought that this was just a form of classifying metrics but without taking it seriously from the physical point of view.
Although I'm not very sure in some details, if I assume the Killing vectors v by
Łvg=0
And assuming the manifold is simply connected (which cannot be made by the whole Schwarzschild spacetime) we have that the vector field can be written in the form using Poincaré lemma
V=grad (r)
which may be depend only on space coordinates due to its spheric symmetry.
Thus we have an external associated "static" field which can simulate the spacetime produced by the Schwarzschild metric.
(Static, is not well defined in this context.)
Suppose that light enters a surface that has time to respond before the light leaves. Then, the boundary layer provides additional dynamics. If the observer is close (which can be modelled with densities), 'he' will detect this.
The red shift in cameras is since light changes its curvature at reflections, and that can't be considered as static. But the photo is indeed a static projection.
(if not many photos on a curved line)
It is more natural/easy to consider local motions in terms of densities and small distances, than the large static projections.
(Is that how nano technology developed, and what's next?)
Hi,
It seems that in this discussion about a non Non Euclidian space structure, automatically an Euclidian space distorted by General Relativistic effects is assumed. But the original question was meant more generally. The question is dedicated to a finite geometry of the whole universe and the distortion is exclusively caused geometrically by finity. In this context "static" is only related to the geometry and not to objects which follow eventually modified laws of nature within that geometry.
I used to discuss statics and dynamics with my professor in Mechanics, who also had a past in chemistry.
I agree to concise definitions. Time invariant, is also descriptive for scalars. Boundaries implies a hyperspace. Is that metric related to cars, in general?
..the Schwarzschild cat gravitating towards car lights
While I don't work in this area, I think that the reason many authors discourage people from thinking about cosmological redshifts as "Doppler" is that thinking in terms of "motion away from us" makes several basic errors attractive (think "we're in the middle" and "the universe is expanding into pre-existing space"). There may not be a definite known answer, but a broader view of the problems surrounding this specific topic may allow an open-minded reader the opportunity of taking their own decision as to whether or not to believe “conventional wisdom”.
Initial interpretations of the redshifts and blueshifts of interstellar objects beyond the Milky Way were based exclusively on the idea of the shift being due to the Doppler effect. However, after Hubble discovered a rough correlation between the increasing redshift and the increasing distances of galaxies, theorists almost immediately realised that these observations could be explained by a different mechanism for producing redshifts.
In the theory of general relativity, there is time dilation within a gravitational well. This is known as the gravitational redshift or “Einstein Shift” . The gravitational redshift of spectral lines is often held to be one of the “crucial tests” of general relativity. However, the result may also be derived with no recourse to the general theory of relativity whatsoever, nor even to the principle of equivalence , as has been shown on several occasions.
In other words, it is more likely that we are not in a special place and the universe is expanding than that everything in the universe is flying away from us. This is also supported by the fact that we cannot find anything else particularly special about our location in the universe: the galaxy we're in is typical, the group of galaxies our galaxy is in is typical : NO FRAME TO BE CRITICALLY SURE OF .. . .. .. .. .. . ....
Moreover , what observational evidence do we have for a gradual (GR) increase of the redshift, disproving the possibility of an instantaneous doppler shift at the moment of emission ????
The stretching out of a signal's wavelengths, with the speed of light held constant, will also cause the duration of the signal to increase, causing an apparent time-dilation. One can play around with slowing down and speeding up audio signals to see how this works - you need only to do some extra work to keep the pitch the same if you do that. The converse is also true - if you just alter all of the pitches in an audio signal, you'll alter the duration, too, if you don't do extra work.
Reza Sanaye
Thank you for this detailed explanation.
Let us consider the possibility that our spacetime is a de Sitter universe with a 3-Sphere as the spatial component.
Then geodesics are circles with a giant radius R. If we now compare the arc length with the secant length, then we see that the difference growths with the arc length. This means that with the path length, the elongation of the whole path growth. This growing elongation just causes a red shift.
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