Dark Energy or a Simple Mistake: Do We Understand How Non-Unique the Einstein Tensor Actually Is? by Jeff Baugher

I agree with the quote in your essay: "nothing short of a revolution in our understanding of fundamental physics will be required to achieve a full understanding of the cosmic acceleration." In addition, I do not believe that such a revolution can originate at any of the so-called top universities.

Now about your essay. I have two issues with it.

1) In §2 on page 2, you mention that when integrating over a specific interval [a, b] two different integrands (functions) can yield the same integral. Of course this is so, e.g. the function f(x) = 2x and the constant function g(x) = 1 yield the same integral over the interval [0,1]. But then you seem to go off on a tangent. You define f1 = C - f2 and then you show that both f1 and C - f2 yield the same integral over the interval [x1, x2]. But how is that an example of two different functions yielding the same integral? The functions "f1" and "C - f2" are, namely, identical by your own definition. And as a side note, the term "C" in this defintion of yours is a constant function, not a constant of integration (which is a number) as you seem to imply in figure 2. So why this example?

2) In General Relativity (GR), the index-free form of the Einstein field equation (EFE) is [math]G+\Lambda G=\kappa T[/math]

I understand that you want to replace this EFE by this equation:

[math]g\Omega-L=\kappa T[/math]

But why is that an improvement? With the EFE, one can calculate the metric tensor (or the metric tensor and the cosmological constant) given the stress-energy tensor T. But what is the physical interpretation of the tensor L in your equation? You might be able to calculate it, but what is it? You refer to figure 3 as an illustration, but what are the two scalar fields in the bottom picture physically? What is their physical source? What is the corresponding theory of gravitation (what is gravitation according to you)? It seems to me that your idea requires some further development.

All in all, the parallell between our essays is that we both question GR. But there are nuances. I believe that GR is correct in its area of application, but emergent instead of fundamental. You, on the other hand, seem to believe that GR isn't even correct in its area of application, as you suggest another field equation - I take it that you propose these as an improvement of the EFE even at macroscopic scale. Am I correct?

Good luck with the contest.

Best regards, Marcoen

@Steve: hail all hail to the revolution (Jon English).

Hi Marcoen and thanks for the excellent questions:

(1) Perhaps I should state this as actually three different functions f1, f2 and C. I do mean that C is a constant (i.e. a number) and not a function of x (i.e. C(x), a constant can still be integrated, so perhaps when I referred to it as a constant function this was misconstrued as meaning it to be a linear constant function of x, which it is not). A very elementary part of calculus, not something which stands out. The point is that from the answer alone, which would just be area, there is no way to tell whether I used an integration of f1 over x1 to x2 or the integration of C-f2. If one finds the antiderivative of a function, there exists an arbitrary constant of integration which could have any value.

I use this specific example, in that if the cosmological constant is a constant of integration (sometimes called the unimodular approach), then it is a bit of a coincidence that one of the biggest paradoxes about it is that it should be extremely large (according to QFT) but that empirically it appears to be rather tiny. In my example, the constant C could be extremely large or it could be zero and there is no way from anti-differentiation to tell which.

(2) Why would my equation be an improvement over Einstein's tensor G? It could also be fairly asking whether I am making the equation more difficult. To answer this question fully, it might be helpful if we consider a model of a cosmological fluid (see the Dark Energy Task Force report on the use of these). With the modified EFE, there is no known way to put in the estimated value of the cosmological constant (10120 higher than observed) and also find a mechanism which makes it very small. However, by using the alternate equation I propose, a full theoretical value of the "vacuum" could be put in. Suppose then there is matter present at a point in this vacuum. If we model this particle as a reduction in the density of this cosmological fluid, the Luv would be equated to the stress energy tensor of the remaining density kTresuv. Since this constant and remaining stress energy tensor together are mathematically equivalent to the Einstein tensor/stress energy tensor of matter, it should approximate Newtonian gravity, although calling that gravity "attractive" would be qualitatively incorrect. The main point of this though, is that it appears that it would not model it exactly, since at larger radii the Omega term would increase linearly to a point where the gravitational vector switches direction from an apparent attraction to a repulsion. Whether this matches what we empirically observe I do not yet know, but there are too many coincidences in the formation of field theory, action at a distance, GR and now a linear accelerating expansion that has been tied to the cosmological constant to not be suspicious. If this model gives us a more accurate picture of macro gravitational effects, I would also become very concerned about how abstract these cosmological fluid models actually are.

Forgot to answer: "You refer to figure 3 as an illustration, but what are the two scalar fields in the bottom picture physically? What is their physical source? What is the corresponding theory of gravitation (what is gravitation according to you)?"

The scalar fields are generic vector examples of Newtonian gravitational fields.

The first scalar field is a vector representation of the Omega (or equivalently the cosmological constant) term. It should actually be a scalar field with linearly increasing values radiating out in a circular fashion, but it complicates the explanation in that by definition gravitational force vector requires at least another mass. It is an isotropic pressure.

The second scalar field represents a vector formed from the gradient of stress energy tensor equated to Luv. The field scalar values are of a much larger magnitude and the gradient is of an opposing direction, but of course the minus sign in front gives us an equivalent force vector to that of the regular Newtonian gradient. The illustration is to point out that the top vector field is what GR reduces down to in the weak limit, whereas the bottom two fields are closer to what is now empirically known.

Regards,

Jeff

Thinking about some of Marcoen's questions and I realize that my explanations aren't coming across clearly. Let's try this:

--In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) is only defined up to an additive constant, the constant of integration.

Suppose we take a measurement of gravitational force, lensing, etc. and we compare that to what our differential equations give us, and they agree for a given input. We are basically assuming that we are using the correct anti-derivative of our function.

Thinking about the hypothesized size of the cosmological constant, if [math]R-G \equiv -4\Lambda[/math], then from the definition of an indefinite integral, how do we KNOW whether this equation should be [math]R-G \equiv -4\Lambda=0[/math] or [math]R\equiv 4\Lambda-G[/math] If the second is not the correct choice, why?

Hello Jeff,

(1) If you write "f1 = C - f2" then this is a relation between functions. That is, the term "f1" on the left hand side is an element of a function space, and the term "C - f2" on the right hand side is an element of a function space. That latter term "C - f2" is the function obtained by adding the function "- f2" to the constant function "C": in a function space, the binary operation addition takes two functions as an argument. If you want C to be a numerical constant (a number), then you can define the function "f1" by defining its function value as in "f1(x) = C - f2(x)": this is a relation between numbers, so here C is a number (a constant of integration if you want). So subtleties in mathematical notation can make a big difference.

(2) I'm still having difficulties with the interpretation of your two scalar fields, illustrated by your figure 3. It seems to me that the first scalar field doesn't occur in the ontology of Newtonian mechanics. Isn't this something you want to add?

As a side note, in the framework of my own theory, the EPT, the observed space is a material substance which is transcended by matter waves (there are, thus, two different kinds of components). A metric may then be defined in terms of a difference between two scalars (close to what you suggest) but at this point it is not possible to conduct tensor algebra in this framework as the necessary mathematics have yet to be developed.

With best regards, Marcoen

Hi Marcoen,

(1) Yes, you are correct in your statements, but what I am pointing out in my essay applies to either a function space or to just numbers. If I obtain an answer that is the derivative of the function spaces, was the correct physical relation from which the answer came, f1' or (C-f2)'? If I obtain a numerical value at x, was the correct physical relation f1(x) or C - f2(x)? This antidifferentiation, at its most basic level, stems from the fact that human senses are not well adapted for constant magnitudes but rather differences in space and time. By this I mean that when we obtain a physical measurement, we antidifferntiate though our understanding of the laws of physics. Perhaps I did not relate this well in my essay, but I wanted to keep it short as possible.

(2) I assume you are referring to the first scalar field at the bottom of Fig.(3). I do have a counter example of the first scalar field as derived in GR textbooks. As it is only the equation, I am not sure how much it would add to the essay to present it also. My version doesn't appear in the standard ontology of Newtonian mechanics, but the concept very much does so in the reduction of General Relativity to the weak field limit of Newtonian gravity. Others state the first scalar field has become the most paradoxical question ever posed to science (dark energy). I guess if I could rewrite the essay I would point out that the scalar field addition was not my idea, but is a representation of the idea Einstein threw out. It is what occurs when a cosmological constant is added to the standard Einstein field equation in order to account for a linearly growing force that opposes attractive gravity.

There are several profound problems with the entire existence of a cosmological constant, for which my substitution appears to be a case never considered previously, but for a general essay like this those would probably not stir much interest.

Unfortunately there is a balancing act between showing the mundane concept of my questions (which stem from (1)) and the inherent complexity of the Einstein field equation. Too difficult and people's eyes glaze over, but I am still searching for the simplest but most effective way to get this across.

If we take (da)²/a)+c²k/a²=8pi/3Gro+the energy of the quantum hole/3 .

Now if we insert the hubble law and my equations, you can correlate with the expansion/contyraction correlated with the maximum volume of universal sphere. Considering that the density can imply a contraction at this critical point.So we can see where the points of equilibrium are.The SR and the GR are ok and if we consider a system closed, it becomes relevant considering the number of quantum entanglement having the serie of uniquness.

See that the parameters must be precise for a correct universal dynamic. The volumes , spherical are essential like the rotations spinal and orbital.The proportions appear with mass. If the mass polarises the light and so if the entropy, physical increases proportional with mass.We can see where is this maximum volume just before the contraction towards the perfect equilibrium between all physical spheres. The friedman lemaître equation can be optimized.

Good work :) with the integrations, substitutions, extrapolations, additions, multiplications, settings,.......

Regards

Hello Spherical Jedi (I like your nickname),

I haven't yet gotten into the Friedmann equations in depth, but I will be certain to re-examine your concepts once I do. Do you have a website where your material is listed?

Regards,

Jeff

"What could be causing this acceleration? Physicists call it dark energy, and it could make up more than 70 percent of the cosmos. But so much remains unknown about dark energy that some scientists are asking whether it exists at all.

What if, instead of a mysterious unseen energy, "there is something wrong with gravity?" asks Sean Carroll, a theoretical physicist at the California Institute of Technology.

Einstein's theory of general relativity represents gravity as the curvature of space and time. Perhaps this idea "is still right, but we're not solving the equations correctly," suggests Carroll. "

Who's Afraid of the Dark? Alternatives to Dark Energy

Jeff

In the paper: Fedosin S.G. Cosmic Red Shift, Microwave Background, and New Particles. Galilean Electrodynamics, Spring 2012, Vol. 23, Special Issues No. 1, P. 3 - 13, I describe the possible reason for dark energy, red shift and microwave background. The explanation is based at the Theory of Infinite Hierarchical Nesting of Matter.

On the other hand, the cosmological constant was explained in the paper: Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, February 2012, Vol. 35, No. 1, P. 35 - 70.

Other ideas about gravitation see my Essay.

I hope it will be useful for you.

Sergey Fedosin Essay

Jeff,

In accordance with the Theory of Infinite Hierarchical Nesting of Matter (my Essay), nuons are similar at the level of star to the white dwarfs, and nucleons to neutron stars and magnetar. Nuons are the result of evolution of cosmic substance under action of strong gravitation at the level of elementary particles. In Standard Model there is muons that almost the same as nuons. But the difference is their origin: muons are born in decays of pions and strong interactions of particles and unstable or have charge and magnetic moment. But the nuons are stable, they are result of natural evolution of matter and have no charge. In the absence of charge it is difficult to detect them. At LHC or similar collider we see muons. To check the idea of nuons: they are the supposed reason of dark matter, of redshift of remote galaxies, of microwave background, of attenuation of light spectra of supernova star and so on.

Sergey Fedosin Essay

Jeff,

I really enjoyed your essay! Certainly if we find "dark energy" to be the dominant effect, it is more natural to "correct a repulsive force at small scales" than to "correct an attractive force at large scales," and the historical fact that we discovered "dark energy" relatively late should not deter us from this. However, I have a couple of questions.

1. Your 5th foundational question on page 2 involves the idea of curvature going to zero as matter content goes to zero, which is indeed a convincing point. However, I don't understand how repulsive gravity with a negative "cosmological constant" achieves this any more than attractive gravity with a positive "cosmological constant;" it seems that the difference is between positive curvature and negative curvature.

2. One argument for attractive gravity is based on the idea that gravity ought to be a "local effect." In other words, it is harder to think of objects at great distances "exerting greater and greater forces on each other" than to think of gravity as a local attractive "force" that simply dies out with distance and is overwhelmed by dark energy, whatever it is. What is your view on this?

3. Does dark matter factor into this in any obvious way?

If you want to know more about the motivation for these questions, you might look at my essay

On the Foundational Assumptions of Modern Physics

Basically, I am interested in the reasons for scale-dependence of the various interactions. Take care,

Ben Dribus

Hi Ben,

Thanks so much for the questions! I will attempt to return the favor in your thread but I am afraid that my questions may not be as pertinent as I am unfamiliar with some of the terminology in your essay. Will do my best though.

As for your questions here;

1. It may be that understanding positive curvature doesn't necessarily mean attractive gravity is a bit counter intuitive. Perhaps I should have prefaced this statement. My assumption is that, simply due to symmetry, the full equation should be able to work equally well with Guv or Omega guv-Luv, they both would give the same positive curvature. Curvature increases positively as Guv increases from zero, or as Luv decreases from the value of Omega guv.

The fascinating thing is though, the approximation to a Newtonian field gradient does not produce the exact same field equation. A symmetry breaking of classical gauge theory if you will. Without a quantitative analysis, it is plausible that this difference would only be detectable at larger radii. It also appears that to approximate this effect with only Guv one would have to add in a very tiny multiple of the metric (dark energy).

So what the 5th point probably should state is that if Guv and Omega guv-Luv are both suitable for the EFE, but only Guv requires an extra multiple of the metric that also breaks the condition of zero curvature with no matter, then Omega guv-Luv should take precedence over Guv-Lambda guv.

What this also means is that positive curvature can be equated to either attractive gravity or repulsive. Guv and Luv should result in very different field magnitudes of Phi. The Newtonian gravitational force, however, only depends on the spatial gradient of this field. The gradient vectors of Phi from Guv and Luv are of opposing directions of course, but vectors from Guv and -Luv are equivalent. What is even more interesting is how a constant multiple of the metric becomes a very tiny gradient vector opposing this.

Here is a small thought experiment and I apologize for the lack of mathematical rigor but it should show you what I mean. Take two vectors A and B, of equal large magnitude but they are placed tip to tip opposing each other. Their vector sum is a point, or a vector of zero magnitude. Now subtract from this sum a vector C pointing left that is smaller than either A or B. The total equivalence of all three vectors is a vector D that is of equal magnitude of C but pointing to the right. Occams Razor states that the simplest answer is only D, but the physical reality may be closer to A, B and C. Parsimony also requires that we choose the simplest anti-derivative to produce Newtonian gravity, and that we choose the simplest tensor for the Einstein field equation. Similarly, as vector D increases, you are increasing an attractive force, as vector C increases, you are decreasing a repulsion.

If I can parallel transport D on a curved surface, can I not also do the same for the sum of A, B and C? My guess is probably not and that we have missed something in manifold theory, although I am a long way from deriving this.

2. (Sorry 1 was so long!) My argument is that gravity is causal. The simplest way to understand what I mean is to state how I think gravity works. I think that matter at a point reduces the available field energy in a sphere surrounding it. The symmetry of the matter also depends on the symmetry of the reduced field energy. (this is not energy in the regular definition of matter-energy, more of a curvature potential) If another point within this sphere of reduced field energy also contains matter, it too reduces the available energy within a certain radius. The matter of these two points interfere with the symmetry of each, and both seeking symmetry (related to principle of least action) is manifested in a force where to tend to move together. The equations for the Newtonian approximation, however, seem to show that if clumps of matter were to spread far enough apart (past the constant within the EFE) then it would result in a purely linear repulsion. This would seem to fit a theory where the accelerating expansion only occurs after enough average distance between clumps of matter but it seems counter to our understanding of "expansion of space".

3. For Dark Matter, suppose that Omega guv can vary within a region not centered on baryonic matter. I suspect that this would add a stress energy tensor to the field equation. Since this tensor did not result from the exact point that matter is located, then the only way we would be able to detect it is through gravitational effects of matter passing through this region. I am a long way from deriving a set of christoffel symbols to account for this though.

Regards,

Jeff

Hello Mr.Baugher,

thanks:) I am a Jedi and proud of being . We fight Darth Vador in fact.And Yoda is my master, my mentor.

I have not a website, I just share my theory on this platfrom mainly because I beleive it is a seious platform of sciences. I beleive that FQXi has a pure responsability for our sciences. They must be universal in fact.If not their system will not on the rational road. FqxI isd young and of course this kind of platform is going to attract several persons loving monney and opulences. They must sort the pseudos ! Because the sciences are so important for our earth. The sciences have the solutions, the business , it, is an under sciences ! Are we going to pay our air soon ?

Mr Tegmark and Mr Aguirre have a responsability for this earth.They must think about the sortings of members.Just for their credibility in fact.Fqxi is a wonderful platform.This platform must be helped for its good road on the entropical arrow of times.

Regards

After conversations on here, it may be useful to have a simple graphical understanding of how positive curvature can produce either attractive or repulsive gravity. Simple sketch picture is attached. Only one of these naturally produces a dark energy effect.Attachment #1: Attractive_or_Repulsive.jpg

Jeff

Excellent. We give very different proofs of very similar things.

(This is a reposed reply from my blog).

I turned from maths to ontological construction testing for the reason you give. Ken Wharton also exposes our foolhardy reliance on maths.

You might test this model; The axial anisotropic CMB flow is a scaled up version of a quasar jet. The CMB anisotropy itself has been resolved by computer into a helix, which precisely matches the quasar model, as the outflow jets from AGN's.

I have shown that AGN's are part of a galactic recycling process, re-ionizing all the dead stars and planetary matter to re-start the process with an open spiral on a perpendicuar axis. The common 'Kinetic decoupling' (perpedicular halo rotation) is thereby also finally explained along with other effects. Take a look at Centurus A (NASA HST) for a scale model of the universe.

Expansion is thus not accelerating but mainly reducing, except from the other end of the axis to the 'great attractor' (nonsense of course) in the direction of Leo. I've determined galaxies recycle every ~10-12Gyrs (a massive quasar peak is at z=1.7) so our 2nd iteration of the Milky Way is in middle age. A better analysis of the CMB anisotropy might constrain the same factor for our universe. (There may then be infinitely many numerically as well as temporally). If you're interested I'll link you to a past paper (new one in review).

Last technical point; I've found algerbaic vector apace cannot model motion as it's based on geometry where motion is an invalid concept, but I do know that to get a 'plus' sign hit; ampersand hash 43 semicolon. Like this;

Peter

Hi Jeff,

In your sketch you say that the repulsive Cosmological term makes no mathematical sense. It's kind of funny, I look at the Cosmological term differently. In my view, anything that doesn't make sense is treated as an intrinsic characteristic. Dark energy is said to be the explanation of why the universe is expanding at an accelerating rate.

"The only way I know to do this is to invert the equation so that instead of solid "particles" moving within a void, the stress energy tensor describes waves moving within a solid. "

A solid is something that I can whack with a hammer. I don't think a solid is the right characteristic for empty space.

Technically it would be closer to a perfect fluid but that too has different connotations to different people. When people hear "fluid" they think of a water, but if I were to state that it was an elastic medium then they might think of rubber. A good qualitative description is difficult to give someone, so I just assume the best is to think of inverting the concept of what a particle is "made of" and what empty space is made of. What it would come down to is that if the hammer were to be made of waves, it apparently could normally only hit other waves.

Last post was me, woops.