A Challenging Question about an Infinite Universe.

[sjastro]

Given the recent thread about the size of the Universe lets assume the Universe is flat and infinite (no flat finite Universes such as 3-torus allowed).

A question that spans both science and philosophy is how is how a Big Bang cosmology which has a finite expansion velocity and occurred in a finite time frame (13.8 billion years ago) leads to an infinite Universe.
In other expansion based models such as the Steady State this is not a problem as an infinite Universe is also infinitely old.

The question arises because of our intuitive association of "infinity" with something "very very big."

Steven

[N1]

I understand the Universe is infinite in a similar way a circle or spherical surface is infinite. Reduce its diameter, and it's still infinite. Reduce it to a dimensionless point and it's still infinite because it lacks dimension. So at no stage was it finite.

[sjastro]

The spherical surface is finite but unbounded. It has a finite surface area for any given radius.
Similarly the circumference of a circle, is finite for any given radius.

For an n-dimensional hypersurface to be infinite means it cannot be enclosed in an n+1-dimensional "ball". In 2 dimensional Euclidean space the ball is a circle, in 3 dimensional Euclidean space a sphere etc. We can make the radius of the ball as large as we want, there will always be points on the infinite surface that lie outside the radius of the ball.

Regards

Steven

[xelasnave]

Well Steven I assume you have an answer but I will use the opportunity to comment.
The reason I find steady state comfortable is infinity needs little explanation if any.
I find inflation difficult to accept so avoiding it in the mix appeals to me.

I would think you can not double up finite to ever become infinite.

But I suspect maths can provide a path.

[xelasnave]

Sorry Steven somehow I missed your last post before I posted.

[julianh72]

There are many concepts in modern cosmology and quantum mechanics which relate to the very big and the very small that my brain simply cannot grasp - yet I still accept them as being (almost certainly) true.

The human brain evolved to deal with human-scale problems like sabre-toothed tigers and rivers and mountains, but there was no evolutionary imperative to develop a brain which can truly understand and visualise fundamental particles, wave-particle duality, quantum entanglement, or finite-but-unbounded n-dimensional space-time.

Analogies such as an inflating balloon or the raisins in an expanding plum pudding help, and I get that "Ah-Hah!" moment when I think I "get it", but that is always followed by a crashing disappointment when my 3-dimensional brain puts the balloon or pudding into the physical context of my larger universe, where I can sit god-like outside the balloon / pudding and see its boundaries, and also what lies outside it. Somehow, my brain can't do the reverse, and put itself inside the pudding, or on the surface of the balloon, and accept that there is no "outside".

I often wonder whether the people who develop these theoretical frameworks can somehow truly "see" in their mind's eye what an n-dimensional hyper-surface looks like, or are they limited by the same limited 3-dimensional visualisation, and a mathematical model of n-dimensions?

[xelasnave]

Yes Juliahn I hold similar thoughts.

I often find myself thinking about the Universe and somehow sitting outside which of course could not be done in any reality.

I find cosmology very interesting but fortunately realise one must embrace difficult concepts often counter intuitive.

Thank goodness for scientific method it gives us supportable theory.

[sjastro]

Alex,

The answer I provided to Mirko is not really what I had in mind in conceptualizing infinity.

I doubt there is a satisfactory answer to the question, only a redefinition of infinity which makes one more comfortable with the concept and doesn't make the question of how a Big Bang with a finite expansion velocity and finite time frame lead to an infinite Universe, a seemingly impossible one to answer.

In the late 19th century transfinite mathematics was developed which revolutionized the notion of infinity. It was so controversial at the time and opposed by mainstream mathematics, it led to its creator Georg Cantor in having a nervous breakdown and being committed to a mental asylum.
Today the subject is textbook pure mathematics.

A simple example is the real number line which is an infinite set. It extends to -infinity and +infinity.
If you take any interval on the real number line such as [1,2], you find it is also contains an infinite number of real numbers.
The intuitive concept of infinity is blown out of the water, you have two infinite sets, one which contains "fewer" numbers and is a subset of the "larger" infinite set.

If you extend the idea of the number line as forming a 1-dimensional space, you have a smaller infinite space "encapsulated" in a larger infinite space.

Hence the intuitive idea that infinity is larger than anything we can imagine is not terribly useful.

Steven

[xelasnave]

Thank you Steven.
I knew about Cantor from a humble TV show.

So unfortunate.

He probably was never able to appreciate his contribution.
I am not sure if he recovered.

Let's try something easier... what is dark energy and how is it causing an accelerated expansion.:D

[Eratosthenes]

Mathematicians work with different conceptual frameworks involving "infinity". Some refer to a "countable" infinity and an "uncountable" infinity.

When infinities appear in derivations or calculations, they are usually a good sign that something is wrong. (Hilbert, the great German Mathematician came up with the Hotel reservation thought experiment). It's useful to get infinities to vanish before they appear in the calculations or derivations, or perhaps cancel each other out. Then they dont need to be dealt with. mathematicians deploy some clever techniques that avoid infinities. On the other hand one can integrate to infinity and arrive or approach a limit which is a valid result. Or Sum up a series that approaches a certain finite number, or perhaps infinity itself.

There are some strange Summation series which can be described as nonsense such as {1+2+3+4+5+6......to infinity} = -1/12

Now does the series {1+2+3+4+5+6......to infinity}, a set of positive integers btw, approach a value -1/12 which is not only less than 1, but also negative (less than zero)???? And yet this result is used in several areas of Physics, including String Theory. (String Theory, Joseph Polchinski, Vol. 1, p. 22). Now one can argue that some tricks were used to arrive at this result which contradict mathematical logic and so result is not a valid result. The result is nevertheless use in other areas of mathematics and Physics. Mathematicians use a method called analytic continuation and when applied to divergent series, its possible to rationalise the -1/12 result. Anyway back to the subject at hand...

Some infinities are "bigger" than other infinities

So what type of infinite Universe are we talking about here Steven?

[xelasnave]

I think of infinity as something without limit on it's dimension.
It has no top, bottom or sides and without a start in time or an end of time.
You can apply math yet it can not grow or shrink
There could be no concept of time.
A set of numbers has a start so by my definition one could not describe a set as infinite.
As cosmology has a starting point for time can we still entertain infinity.
On this basis is it unreasonable to say the Universe must be finite.

[sjastro]

The correct term is countable and uncountable infinite sets.

A countable set is equivalent to the set of natural numbers N, if one can define a mapping function which is one-one and onto between the sets.
An infinite set that is not equivalent to N is uncountable.
Given that N is the common denominator in defining countable and uncountable sets you are wrong in asserting that mathematicians are using different conceptual frameworks involving infinity.



What has this got to do with the subject?
Do you seriously think that cosmologists have calculated the Universe as being infinite?
I suggest you try looking at http://en.wikipedia.org/wiki/Friedmann_equations.

Whether the Universe is infinite or not depends on the value of the curvature k used in the cosmological model under study.




What absolute hogwash.
Analytic continuation applies to analytic functions which are functions of the complex variable z.
Sorry to disappoint you but {1+2+3+4+....} are not complex numbers, they don't even form a power series on which analytical functions of both complex and real variables are based on.

As far as {1+2+3+4+.....} converging to -1/12 well I think any reasonably smart high school student will tell you otherwise.

I suspect Mr Polchinski is stating something very different.
Why don't you provide the relevant source.



Try reading the thread instead of cherry picking statements.

[N1]

Steven, how do you know that a spherical surface is finite when your entire existence is inside same surface?

[Eratosthenes]

Wow, you're response is very impressive.

Its strange that there are areas of Physics that actually use the result "Summation of positive integers, to infinity equaling -1/12"

If it's hogwash, then so are parts or all of String Theory (which is probably hogwash anyway)

What have you got against Cherries? They contain anthocyanin antioxidants - may well be unique to this fruit. Lets not forget the Vitamin A, C, calcium and iron

love cherries:D

[tonybarry]

Dear Steven,

Eratosthenes may not be presenting hogwash.

{1+2+3+4+... to infinity} = -1/12

http://www.smithsonianmag.com/smart-news/great-debate-over-whether-1234-112-180949559/?no-ist

Phil Plait does a good rendition of the series according to Ramanujan. He also offers an apology to the conventional mathematicians among us, who take umbrage at the novel use of = .

Would you care to comment ?

Regards,
Tony Barry

[sjastro]

Mirko,

We live on a (near) spherical surface. We can travel around along a great arc on the Earth's surface and end up in the same spot we departed. Clearly this is not possible if the surface is infinite.

Cosmologists have used a similar principle to test if the Universe in finite and spherical. A photon emitted could traverse a spherical Universe and end up near the same spot of emission.
Since the anisotropic regions of the cosmic radiation background can now be well resolved, double images in the background would indicate this phenomena.

Regards

Steven

[sjastro]

Hello Barry,

Its interesting that Phil Plait ended up back tracking on this.
http://www.slate.com/blogs/bad_astronomy/2014/01/18/follow_up_the_infinite_series_and_t he_mind_blowing_result.html

To put this {1+2+3+4+5+......}= -1/12 into perspective, the series is clearly divergent so mathematicians are looking looking for a corresponding convergent geometric series.

Mathematicians start off with an oscillating series
S1={1-1+1-1+1-.......}

And
S2={1-2+3-4+5-6.....}

If you add S2 to itself with terms shifted one place ie -2+1, 3-2, -4+3 etc
you get S2+S2={1-1+1-1+1-1....}= S1
S2=S1/2

If S3={1+2+3+4+5+6...} and you subtract S2 from S3
You get (0+4+0+8+0+...}=4S3
S3=-S2/3

Now S1={1-1+1-1+1-.......}=1/2, hence S2=1/4 and S3=-1/12 the desired result.

You might wonder where S1={1-1+1-1+1-.......}=1/2 comes from.
Its an oscillating series which is neither divergent or convergent.
In order to calculate this value one needs to compare it to a convergent series.

This is where the analytic continuation comes into the picture involving complex numbers which involves a geometrical interpretation.
The oscillating series sits on a boundary of a circle. A geometric series outside the circle is divergent, a geometric series inside the circle is convergent.
When a convergent geometric series approaches the boundary, its summed value is found to be S1=1/2.

Hence S1=1/2 S2=1/4 and S3=-1/12.

This shows we are not evening summing the series, we are summing a convergent series that is mathematically close to it.

Regards

Steven

[sjastro]

Your whole argument is based on a wonky video where the only reference to String Theory is a guy who points to an equation in a book.
That's all there is?
String Theory has been caught out by an ill directed finger?

Since you didn't bother with my request to provide the source, a simple google search found the appropriate document.

http://stringworld.ru/files/Polchinski_J._String_theory._Vol._1 ._An_introduction_to_the_bosonic_st ring.pdf

As expected cherry picking or in this case finger pointing doesn't tell the whole story.
It doesn't tell you that the author already knows that {1+2+3+4+5+6....} is a divergent series.

The story leading up to the infamous finger pointed equation 1.3.32, involves treating the energy levels of a bosonic string like a quantum mechanical harmonic oscillator. In this case summing over all the harmonics leads to a divergent value.
String Theorists use a mathematical technique from Quantum Field Theory known as renormalization. In this case the higher frequency harmonics are cut out of the sum as they are above the energy threshold level of the system.
Equation 1.3.32 basically indicates the summed energies after renormalization.

Rather than being a mathematical trick, renormalization has a physical significance and is related to the Casimir effect where there has been a shift in the ground energy state of the bosonic string.

All of this is of course totally irrelevant to a thread involving Cosmology.
Once again you are in engaging in Gish Gallop tactics.

[Eratosthenes]

Can you provide a source for any evidence for the existence of strings, Bosonic or otherwise.

You are swimming in String Theory which isn't even a science, but rather a Mathematical Philosophy as it stands today.

In any case Bosonic String Theory isn't even a viable Physical Model.

You seem to hop from Metaphysics to Physics to Philosophy and back again - which isnt necessarily a negative, but at least inform posters when you do this otherwise the demarcation between solid mainstream Physics (which can be challenged incidentally) and mysticism and metaphysical conjecture becomes shiftingly blurry. Very shiftingly blurry indeed Steven

:D

[sjastro]

I have made it clear in the past that I don't accept String Theory as is not falsifiable. My comments relate to how a String Theorist sees the physical significance of renormalization.
What you don't seem to understand is your song and dance routine over a set of numbers that supposedly adds up to -1/12, relates to the mathematics of renormalization, not string theory per se.



You have got to be joking.
I made it perfectly clear in the very first post of the parallels between Science and Philosophy in this topic.

In case you missed it.


Where does the metaphysics and mysticism show up in this thread?

[Eratosthenes]

I see.

And yet you make this statement concerning the Casimir effect and bosonic strings :shrug:

Rather than being a mathematical trick, renormalization has a physical significance and is related to the Casimir effect where there has been a shift in the ground energy state of the bosonic string.

(incidentally, I don't accept String Theory as a science. It's a concept more akin to a Mathematical Philosophy. This of course can change, should experimental evidence emerge that supports the predictions that String Theory claims.)

Cheers

:)

[sjastro]

Argument by repetition.
I addressed this point in my last post.
Do you actually bother to read posts or do you simply want to be argumentative.

[Eratosthenes]

:question:

[N1]

Steven,



We only know that because we can rise above it (in both theory and practice), but by doing that, we are introducing an external system of reference.



But we could keep going indefinitely, couldn't we? We'd only



because we, not physical constraints, have decided to "end" our journey there.

By nominating a departure point, and by further stating that we could only be there once, we, not "the Universe" have defined the limits.



Just because this hasn't been observed doesn't mean it' not happening, since "well resolved" is clearly less than 100% resolved.

Edit: I might add that the further a photon travels, the less likely it is to maintain its original direction due to gravitational and other factors along the way. It would not seem unreasonable to think that given enough spacetime, a photon would eventually be turned back onto its origin, if not somehow rendered non existant in the process. As a result, perhaps the question of infinity is irrelevant because all possible trajectories of our photon are interlinked loops between which it can change. That way, it could avoid returning to the keen observer for a very long time but still be in a loop at any given time.

[sjastro]

Mirko,





Our frame of reference is the same as the Earth's.
In this case it is in spherical coordinates defined by a polar and azimuthal angle which also locates our position on the surface.






The pathway is defined by the circle inscribed on the sphere. There are no limits. It doesn't matter how many times we travel around this circle, the radius of the circle remains the same. By calculating the radius through the circumferential distance travelled, we can find the surface area of the sphere which is finite.





When I mean resolved, I'm referring to the angular resolution of the anisotropic detail in the cosmic radiation background.
There has been a progressive increase in detail from COBE to WMAP to the Planck probe.
Since the detail is now well resolved, cosmologists are able to measure the angular size of individual structures with greater precision.

The measurements are consistent with a flat univese.

Steven

[xelasnave]

I can't help it.
I find in my imagination the Universe as a ball and I am on the outside, without though as to what this outside is and the dimension is infinite.
I find it funny that there is no outside.
It is a wise man who can imagine a stick with no ends.
Steven I bet you have a formula for a stick with no ends.

[xelasnave]

I can't help it.
I find in my imagination the Universe as a ball and I am on the outside, without though as to what this outside is and the dimension is infinite.
I find it funny that there is no outside.
It is a wise man who can imagine a stick with no ends.
Steven I bet you have a formula for a stick with no ends.

[xelasnave]

I have just returned from a very stressful week in sydney.
Nine hour drive after mowing the lawns with gammy legs before I left.
I back on my mountain and got in some viewing thru the 80 mm binnos.
Came in somehow calmed..I could not see any outside.
Great to be home.

[Eratosthenes]

...a stick with only one end....:question:

[xelasnave]

Well you start with one of those.

I have been outside imaging what my stick passes ..

[Eratosthenes]

What sort of shadow does a one ended stick cast?

.....can shadows move at speeds greater than the speed of light?;)

[julianh72]

A Hula Hoop?

[xelasnave]

The shadow would be all over the place depending on the environment of that part of the stick.
The answer is complex.
A hoop qualifies.
But it must be a straight stick.

[sjastro]

Alex,

The best I can do is to give the topological edge equation of a one sided sheet of paper.;)

If you are on the outside of this Universe you must be at a higher dimension than this Universe. An analogy is the surface of a ball embedded in 3-D space.
If the observer however exists in the same dimension as the Universe, in your example the surface of the ball, there is no outside.

Steven

[Eratosthenes]

I just had a ridiculous idea float by.

What if a stick falls into a black hole whereby, one end of the stick has breached the event horizon and the other is still "outside" the event horizon? How many ends does this stick have?

In we consider a very long stick that spans the distance between two black holes, and the ends of the stick breach the event horizons of the two black holes, the question must be asked "does this stick have any ends"?

:P

[xelasnave]

There will be only one end and that is the absolute destruction of all the stick.

[xelasnave]

If a hoop was huge one standing on it would not be able to notice any curvature, the stick would appear straight. So we can have a straight stick and a hoop.

[Eratosthenes]

what if two black holes are placed at the ends of a very long stick?

Is this a stick with no ends?


;)

(a flexible stretchy sort of stick)

[xelasnave]

I think one may find tidal forces would start demolition of the stick well before it could approach the event horizon, unless it was an imaginary stick.
And so we could observe the ends of a stick as it's ends are eaten away...it would be dust that would pass the event horizon.

[xelasnave]

It would be dust well before the event horizon gravity does not start at the event horizon.

[sjastro]

It depends on the mass of the black hole.
The event horizon radius is proportional to the black hole mass.
The larger the mass, the greater the radius, the lesser the tidal forces.
For a supermassive black hole, tidal forces at the event horizon are insignificant.
In the sticks frame of reference nothing significant happens as it crosses the event horizon of a supermassive black hole.

Steven

[xelasnave]

An imaginary stick with no ends would of course pass right thrue unaffected by physics and out the other side leaving that black hole sitting on our never ending stick like meat on a skewer and it travels onward collecting galaxies etc into eternity.

[xelasnave]

Sorry Steven I just saw your post.
I thought I was on safe ground.
I find that extraordinary.
Thanks

[xelasnave]

I was following a black hole thread recently which covered this..
My memory is going unfortunately.

[xelasnave]

On the positive I remember after being reminded.
The stick with out ends is like an exercise to imagine the difficulty of grasping the unimaginable enormity of the universe.
Last night I sat outside under a perfect sky thinking about what was before me.
Thinking of the Hubble Deep field knowing everywhere there are so many galaxies.
Makes the problems of the day trivial.

[Eratosthenes]

yes indeed xelasnave,

.....a humble reminder of how "universally trivial" life is on this planet.

It's astonishing to see the huge numbers of people who place the Earth at the center of "everything" in terms of importance. In fact this arrogance and ignorance extends to spiritual and moral importance. Is it any wonder that the behaviour of humans collectively and individually is sometimes so self destructive and cruel?

A stick without any ends may well be an imaginary artifact of the mind, but it appears that this does not apply to human arrogance and ignorance. If you divide the actual knowledge that humans claim to possess by the "absolute truth" and total knowledge out there, you would get something approaching zero. Not a bad approximation of zero. That Socrates chap was one clever bugger

:D

[xelasnave]

I found that even when I knew everything there was still more to learn.

[N1]

That's exactly what I mean by rising above the surface. As soon as you do that, you see the curve. You have adopted an outside frame of reference. The point is to stay within the surface, a 2-dimensional universe. It's an analogy after all. Perhaps the problem of infinity can always be overcome by adding another dimension and looking at it from the "outside". You would need to step outside the universe though, as you've said to Alex below.



The radius does not need to change for an infinitely long journey to be possible.



Granted, this does not necessarily form part of the agument, but are we actually able to calculate the radius based on distance travelled, as opposed to obtaining an approximate value?



That's the status as at today. I know of something else that was observed as being flat but later found to be (nearly) spherical ;)

[Eratosthenes]

xela, you would made a good disciple of the great Socrates

although Socrates was condemned to self administer hemlock after being found guilty of corrupting the youth of Athens. An interesting offense, seeing as the Socratic method of gaining wisdom by "posing questions" and continually probing and questioning the infinite ether of knowledge is a teaching philosophy used today in almost every school. (some argue that education systems today avoid the Socratic method, especially with young children, and prefer the early schooling years to a military training type experience where the natural creativity and curiosity of children is "drummed" and hammered out of them. Out the other end are nice obedient corporate citizens who rarely question anything they are told and are driven by individualism and materialism.)

Science, as a grand pursuit of uncovering knowledge has been contaminated by corporate greed and market forces. Our political system has also suffered the same contamination, yielding to short term profits and thrills. I was particularly touched by a recent Nobel Prize in Physics handed out to inventors of the blue LED. Whilst this is a great invention which finally enables a white LED to be manufactured, how does this compare to say Paul Dirac's PhD thesis completed in his early 20's? or one of the 3 great papers published by Einstein in 1905?

Dreamless pathetic politicians and leaders controlled by greedy sociopathic and narcissistic bankers and corporate dead beats. A compliant corporatised media does the rest....

an endless stick indeed

[xelasnave]

You are right it is indeed a wonderful world.
I suggest expanding your observations perhaps in a new thread.
Out of respect to Steven I think we should engage the question he has founded this thread upon.
I

[xelasnave]

I had not intended the stick to lead us so far off course.

[sjastro]

Given that I am walking on the surface, not jumping or flying off it, nor drilling a hole through the Earth and travelling through it, my motion is purely confined to the surface.
There is no outside frame of reference in this example. I don't have to be outside the sphere to know its a sphere. I can measure its intrinsic curvature.



That doesn't define whether a surface is finite or infinite.
A spherical surface is finite and unbounded. The surface of the Earth is finite irrespective how many times we travel around the circumference.




Assuming the Earth is a true sphere we can measure the circumferential distance C travelled. Since the pathway takes on a great circle, the radius r is simply C/2*pi.
This also the radius of the Earth which has surface area of 4*pi*r^2 which is finite.



The difference here is the evidence itself rules out the Universe as a 3-sphere. Cosmology doesn't suggest the entire Universe has to be of a particular geometry to start with.
The options are flat, closed (spherical) or open (hyperbolic).

Steven

[N1]

I'm aware of the formula. Do we know what exactly pi is?



Sounds like it's all done and dusted then & any researchers still working on this particular problem would be wasting their time. :question:

[Eratosthenes]

...pi is a transcendental number, which also makes it an irrational number

(and so is "e" - there's two elements in Euler's Identity already - you cant ask for more than that ;))

[sjastro]

PI is irrational but it is a finite number.
The surface area of the sphere is still finite.



A misconceptions about science.
Scientific theory is only as good as the technology performing the observation or experiment. With improvements in technologies, scientific theories undergo retesting and are never done and dusted.

The Newtonian model of the moon's orbit as being in freefall around the Earth is a good example. For 250 years theory agreed with observation until Apollo astronauts put mirrors on the lunar surface which allowed a far more precise measurement of the Earth Moon distance.
It was found that the moon is not in freefall but is moving away from the Earth.

The successive improvements in measuring the anisotropic detail in the cosmic radiation background has only reinforced that the universe is flat.

Steven

[N1]

Thank you. :thumbsup:

[xelasnave]

Steven does this mean that the Universe was flat at the time the background radiation started it's journey.

[sjastro]

Yes.

In fact the Universe was flat from the Planck era some 380,000 years before the cosmic radiation background.


Steven

[xelasnave]

Thank you Steven.

[Eratosthenes]

The GUT (10^-43 to 10^-38 sec) and Electroweak (10^-38 to 10^-12 sec) eras followed the Planck era (0 to 10^-43).

The recent discovery of the Higgs Boson supports a cosmological model which includes the GUT era followed by the Electroweak era.

When the GUT era ended the strong force separated from the other forces. This decoupling event resulted in a huge release of energy which cause the of "inflation" period. Just to put "inflation" in perspective, the Universe actually expanded by a factor of 1035 in only 10-32 seconds.The Universe exploded from the size of a single electron to the size of a table tennis ball in only 10^-32 seconds. (that's the theory anyway)

There are many unanswered questions about the early Post Planck eras let alone what happened prior to 10^-43 seconds - where there is absence of any Physical theory or observation.

Even though Inflation theory does help explain both the Horizon Problem and the Flatness problem, its by no means settled. (Physicists dont even understand what the bulk of the Universe is made up of with modest understanding of the Baryonic matter that makes up our reality - whatever that it. I would prefer to give Physicists and philosophers another 20 or 30 thousand years of hard work and serious thought - maybe more;). Imagine how long Chemists will need to sort things out, a discipline which is more complex than Physics. Biology more complex than Chemistry and then we have psychology)

[xelasnave]

I have never made it a secret that I don't like the big bang theory.
I don't care what science concludes but how the universe could grow from a speck to all there is, as huge as it is, for me it seems unlikely.
Thought up initially by a priest who presumably believed in God and wanted to give creation a scientific base...but the big bang does not cover the moment of creation but chimes in a split second after leaving open the creation question which one could easily attribute to God.
I really think our cosmology may be driven by philosophy.
Nevertheless this is our best theory.

[xelasnave]

The only alternative theory was the steady state theory which is also difficult to comprehend with it's implication of the universe being timeless with no start in time.
I am not happy with that either.

[sjastro]

A further point on the reference to {1+2+3+4.....}=-1/12 in the book on String Theory by Joseph Polchinski is that the mathematical formalism is based on the operator method.
To save space, equations are truncated to only include the actions of the operator on the quantum states. The quantum states are not included in the equations.

The equation 1.3.32 in the book (http://stringworld.ru/files/Polchinski_J._String_theory._Vol._1 ._An_introduction_to_the_bosonic_st ring.pdf) does not illustrate the summing of the numbers from one to infinity, but the summing of the energy levels of the quantum states which are multiples of (n+1/2), where n=1,2,3........

Steven

[sjastro]

Alex,

Isaac Newton used the Bible to date the creation of the Universe to around 4,000 BC but this doesn't detract from his work as a scientist.
Interesting how Lemaitre was first and foremost a scientist, but being a priest has somehow muddied the waters.
If Lemaitre wasn't a priest would it have added more credibility to the theory?
Even Albert was initially concerned about a possible covert religious content in the theory but came about to accept it.

Steven

[xelasnave]

Thanks for that Steven.
Yes it would be more acceptable if there were no priests involved.
I don't trust them at all.
Fortunately unlike other crack pots I can nevertheless embrace mainstream.

[Eratosthenes]

...in any case it was Friedman who proposed the first expanding universe model (based on GR). Lemaitre's model was proposed later but nevertheless independently (apparently).

Whilst it's true that Georges Lemaitre was a Catholic Priest that dressed the part, he worked as an astronomer, cosmologist and Physics professor. He was by no means some sort of Vatican sponsored Theologian dabbling in cosmology in order to validate the Book of Genesis.

One would assume that Hubble's Constant was calculated and published by Edwin Hubble himself - it wasn't. Lemaitre was the first to derive Hubbles law and also estimate a value for Hubble's constant. The fact that he wore a Catholic frock and white band around his neck is irrelevant - he was one serious Physicist who contributed in a huge way, during a period where the great Physics Icons where doing their great work.

.........some quantum field effects involve negative energy ;)

[xelasnave]

One question I would like help with is if GR was developed, I think but am unsure, when it was thought the Universe was static but it happily described an expanding Universe
I do respect our priest it's good that you stood up for him.

[Eratosthenes]

Not only was the Static model of the Universe the major cosmological paradigm at the time of Einstein's GR theory, Einstein had to input that famous parameter (cosmological constant, Λ) into his GR tensor equations so that the Universe wouldn't expand itself to Smitherenes but rather behave like a static entity. (Einstein's biggest blunder as it turned out)

With the relatively recent discovery that the Universe is not only expanding but the expansion rate is accelerating, (attributed to the mysterious dark Energy repulsion force) that cosmological constant, Λ has made a big come back into Physics and cosmology.

It appears that Einstein himself didn't quite appreciate the ramifications of his Special and General Relativity theories. Einstein initially refused to accept the work of Friedman, Hubble and even fought against the stochastic nature of Quantum mechanics. I recall reading that Einstein had to meet with Edwin Hubble to discuss his red shift measurements and by the time he left, he was convinced that the Universe was indeed expanding.

Cosmology is a fantastic area of Science/Philosophy/mathematics to be involved with professionally, or even as an amateur - there is certainly something deep involved. Its like chasing God around, if you believe in God, trying to corner her and tell her "I know what you did, great work". Or if you are an atheist, there is the equally important search for meaning and understanding.:question:

[xelasnave]

He did not buy the idea of black holes..or so I believe.
You mentioned it started with something the size of an electron.
Lately I get the impression that at the point of singularity it is perhaps not correct to to see it as necessarily a miniscule object.
I wonder have you a view of what it means this singularity.

[xelasnave]

I am an atheist but I am sure god did it.

[xelasnave]

Hubble would have been a powerful and convincing man.
Ex lawyer and I believe he had opportunity to be a heavy weight boxer looking to a world title attempt.
I probably would not argue with him..if he said the universe is expanding I would go along with him...maybe.

[Eratosthenes]

I agree - I tend to view a singularity in its mathematical context. A point of infinite density and zero size. Its physical significance becomes meaningless, but mathematicians recognise a singularity as an undefined concept/entity.

Plenty of evidence exists that Black Holes are real physical objects in the Universe - whether they "actually" have at their centers, points of infinite space time curvature or density is another matter. It may well be just a very compressed object which creates such a high gravitational field that light itself cannot escape.

Who knows xela :question:

[xelasnave]

If we could make an ingot of dark matter say like a gold bar for example and sat it on the desk would we be able to see it?

[Eratosthenes]

:question:we may be able to see some secondary effects caused by this ingot of yours

we certainly would be able to feel it if it is concentrated enough. it is after all pushing outwards...or is it?

on the other hand dark energy may not be acting upon Baryonic matter or 'normal' energy directly or in any way at all.:question:

[xelasnave]

I don't know if it is pushing outwards or what it's doing.
I don't know anything about it other than what I have learned when reading about it.
It seems to me we assume the laws of physics to be universal and our current laws of gravity tell us there should be mass where we do not observe it.
As we can't see it and because we can't explain this situation without more matter than we observe the unseen matter has been called dark matter,
If our gravity laws are valid, and it is somewhat reasonable to expect they are valid then something is interacting gravitationally with matter we can see (stars that are moving faster than our gravity laws predict )
So it is a curious matter.

[xelasnave]

Could dark matter be found in clumps or is dust like.
Could there be clumps as big as a house or even the size of a planet.
Could our gravity laws be questioned given dark matter was not predicted..or was it

[xelasnave]

I think to be fair calling these people such is not fair really.
Mind you in the past I have made similar comments.
However I see this as no more than personal frustration that we still do not know everything.
I think it great there are so many people engaged.
All levels from crank to leading physicist .

[xelasnave]

Sorry Peter I forgot to thank you for your good explanations.

[Eratosthenes]

....so the "-1/12" result has units of energy? (ie eV or J)

The ground state energy, ie n=0, is not equal to zero (1/2hώ) and yet the summation in the Polchinski reference is from n=1 to infinity. Is the ground state energy omitted?

The quantum energy states are distinctly separated by an equal energy level equalling hώ, which is a small number, but nevertheless a finite number. So what happens when you sum an infinite number of finite numbers that increase by the same amount?

The negative sign as well as the 1/12 result needs to be physically explained. What exactly does it represent?

[sjastro]

Neither. The units are in hw where h is Planck's constant and w is the frequency of the oscillator.



The mathematics behind both Quantum Field Theory and String Theory involves the use of mathematical operators acting on the energy state which brings about a particular change.
One such operator for the Hamiltonian H in equation 1.3.30 involves a general term aⁿₓ. This operator drops the bosonic string into the next lowest energy level.
If you start off with n=0, the operator will drop the string into an n=-1 energy level. But there cannot be an n=-1 level as the ground state exists for n=0. Hence you start from n=1 which also includes the ground state for the string. Note that n=0 is the vacuum state.



This has already been explained through renormalization.
The energy levels above a particular level are cut off as they exist above the energy threshold.
This is handled mathematically by multiplying each term in the sum 1.3.31 by an exp(-n) factor. As n becomes larger, the exp(-n) factor becomes smaller. For large n, exp(-n) is approximately zero.
The infinite series is truncated. Remaining terms can be further cancelled out by symmetry leaving 1.3.35 which equals 1.3.31.



It's been explicitly explained in the reference as.

[Eratosthenes]

no need Alex

It's a bit like my 27 year old Newtonian reflector and collimation

[Eratosthenes]

"hw" has units of energy

What is the physical significance of -1/12 (energy units)?

[sjastro]

Try this.
http://en.wikipedia.org/wiki/Natural_units

[sjastro]

Well then by the same token why believe in Polchinski's book.
Planck's constant doesn't explicitly appear because it has been assigned a natural unit equalling one.:)

[Eratosthenes]

Everything written down or spoken is inherently corrupt.

Including this

[sjastro]

To make such a statement you must be one hell of an omnipotent being whose knowledge and understanding exceeds that of every other individual, or the more likely explanation is the argument from personal incredulity at work.

[xelasnave]

For the boys and girls watching and not familar with logical falacies here is a link for the one Steven mentioned.

http://rationalwiki.org/wiki/Argument_from_incredulity

[xelasnave]

I don't want to say anything but I want a copy of this post in case it becomes a virtual post and pops out of existence.

But back to the thread subject matter with a question.

If the universe is really really big would we be able to observe curvature as it I presume may not be measurable but nevertheless not flat.

[xelasnave]

There we go it's gone but it was there.

[sharpiel]

Don't worry Alex Evans. You saved it for us all...

[Eratosthenes]

You make a good point Alex

If we assume that the Universe is extremely large - well beyond the detection limits we are capable of today - then we may observe a near flat Universe even though the Universe could have any shape or curvature imaginable.

If the Universe is indeed infinite in size, wouldn't that imply that any measurement of its curvature would be flat, irrespective of its actual shape or geometry (if shape and geometry can be applied to a truly infinite object)

[sjastro]

Flatness is not a criteria for an infinite surface. As has already been discussed, a 3-torus is flat but finite.

A zero curvature flat Universe is supported by the angular size of the anisotropic structures in the CMB.
If the Universe has a positive curvature the anisotropic structures would appear larger.
Conversely for a negative curvature, structures would appear smaller.

[xelasnave]

Hi Les I notice you save animals and birds you are a good man.

[Eratosthenes]

Let us extract ourselves from the rudimentary sermons depicted in the dusty text books and lecture rooms. We know that cosmologists are NEVER wrong but always in doubt.

An Alcmaeonistic revisiting of the boundary limits of a pseudo non Euclidean membrane type existence offers some very interesting insights in what may be occurring at the transient interfaces that form as a direct result of instabilities in the core-film itself.

Now, you may think that these instabilities would collapse to a point where they don't influence the "bulk" phase. This would be a sound assumption to make if one could fully describe the phase space that is applicable in this situation. Of course one cannot

So that returns us to the original question posed.

[xelasnave]

One does not have to be a cosmologist to never be wrong.
It is a human condition few are immune.
I hold beliefs which are not mainstream but I respect the approach of mainstream.

[Eratosthenes]

I stir fry "mainstream" and "non-mainstream" vegetables with the same oil, garlic and onion before I add my secret ingredients

[xelasnave]

My speciality is a spiced up word salard.

[Eratosthenes]

So Alex, is salard a type of seafood?:D

[xelasnave]

Rather than admit I was wrong...yes, yes it is a fish

[sharpiel]

Thank you Alex.

I must say I have enjoyed this thread much more than any other I have ever read. Even those posts which slipped beyond the event horizon and became lost to view...

[Eratosthenes]

I thought so,

nothing like a grilled salard with a side salad and chips and a chilled Leffe:)

(the important thing to remember is that the B-mode polarisation is a signature of inflation in the early Universe. Are we really seeing gravitational waves?)

[sjastro]

My goodness you suffer a lack of basic comprehension skills.




What is absolutely laughable this was a response to one of your own posts.:rofl::rofl::rofl::rofl:

Evidently the answer went way over your head because you failed to see the association of gravitational waves with inflation, or you simply refuse to accept anything that deviates from your narrow minded view of events.

[xelasnave]

It was my Impression that when BICEP 2 results were hot news there were doubts being expressed within the mainstream community that the data may not be reliable.

In any event given the truth came out a short time later I would not call the situation an embarassment but rather a good example of science working openly and effectively.
News reached us via science journalists and I wonder if the matter was sensationalised .
And even if the team went off half cocked the system was shown to work.

[sjastro]

Alex,

From day one the results were deemed suspicious particularly when they were released prior to the Planck data.
The BICEP2 team had effectively jumped the gun in proclaiming to the world they had found gravitational waves through polarization.
It was an embarrassment as they had to retract their statement when it was found through the Planck data, the contribution of a similar polarization effect through magnetized dust in our galaxy was greater than the BICEP2 team had accounted for.

The BICEP2 case is a good example of how science works.

As far as the cosmologists "never being wrong........", the cosmologist Laurence Krauss stated it eloquently on ABC's Q&A program, that cosmologists usually do get it wrong. A point that is conveniently omitted is the relentless peer review process where many ideas fall by the wayside and even if they survive that stage, observation and experiment can kill off the hypothesis.

Steven

[xelasnave]

Yes Steven you covered the situation very well.
I don't fully understand the system but I am confident that it works as good as we can hope for.
The point to emphasise is that the team made the mistake not the whole body of mainstream.
Reading at cosmoquest tells me that one has little chance of presenting material that will not go without the most critical review.
I obviously have no experience with peer review but I can't imagine it is not effective.

[Eratosthenes]

Alex is a real gentlemen - and open minded.

And also likes salards - cant ask for more than that

Unlike the pseudo intellectual scientist Lawrence Krauss who prances around the planet spewing out his materialistic corporatized version of Atheism and selectively bagging Christianity....Pity he doesn't spend more time researching some vibrant areas of Physics rather than interviewing Chomsky on stage for big dollars and pretending to sound important - Morally Sour Krauss is perhaps in the top 7 clowns in the western hemisphere today

[RB]

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