The big question when it comes to dark energy is whether the Einstein Field Equations (EFE) are subject to the Fundamental Theorem of Calculus. Can G_{\mu\nu} be negative? The problem with subtracting the curvature tensor from the cosmological constant is that it means the perfect fluid stress energy tensor denotes a reduction in density (instead of an increase) and is aether theory. This, however, matches with the Lagrangian of the cosmological data.
If you would like to see the Tex versions of the equations, you can go to the BAUT forum at
www.bautforum.com/showthread.php/135626-Resurrection-of-the-Lorentzian-Aether
Let's assume that the curvature tensor can be negative and I will explain my view of what that would physically mean compared to a positive curvature tensor. This is simplified but hopefully will encourage some criticism.
In both cases, positive or negative, a difference in the density (and pressures) of a perfect fluid is equated to the curvature tensor. This is the stress energy tensor. This difference at a point causes stress outside the immediate vicinity of it, which is reflected in a change of the coordinate system. Where the coordinate system was Euclidean (flat), it no longer is. The greater the change in the density of the perfect fluid, the greater the change away from a flat coordinate system. The equations state that the curvature away from flat would be interpreted for two regions that contain a derivative of perfect fluid density into a "force" where they tend to come together. One case would consider this as an attraction between the two regions, the other would consider that the repulsion between the two regions has decreased. The defining difference between the two is the accelerating expansion. One case considers the accelerating expansion as most likely caused by an underlying energy that opposes the attractive gravity, whereas the other case simply considers the constant of integration as the point where the repulsion is no longer reduced by the presence of a change in density of a perfect fluid in another region.
AFAIK, particle-wave duality is an accepted phenomenon, as is now the quantum vacuum. In addition, most of the "mass" seems to not be within the nucleus of an atom but in empty space. The density of that perfect fluid is directly tied to the density of Newtonian matter. Therefore the majority of gravitational mass is from nothing. See starting at 19:00 www.youtube.com/watch?v=7ImvlS8PLIo I know of no way to integrate these two concepts with a positive curvature tensor, but with a negative I do.
In both cases, positive or negative, a difference in the density (and pressures) of a perfect fluid is equated to the curvature tensor. This is the stress energy tensor. This difference at a point causes stress outside the immediate vicinity of it, which is reflected in a change of the coordinate system. Where the coordinate system was Euclidean (flat), it no longer is. The greater the change in the density of the perfect fluid, the greater the change away from a flat coordinate system. The equations state that the curvature away from flat would be interpreted for two regions that contain a derivative of perfect fluid density into a "force" where they tend to come together. One case would consider this as an attraction between the two regions, the other would consider that the repulsion between the two regions has decreased. The defining difference between the two is the accelerating expansion. One case considers the accelerating expansion as most likely caused by an underlying energy that opposes the attractive gravity, whereas the other case simply considers the constant of integration as the point where the repulsion is no longer reduced by the presence of a change in density of a perfect fluid in another region.
AFAIK, particle-wave duality is an accepted phenomenon, as is now the quantum vacuum. In addition, most of the "mass" seems to not be within the nucleus of an atom but in empty space. The density of that perfect fluid is directly tied to the density of Newtonian matter. Therefore the majority of gravitational mass is from nothing. See starting at 19:00 www.youtube.com/watch?v=7ImvlS8PLIo I know of no way to integrate these two concepts with a positive curvature tensor, but with a negative I do.
Here is a good explanation of why adding terms to the EFE has not worked so far. This is by Sean Carroll and the whole lecture series is on Youtube. Highly recommended.
www.youtube.com/watch?v=hlWty...feature=relmfu
www.youtube.com/watch?v=hlWty...feature=relmfu
Here is a paper giving the reason for why a cosmological constant is a leading candidate for dark energy: arxiv.org/pdf/1201.3608v1.pdf .
Thus, according to the theory in this thread, the problem with the Lagrangians on Wikipedia's page en.wikipedia.org/wiki/Lagrangian#General_relativistic_test_particle is that the "m" in the equations is directly derived from the change in density of a perfect fluid within a volume from Poisson's equation (density comes from g00). Instead of ad-hoc throwing the cosmological constant into the equation to account for dark energy, it should come naturally from the fundamental theorem of calculus so that Lamda-rho V=m(I am removing the other terms that are with the cc in order to make it more illustrative on what the paper is stating) .
From
Thus, according to the theory in this thread, the problem with the Lagrangians on Wikipedia's page en.wikipedia.org/wiki/Lagrangian#General_relativistic_test_particle is that the "m" in the equations is directly derived from the change in density of a perfect fluid within a volume from Poisson's equation (density comes from g00). Instead of ad-hoc throwing the cosmological constant into the equation to account for dark energy, it should come naturally from the fundamental theorem of calculus so that Lamda-rho V=m(I am removing the other terms that are with the cc in order to make it more illustrative on what the paper is stating) .
From
The resulting constraints on the parameters of the Lagrangian are shown in Fig. 1, where we show the scatter plot of the models sampled by our MCMC. The results do not display any sign of the existence of discrete symmetries, as for instance , that would signal the spontaneously breaking of a gauge symmetry [21].
rho ---> -rho would seem to fit this definition of gauge symmetry breaking.
For every one of the 16 elements of the of Einstein curvature tensor, the cosmological constant is under debate how it relates to each and every element, and is the biggest mystery in physics. What is obvious to waitedavid would seem to be a paradox to everyone else.
For every one of the 16 elements of the of Einstein curvature tensor, the cosmological constant is under debate how it relates to each and every element, and is the biggest mystery in physics. What is obvious to waitedavid would seem to be a paradox to everyone else.
Well at least this much is true that there are a lot of things that seem obvious to me that others don't understand.
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious. It isn't meaningful to talk about the Einstein tensor as being positive or negative.
Oh dear, you are stating Perlmutter, Reiss and Schmidt were fraudulently given the 2011 Nobel prize?
www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html/
Where have you been for the last 15 years?
Oh dear, you are stating Perlmutter, Reiss and Schmidt were fraudulently given the 2011 Nobel prize?
www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html/
Where have you been for the last 15 years?
No, I'm saying that you don't know what you are talking about.
Accelerated expansion has been understood almost as far back as when Einstein wrote the cosmological constant in the field equations decades before the effect was observed.
The catch is that you haven't paid attention to history and motives.
Lets say for example and agency like nasa comes along and says we want you to give us money. Einstein's field equations with the cosmological constant predict accelerated expansion. And in accordance with the theory sure enough we've already observed what we understood all along to be the outcome. But we want you to give us money to research what we already know and have observed anyway.
How effective do you think this would be?
Or they can neglect all that history and say we've observed something mysterious. The expansion is accelerating. We will call this spooky thing dark energy as it is sooo mysterious. We'd like some research money to come to understand this mystery.
Now which approach do you think is going to be more effective on the ignorant masses holding the purse string? About the only thing they could have done better was throw the term "god" in it somehow by saying the energy must be mediated by some kind of god particle, and then maybe more of those looney tune republicans would pitch in their dollars.
QUOTE
Our main finding is that the expansion history is consistent with that predicted by a flat potential. The data do not require extra parameters beyond a constant term in the Lagrangian to explain the current accelerated expansion. Further, we have shown that the potential deviations from a constant are constrained to be below 6%. Observational constraints allow the parameters describing the Lagrangian to vary only within certain limits; the relative range of the allowed variation of the parameters confirms a well defined hierarchy where the linear and quadratic terms dominate over higher-order terms, justifying the basic assumption of the effective theory approach. Observational constraints also give some indications of the relevant energy scales involved. Because a direct determination of a Lagrangian allows us to determine the underlying symmetries in the theory, our results
can be used to shed light on this as well.
can be used to shed light on this as well.
Thus, according to the theory in this thread, the problem with the Lagrangians on Wikipedia's page en.wikipedia.org/wiki/Lagrangian#General_relativistic_test_particle is that the "m" in the equations is directly derived from the change in density of a perfect fluid within a volume from Poisson's equation (density comes from g00). Instead of ad-hoc throwing the cosmological constant into the equation to account for dark energy, it should come naturally from the fundamental theorem of calculus so that Lamda-rho V=m(I am removing the other terms that are with the cc in order to make it more illustrative on what the paper is stating) .
From
QUOTE (->
| QUOTE |
| Our main finding is that the expansion history is consistent with that predicted by a flat potential. The data do not require extra parameters beyond a constant term in the Lagrangian to explain the current accelerated expansion. Further, we have shown that the potential deviations from a constant are constrained to be below 6%. Observational constraints allow the parameters describing the Lagrangian to vary only within certain limits; the relative range of the allowed variation of the parameters confirms a well defined hierarchy where the linear and quadratic terms dominate over higher-order terms, justifying the basic assumption of the effective theory approach. Observational constraints also give some indications of the relevant energy scales involved. Because a direct determination of a Lagrangian allows us to determine the underlying symmetries in the theory, our results can be used to shed light on this as well. |
Thus, according to the theory in this thread, the problem with the Lagrangians on Wikipedia's page en.wikipedia.org/wiki/Lagrangian#General_relativistic_test_particle is that the "m" in the equations is directly derived from the change in density of a perfect fluid within a volume from Poisson's equation (density comes from g00). Instead of ad-hoc throwing the cosmological constant into the equation to account for dark energy, it should come naturally from the fundamental theorem of calculus so that Lamda-rho V=m(I am removing the other terms that are with the cc in order to make it more illustrative on what the paper is stating) .
From
The resulting constraints on the parameters of the Lagrangian are shown in Fig. 1, where we show the scatter plot of the models sampled by our MCMC. The results do not display any sign of the existence of discrete symmetries, as for instance , that would signal the spontaneously breaking of a gauge symmetry [21].
rho ---> -rho would seem to fit this definition of gauge symmetry breaking.
QUOTE (JMessenger+Jul 8 2012, 06:37 PM)
The big question when it comes to dark energy is whether the Einstein Field Equations (EFE) are subject to the Fundamental Theorem of Calculus. Can G_{\mu\nu} be negative? ...
The Einstein curvature tensor has sixteen elements and as such it doesn't make sense to refer to it as positive or negative. Since this is right at the beginning and should be obvious the rest of the speculations about aether etc will be ignored.
The Einstein curvature tensor has sixteen elements and as such it doesn't make sense to refer to it as positive or negative. Since this is right at the beginning and should be obvious the rest of the speculations about aether etc will be ignored.
QUOTE
The Einstein curvature tensor has sixteen elements and as such it doesn't make sense to refer to it as positive or negative. Since this is right at the beginning and should be obvious the rest of the speculations about aether etc will be ignored.
For every one of the 16 elements of the of Einstein curvature tensor, the cosmological constant is under debate how it relates to each and every element, and is the biggest mystery in physics. What is obvious to waitedavid would seem to be a paradox to everyone else.
QUOTE (JMessenger+Jul 9 2012, 12:14 PM)
For every one of the 16 elements of the of Einstein curvature tensor, the cosmological constant is under debate how it relates to each and every element, and is the biggest mystery in physics. What is obvious to waitedavid would seem to be a paradox to everyone else.
Well at least this much is true that there are a lot of things that seem obvious to me that others don't understand.
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious. It isn't meaningful to talk about the Einstein tensor as being positive or negative.
[QUOTE}Well at least this much is true that there are a lot of things that seem obvious to me that others don't understand.
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious.[/QUOTE]
Your Nobel prize awaits you sir for your explanation of the accelerating expansion.
The rest of us will have to play catch up:
wfirst.gsfc.nasa.gov/science/DETF_Report.pdf
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious.[/QUOTE]
Your Nobel prize awaits you sir for your explanation of the accelerating expansion.The rest of us will have to play catch up:
wfirst.gsfc.nasa.gov/science/DETF_Report.pdf
QUOTE (JMessenger+Jul 9 2012, 12:41 PM)
[QUOTE}Well at least this much is true that there are a lot of things that seem obvious to me that others don't understand.
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious.[/QUOTE]
Your Nobel prize awaits you sir for your explanation of the accelerating expansion.
The rest of us will have to play catch up:
wfirst.gsfc.nasa.gov/science/DETF_Report.pdf
What are you talking about? Accelerated expansion was understood before it was even observed.
How the cosmological constant relates to the Einstein tensor isn't under debate amongst people like me to see it at blatantly obvious.[/QUOTE]
Your Nobel prize awaits you sir for your explanation of the accelerating expansion.The rest of us will have to play catch up:
wfirst.gsfc.nasa.gov/science/DETF_Report.pdf
What are you talking about? Accelerated expansion was understood before it was even observed.
QUOTE
What are you talking about? Accelerated expansion was understood before it was even observed.
Oh dear, you are stating Perlmutter, Reiss and Schmidt were fraudulently given the 2011 Nobel prize?
www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html/
Where have you been for the last 15 years?
QUOTE (JMessenger+Jul 9 2012, 12:51 PM)
Oh dear, you are stating Perlmutter, Reiss and Schmidt were fraudulently given the 2011 Nobel prize?
www.nobelprize.org/nobel_prizes/physics/laureates/2011/press.html/
Where have you been for the last 15 years?
No, I'm saying that you don't know what you are talking about.
Accelerated expansion has been understood almost as far back as when Einstein wrote the cosmological constant in the field equations decades before the effect was observed.
The catch is that you haven't paid attention to history and motives.
Lets say for example and agency like nasa comes along and says we want you to give us money. Einstein's field equations with the cosmological constant predict accelerated expansion. And in accordance with the theory sure enough we've already observed what we understood all along to be the outcome. But we want you to give us money to research what we already know and have observed anyway.
How effective do you think this would be?
Or they can neglect all that history and say we've observed something mysterious. The expansion is accelerating. We will call this spooky thing dark energy as it is sooo mysterious. We'd like some research money to come to understand this mystery.
Now which approach do you think is going to be more effective on the ignorant masses holding the purse string? About the only thing they could have done better was throw the term "god" in it somehow by saying the energy must be mediated by some kind of god particle, and then maybe more of those looney tune republicans would pitch in their dollars.
Ah, wow, crank alert.
If anybody is interested, here is the text from the Nobel website:
Accelerated expansion has been understood almost as far back as when Einstein wrote the cosmological constant in the field equations decades before the effect was observed.
Please read up on some history and how GR was derived. Specifically how Friedmann determined that the universe could not be static, it either had to be collapsing or a decelerating expansion. (NOT ACCELERATING)
If anybody is interested, here is the text from the Nobel website:
QUOTE
For almost a century, the Universe has been known to be expanding as a consequence of the Big Bang about 14 billion years ago. However, the discovery that this expansion is accelerating is astounding. If the expansion will continue to speed up the Universe will end in ice.
The acceleration is thought to be driven by dark energy, but what that dark energy is remains an enigma - perhaps the greatest in physics today. What is known is that dark energy constitutes about three quarters of the Universe. Therefore the findings of the 2011 Nobel Laureates in Physics have helped to unveil a Universe that to a large extent is unknown to science. And everything is possible again.
The acceleration is thought to be driven by dark energy, but what that dark energy is remains an enigma - perhaps the greatest in physics today. What is known is that dark energy constitutes about three quarters of the Universe. Therefore the findings of the 2011 Nobel Laureates in Physics have helped to unveil a Universe that to a large extent is unknown to science. And everything is possible again.
QUOTE (->
| QUOTE |
| For almost a century, the Universe has been known to be expanding as a consequence of the Big Bang about 14 billion years ago. However, the discovery that this expansion is accelerating is astounding. If the expansion will continue to speed up the Universe will end in ice. The acceleration is thought to be driven by dark energy, but what that dark energy is remains an enigma - perhaps the greatest in physics today. What is known is that dark energy constitutes about three quarters of the Universe. Therefore the findings of the 2011 Nobel Laureates in Physics have helped to unveil a Universe that to a large extent is unknown to science. And everything is possible again. |
Accelerated expansion has been understood almost as far back as when Einstein wrote the cosmological constant in the field equations decades before the effect was observed.
Please read up on some history and how GR was derived. Specifically how Friedmann determined that the universe could not be static, it either had to be collapsing or a decelerating expansion. (NOT ACCELERATING)
Ya wanna know the funny thing, throw out your conspiracy theories and you and I may not actually disagree. Have a website or some links?
QUOTE (JMessenger+Jul 9 2012, 01:13 PM)
...Please read up on some history and how GR was derived....
So where are your links describing accelerating expansion? And I do mean your mathematical derivation.
QUOTE (JMessenger+Jul 9 2012, 01:18 PM)
So where are your links describing accelerating expansion? And I do mean your mathematical derivation.
I mentioned how to do it in a video actually, but I didn't go into detail so I'll give it here.
From Einstein's field equations with the cosmological constant included it is a trivial matter of contracting the indexes to find the relation between the Ricci-scalar and the cosmological constant from equation 6.3.15. Interpreting the cosmological constant as the amount of T one actually had when taking it to be 0, you get
Ricciscalar = -4λ
In the video I was using a time coordinate transformation of the FRW metric.
ds² = [R(ct)/R₀]²dct² - {[R(ct)/R₀]²/[1+k(πr/cT)²]²}[dr² + r²dθ²+r²sin²θdφ²]
Our universe is very nearly flat. It may not be exactly, but one makes a very good approximation of it with k=0. So this is a step that is an approximation but is worth while because it makes for a simple result and very closely models what we observe. In this case the Ricci-scalar for that line element becomes
Ricciscalar = -6R₀²(d²R(ct)/dt²)/R(ct)³
So from above we have
-6R₀²(d²R(ct)/dt²)/R(ct)³ = -4λ
(d²R(ct)/dt²) = [2λ/(3R₀²)]R(ct)³
To solve this second order differential equation Let v = dR(ct)/dt
(d/dt)v= [2λ/(3R₀²)]R(ct)³
Chain rule:
(dv/dR)(dR/dt) = [2λ/(3R₀²)]R(ct)³
(dv/dR)v = [2λ/(3R₀²)]R(ct)³
This is separable first order
v² = [λ/(3R₀²)]R(ct)⁴ + k²
So we have
(dR(ct)/dt)² = [λ/(3R₀²)]R(ct)⁴ + k²
now you see that if the cosmological constant were 0, your only expansion rate would be the constant k, no accelerated expansion. But if your cosmological constant were positive than the expansion rate increases with the "size" R of the universe as
(dR(ct)/dt)² = [λ/(3R₀²)]R(ct)⁴ + k², thus accelerated expansion.
Positive λ also corresponds to a positive energy density, λg₀₀ > 0, (sign convention). So aside from the large pressure terms the matter isn't all that interesting.
QED
I mentioned how to do it in a video actually, but I didn't go into detail so I'll give it here.
From Einstein's field equations with the cosmological constant included it is a trivial matter of contracting the indexes to find the relation between the Ricci-scalar and the cosmological constant from equation 6.3.15. Interpreting the cosmological constant as the amount of T one actually had when taking it to be 0, you get
Ricciscalar = -4λ
In the video I was using a time coordinate transformation of the FRW metric.
ds² = [R(ct)/R₀]²dct² - {[R(ct)/R₀]²/[1+k(πr/cT)²]²}[dr² + r²dθ²+r²sin²θdφ²]
Our universe is very nearly flat. It may not be exactly, but one makes a very good approximation of it with k=0. So this is a step that is an approximation but is worth while because it makes for a simple result and very closely models what we observe. In this case the Ricci-scalar for that line element becomes
Ricciscalar = -6R₀²(d²R(ct)/dt²)/R(ct)³
So from above we have
-6R₀²(d²R(ct)/dt²)/R(ct)³ = -4λ
(d²R(ct)/dt²) = [2λ/(3R₀²)]R(ct)³
To solve this second order differential equation Let v = dR(ct)/dt
(d/dt)v= [2λ/(3R₀²)]R(ct)³
Chain rule:
(dv/dR)(dR/dt) = [2λ/(3R₀²)]R(ct)³
(dv/dR)v = [2λ/(3R₀²)]R(ct)³
This is separable first order
v² = [λ/(3R₀²)]R(ct)⁴ + k²
So we have
(dR(ct)/dt)² = [λ/(3R₀²)]R(ct)⁴ + k²
now you see that if the cosmological constant were 0, your only expansion rate would be the constant k, no accelerated expansion. But if your cosmological constant were positive than the expansion rate increases with the "size" R of the universe as
(dR(ct)/dt)² = [λ/(3R₀²)]R(ct)⁴ + k², thus accelerated expansion.
Positive λ also corresponds to a positive energy density, λg₀₀ > 0, (sign convention). So aside from the large pressure terms the matter isn't all that interesting.
QED
Just so you can see how it is done with the standard time coordinate for the FRW line element I've added equations 8.1.17 to 8.1.22 to section 1 of chapter 8.
Like his contemporaries, Albert Einstein initially thought that the universe was static: that it neither expanded nor shrank. When his own Theory of General Relativity clearly showed that the universe should expand or contract, Einstein chose to introduce a new ingredient into his theory.
His "cosmological constant" represented a mass density of empty space that drove the universe to expand at an ever-increasing rate.
When in 1929 Edwin Hubble proved that the universe is in fact expanding, Einstein repudiated his cosmological constant, calling it "the greatest blunder of my life."
This excerpt is from http://www.spacedaily.com/news/cosmology-05n.html
I found this with Google this was the first one I pulled up. the 3rd one in the list. I am sure there are better articles with this information, However I kind of am lazy.
Turns out Einstein Greatest Blunder was correct.
His "cosmological constant" represented a mass density of empty space that drove the universe to expand at an ever-increasing rate.
When in 1929 Edwin Hubble proved that the universe is in fact expanding, Einstein repudiated his cosmological constant, calling it "the greatest blunder of my life."
This excerpt is from http://www.spacedaily.com/news/cosmology-05n.html
I found this with Google this was the first one I pulled up. the 3rd one in the list. I am sure there are better articles with this information, However I kind of am lazy.
Turns out Einstein Greatest Blunder was correct.
QUOTE (Ed Wood+Jul 10 2012, 06:57 AM)
Like his contemporaries, Albert Einstein initially thought that the universe was static: that it neither expanded nor shrank. When his own Theory of General Relativity clearly showed that the universe should expand or contract, Einstein chose to introduce a new ingredient into his theory.
His "cosmological constant" represented a mass density of empty space that drove the universe to expand at an ever-increasing rate.
When in 1929 Edwin Hubble proved that the universe is in fact expanding, Einstein repudiated his cosmological constant, calling it "the greatest blunder of my life."
This excerpt is from http://www.spacedaily.com/news/cosmology-05n.html
I found this with Google this was the first one I pulled up. the 3rd one in the list. I am sure there are better articles with this information, However I kind of am lazy.
Turns out Einstein Greatest Blunder was correct.
Its inclusion is correct, but his motivation for its inclusion was wrong. I think really he realised that if he just trusted the expanding universe implication of his field equations as they were, he would have been the one to predict the expansion himself.
His "cosmological constant" represented a mass density of empty space that drove the universe to expand at an ever-increasing rate.
When in 1929 Edwin Hubble proved that the universe is in fact expanding, Einstein repudiated his cosmological constant, calling it "the greatest blunder of my life."
This excerpt is from http://www.spacedaily.com/news/cosmology-05n.html
I found this with Google this was the first one I pulled up. the 3rd one in the list. I am sure there are better articles with this information, However I kind of am lazy.
Turns out Einstein Greatest Blunder was correct.
Its inclusion is correct, but his motivation for its inclusion was wrong. I think really he realised that if he just trusted the expanding universe implication of his field equations as they were, he would have been the one to predict the expansion himself.
good point.
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