Are Lawrence Krauss' theories accepted in physics academia?

In summary, there are multiple cosmological models proposed to explain the origin of the universe, including ideas from quantum gravity and cyclic models. Alex Vilenkin's theory, dating back to 1982, suggests that space and time can emerge from a state of "nothing." However, there is no consensus on the origin of the universe and many models are still being researched. The concept of a zero-energy universe, first proposed by Edward Tyron, has faced criticism for its reliance on pseudo-tensors and coordinate dependence. Overall, the topic remains a subject of debate and further research in the field of theoretical physics.
  • #1
thetaobums
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How could something possibly come from nothing? He reminds me of the Sokal Affair.
 
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  • #2
They are not his theories, the idea came from a cosmologist called Alex Vilenkin and its far from new, dates back to 1982 in this paper:
http://www.sciencedirect.com/science/article/pii/0370269382908668

Vilenkin's is one of many many ideas as to the origin of the universe, including ideas that it may be eternal. There is no consensus and certainly no observational support for anyone idea
 
  • #3
There are many, many cosmological models that attempt to explain the origin of the universe. Most recently, models from quantum gravity have grown in popularity, such as loop quantum cosmology's 'bounce' models (marcus is the expert on this, I don't know enough about LQG to comment on them). There are various cyclic models, and loads of models where the universe can fluctuate out of a vacuum.

If you're asking if Krauss's ideas are accepted to be possible, then yes. That may be how the universe began. However, there is no consensus. There are dozens of non-singular cosmologies that are researched by many people.

BTW, as pointed out by skydivephil, these aren't Krauss's ideas. Actually, Edward Tyron was the first to start taking seriously a zero-energy universe.
 
  • #4
thetaobums said:
How could something possibly come from nothing? He reminds me of the Sokal Affair.

Krauss is IN physics academia, so obviously for at least one person the answer is yes.
 
  • #5
Mark M said:
There are many, many cosmological models that attempt to explain the origin of the universe. Most recently, models from quantum gravity have grown in popularity, such as loop quantum cosmology's 'bounce' models (marcus is the expert on this, I don't know enough about LQG to comment on them). There are various cyclic models, and loads of models where the universe can fluctuate out of a vacuum.

If you're asking if Krauss's ideas are accepted to be possible, then yes. That may be how the universe began. However, there is no consensus. There are dozens of non-singular cosmologies that are researched by many people.

BTW, as pointed out by skydivephil, these aren't Krauss's ideas. Actually, Edward Tyron was the first to start taking seriously a zero-energy universe.

Tyrons idea is the universe arose form a vacuum fluctuation. But there is still a vacuum and an underlying space time there in the first place. Vilenkin's idea has a better to claim to the word "nothing". In Vilenkin's model space and time themselves tunnel into existence from a state where there is no state or time. Vilenkin is a well respected cosmologist so i really don't htink one can compare him to the subject of the Sokal affair. But being respected and getting your stuff into proper physics journals is just the beginning of a long road to acceptance, not the end point if there ever is one. Until your model makes predictions that can be tested it won't gain full acceptance. To be fair to Krauss he's not arguing anything more than the idea is plausible, not correct.
 
  • #6
Mark M said:
Edward Tyron was the first to start taking seriously a zero-energy universe.
What does the zero energy universe mean though? From what I've read on this there seem to be some formidable obstacles:
1. Energy is not a globally defined quantity in GR, so it means nothing to say the total energy is zero.
2. The simple description of the concept seems to rely on the fact that gravitational potential energy is usually presented as negative, with zero being achieved only at infinity, and offsetting this negative against the positive energy of matter and radiation. But this seems purely arbitrary, as it is only relative, not absolute levels of potential energy that have any physical meaning.

Is there any physically sound way around these objections?
 
  • #8
thetaobums said:
How could something possibly come from nothing? He reminds me of the Sokal Affair.
Krauss basically presents the mainstream view of theoretical physics within academia to the public.
 
  • #9
Thanks for that link Mark. I see from the abstract that the paper is based on pseudo-tensors. I have avoided pseudo-tensors thus far, having gained the impression that many physicists do not consider them respectable, such as in this FAQ post. The coordinate dependence does seem to be a problem. Does the total energy, as defined by these pseudo-tensors, sum to zero in all reference frames, or only in a particular class of frames?
 
  • #10
Thanks for the link to the FAQ.

Yes, I believe all of Berman's papers use pseudo-tensors. The energy sums to zero for only a closed universe, see from the FAQ you posted:

"For certain pseudo-tensor definitions of mass-energy, the total energy of a closed universe can be calculated, and is zero.[Berman 2009] This does not mean that "the" energy of the universe is zero, especially since our universe may not be closed."

Since he obtains the 'general relativistic version of the energy-momentum four vector' (by contracting the Einstein pseudo-tensor with the stress-energy tensor), I would assume this is independent of the FoR.

However, there is a coordinate dependence - on Cartesian coordinates. If you use spherical coordinates, there isn't any way to guarantee energy is conserved, but instead varies in time. So, you must use Cartesian coordinates. However, I seem to remember Berman writing another paper where he derived the result with spherical coordinates, but I'm not sure.

And of course, these only apply for a finite universe (closed or flat). If the universe has negative curvature, or an infinite flat topology, then of course there is no consistent way you can integrate the energy density over an infinite space and receive sensible results.
 
  • #11
thetaobums said:
How could something possibly come from nothing? He reminds me of the Sokal .

a good review at

INTRODUCTORY LECTURES ON QUANTUM COSMOLOGY
Jonathan J.Halliwell
http://arxiv.org/pdf/0909.2566v1.pdf

...Boundary Condition Proposals and Tunneling
Hartle and Hawking (Hartle and Hawking, 1983; Hawking 1982, 1984a), Linde (1984a, 1984b, 1984c) and Vilenkin (1982, 1983, 1984, 1985b, 1986, 1988), but there are others (see for example, Suen and Young (1989)).

Quantum Creation of the Universe
Some of the older papers on quantum creation of the universe are those of Atkatz and Pagels (1982), Brout, Englert and Gunzig (1978, 1979), Brout, Englert and Spindel (1979), Casher and Englert (1981), Gott (1982) and Tryon (1973). Various aspects of the quantum creation of the universe as a tunneling event have been explored by Goncharov et al. (1987), Grishchuk (1987), Grishchuk and Sidorov (1988, 1989), Grishchuk and Zel’dovich (1982), Lavrelashvili, Rubakov, Serebryakov and Tinyakov (1989), Lavrelashvili, Rubakov, and Tinyakov (1985), Rubakov (1984) and Rubakov and Tinyakov (1988)...

Creating a Universe in the Laboratory
The possibility of quantum creation of an inflationary universe in the laboratory has bee studied by Farhi et al. (1989) and Fischler et al. (1989). See also Hiscock (1987) and Sato et al. (1982)...

vilenkin paper
http://mukto-mona.net/science/physics/a_vilinkin/universe_from_nothing.pdf [Broken]
i asked personally to vilenkin "why exist the universe" , and he answered:
"better ask to the Dalai Lama" and we laugh a while...how can be tested ? Quantum creation scenarios produce gravitational waves of a calculable form and magnitude, models differ in predictions

The Quantum Creation of the Universe can be Observationally Verified.
Grishchuk, L.P. (1987), Mod.Phys.Lett. A2, 631
http://gravityresearchfoundation.org/pdf/awarded/1987/grishchuk.pdf [Broken]

QUANTUM EFFECTS IN COSMOLOGY
http://arxiv.org/pdf/gr-qc/9302036v1.pdf
 
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  • #12
thetaobums said:
How could something possibly come from nothing? He reminds me of the Sokal Affair.

Vacuum is not nothing.

Also the situation with Krauss ...

1) as far as I can tell his peer-reviewed papers have nothing to do with quantum cosmology
2) he presents one possible scenario for the creation of the universe, but doesn't mention
that this is one of about fifty others
3) once he presents that one scenario he goes *way* overboard into theology and philosophy in ways I personally find very objectionable since it give the impression that all physicists support those views which they don't.

A lot of this depends on "theories of what?"
 
  • #13
skydivephil said:
To be fair to Krauss he's not arguing anything more than the idea is plausible, not correct.

He gives people the impression otherwise. If that's not his intention, he needs to be much more careful.

The description from his book on Amazon says:

A fascinating antidote to outmoded philosophical and religious thinking, A Universe from Nothing is a provocative, game-changing entry into the debate about the existence of God and everything that exists. “Forget Jesus,” Krauss has argued, “the stars died so you could be born.”

Personally, I find that nutty, and well beyond the data. The problem with Krauss is that not that he is a crank. The problem is that he isn't a crank. He's written some excellent papers, and if you cut out the philosophical and religious commentary, there is a decent introduction to some of the current ideas in cosmology. That's what makes this dangerous. Someone that doesn't know better can't easily separate the science from the non-science and assume that physicists agree with his religious views.

It's actually very hard to figure out what to do with Krauss. I have different religious views which are quite different to his, but I don't want to go into details because I like to keep my religion and my science separate. On the other hand, when he ends up giving people the impressions that because physicists often try to avoid talking about God, that means they are all atheists, then I just have to mention that this just isn't true.

One other problem is that Krauss doesn't emphasize the extent to which theorists are *guessing* and that what guess ends up to be the solution is based on observation. Something that could very well happen is that we see a certain CMB signature or find primordial gravity waves at which point, Krauss would have to write another book, remember what I wrote 10 years ago, well it was wrong. Not that this is a bad thing. One of the reason I find science more interesting than religion is that science tends to say "remember what we thought ten years ago. HA! it was wrong."

Once you remove the element of self-doubt, observation, and uncertainty from science, you end up with a mess.
 
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  • #14
I totally agree that a more interesting book on the early universe would include the many different scenarios, quanutm bounce, brane clash, CCC, VSL, etc . and yes I think he needs to be louder on his caveats. The model he's basically putting forward, Vilenkins tunneling form nothing model, is just one idea out of many. Nevertheless those caveats are there and I guess perhaps people read into what they want to, ignoring them or noticing them depending on what they want to see.
What I think Krauss is saying is that a quantum nucleation event from "nothing" has to have undergone inflation to get the unvierse we see. The evidence points to inflation and so his scenario is "more plausible", I don't think he ever claims its verfied. My problem is that a bouncing unvierse also undergoes inflation and in this case its a universe from "something". So whilst Krauss seems to be right, inflation does make Vielnkins model more plausbile ; its a very incomplete picture. You can't disntinguish Vilenkin model form say LQC bounce by confirming inflation happened.
Im not sure why, but popular accounts of cosmology seem to like inflation, like the brane clash model, but LQC boucne seemes to be ignored by many popularisers. Krauss is guilty of this, but he's far from the only one.
I don't see why Krauss can't promote his religious beliefs, there's plenty of religious people claiming cosmology supports their religious position. Somebody needs to respond to that. Krauss reposnse is not the way I'd respond. I think its better to say we don't know the origin of the unvierse and the religious often imply we do, rather than - hey it might have come from nothing. but perhaps its a better response than silence.
 
  • #15
Mark M said:
And of course, these only apply for a finite universe (closed or flat). If the universe has negative curvature, or an infinite flat topology, then of course there is no consistent way you can integrate the energy density over an infinite space and receive sensible results.
I just noticed this. What do you have in mind for a finite flat universe? Wouldn't it either have to be a manifold with boundary or else be an infinite flat manifold with the mass energy all lying within a bounded region (which would not be consistent with the cosmological principle, not that that's necessarily a show-stopper).
 
  • #16
skydivephil said:
Nevertheless those caveats are there and I guess perhaps people read into what they want to, ignoring them or noticing them depending on what they want to see.

The issue is that this sort of thing tends to be expected when you are talking with a popular audience. People and the press will react to statements in rather predictable ways, so if you spend two hours talking about something and then mention "God" gets what gets quote. We've had a lot of this recently with the "God Particle" non-sense.

The danger is that people will end up with sensationalism burnout if you aren't careful.

Also talking about "something from nothing" also runs into the possibility of oversensationalism.

What I think Krauss is saying is that a quantum nucleation event from "nothing" has to have undergone inflation to get the unvierse we see.

Which is very common scenario.

You can't disntinguish Vilenkin model form say LQC bounce by confirming inflation happened. I am not sure why, but popular accounts of cosmology seem to like inflation, like the brane clash model, but LQC bounce seemes to be ignored by many popularisers. Krauss is guilty of this, but he's far from the only one.

The reason why LQC gets ignored is that it's "boring". None of this cool talk about "something from nothing" or "God stuff."

I don't see why Krauss can't promote his religious beliefs, there's plenty of religious people claiming cosmology supports their religious position.

I don't object to Krauss talking about his religious beliefs. I also don't mind if he supports various candidates or plugs diet Cola. What I do find a bad thing is that lots of people seem to have the impression that science supports his religious beliefs rather than being some personal preference.

There are plenty of religious people claiming cosmology supports their religious position. Usually anyone that argues that ends up with rotten science. The fact that I have big issues with religious people using cosmology to support their philosophy is pretty much the same reason I have big issues with atheists do the same thing.

I think its better to say we don't know the origin of the unvierse and the religious often imply we do, rather than - hey it might have come from nothing.

The trouble is that we pretty soon will figure it out. The cool thing about cosmology is that we are making a *lot* of progress in figuring out the origins of the universe. Unknown is not unknowable.
 
  • #17
andrewkirk said:
I just noticed this. What do you have in mind for a finite flat universe? Wouldn't it either have to be a manifold with boundary or else be an infinite flat manifold with the mass energy all lying within a bounded region (which would not be consistent with the cosmological principle, not that that's necessarily a show-stopper).

The thing about inflationary models is that a lot of them have the universe being some constantly expanding field, and then a "bubble" drops out of it and stops expanding. That "bubble" forms our entire universe.

There are some big philosophical issues here involving how much you can extrapolate from the data. The good news is that we are ending up with so much data, that it looks like we can figure out a great deal.

One other thing to think about. If we can produce a child universe in the laboratory, then presumably some transdimensional being who is bored can produce one in their laboratory ending up with us.
 
  • #18
Having thought about it a bit more, I wonder if the reason LQC bounce scenarios are "boring" is that they are less of an ultimate answer. Why do you think? Suppose we found the universe was eternal say like in the Caroll/Chen model or something like that or suppose we found it did come from nothing. Now that's a lot closer to some ultimate answer than our universe bounced from another one that collapsed, where did that unvierse come from? Answer : no idea.
The latter I personally find v exciting becuase it means there's more to learn. But perhaps others don't and maybe that's the probelm. Although if it turns out there's some comsic censorship and we can never know what happened on the other side of the bounce that will be very unsatsifying. But whilst LQC may be less satisfying it seems more liklely to get into a testable format and let's hope someone should be making more noise about that.
I agreer with most of what you said. When you say you have a problem with atheists doing the same thing. Doesnt it depend what you mean by atheist? There are postive and negative atheists. Positive atheism asserts god does not exist. Negative atheism is the rejection of belief in god usually on the grounds of insufficient evidence. Since the latter is simply the null hypothesis applied to the god question this form of athiesm seems entirley consistent with the scientific endevour. But if you want to coontinue to discuss this issue I imagine there's a different forum area to go forward.
 
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  • #19
skydivephil said:
Having thought about it a bit more, I wonder if the reason LQC bounce scenarios are "boring" is that they are less of an ultimate answer. Why do you think?

It's also that the LQC approach is "worry about gravity and don't try to come up with a theory of everything."

Now that's a lot closer to some ultimate answer than our universe bounced from another one that collapsed, where did that unvierse come from? Answer : no idea.

On the other hand, the quantum fluctuation models tend to posit that the universe "dropped" out of an expanding field.

Doesnt it depend what you mean by atheist? There are postive and negative atheists. Positive atheism asserts god does not exist. Negative atheism is the rejection of belief in god usually on the grounds of insufficient evidence.

If someone thinks that God is unnecessary, I don't have a problem with that. I do have a problem with someone calling me "delusional" because I happen to believe in God, which is the position that Richard Dawkins takes, and part of the reason that Krauss has gotten a lot of press is that Dawkins wrote a section of Krauss's book.

It's odd but I have some young Earth creationists that I have to apologize to. Way back when when I was arguing with young Earth creationists, I found that they really didn't care that much about the age of the universe. The thing that worried them was the idea that scientists had this agenda of "getting rid of God everywhere" and so they were resistant to accepting scientific evidence because "first you get us to accept that the universe is 13.6 years old, then next you get rid of the crosses and Bibles from our houses and tell us that how to raise our kids." Part of getting them to accept scientific evidence, was to assure them that scientists weren't trying to "abolish God."

It turns out that I wasn't, but Dawkins clearly is, and since Krauss has Dawkins write a section in his book, I don't think it is opposed to Dawkin's agenda. I wish Stephen Jay Gould was still around.

Part of this is a US/UK thing. Religion is very strong in the United States, whereas in the UK, I'm told that it's sort of dying out.
 
  • #20
andrewkirk said:
I just noticed this. What do you have in mind for a finite flat universe? Wouldn't it either have to be a manifold with boundary or else be an infinite flat manifold with the mass energy all lying within a bounded region (which would not be consistent with the cosmological principle, not that that's necessarily a show-stopper).

A universe with the topology of a 3-torus would have Euclidean geometry, but no boundary.

Essentially, a 2-torus is the familiar surface of a doughnut. This may seem to curved, but it is still Euclidean. Parallel lines stay parallel, the angles of a triangle added up to 180 degrees, etc.

A 3-torus would be the higher dimensional generalization.
 
  • #21
twofish-quant said:
Part of getting them to accept scientific evidence, was to assure them that scientists weren't trying to "abolish God."
But that would be lying.
 
  • #22
Not to wander off topic, but, I've always felt gratuitous slaps at theism has no place in science. Science seeks explanations that require no hidden variables [i.e., supernatural elements], which is inoffensive to almost everyone: using it to trivialize belief systems [or practitioners thereof] is not. Let the facts speak for themselves. Inciting or fueling public sentiment that science is anti-theistic is distracting and counter productive.
 
  • #23
Chronos said:
Inciting or fueling public sentiment that science is anti-theistic is distracting and counter productive.
Counter-productive to what, precisely?
 
  • #24
twofish-quant said:
It's also that the LQC approach is "worry about gravity and don't try to come up with a theory of everything."



On the other hand, the quantum fluctuation models tend to posit that the universe "dropped" out of an expanding field.



If someone thinks that God is unnecessary, I don't have a problem with that. I do have a problem with someone calling me "delusional" because I happen to believe in God, which is the position that Richard Dawkins takes, and part of the reason that Krauss has gotten a lot of press is that Dawkins wrote a section of Krauss's book.

It's odd but I have some young Earth creationists that I have to apologize to. Way back when when I was arguing with young Earth creationists, I found that they really didn't care that much about the age of the universe. The thing that worried them was the idea that scientists had this agenda of "getting rid of God everywhere" and so they were resistant to accepting scientific evidence because "first you get us to accept that the universe is 13.6 years old, then next you get rid of the crosses and Bibles from our houses and tell us that how to raise our kids." Part of getting them to accept scientific evidence, was to assure them that scientists weren't trying to "abolish God."

It turns out that I wasn't, but Dawkins clearly is, and since Krauss has Dawkins write a section in his book, I don't think it is opposed to Dawkin's agenda. I wish Stephen Jay Gould was still around.

Part of this is a US/UK thing. Religion is very strong in the United States, whereas in the UK, I'm told that it's sort of dying out.


I think there is a great divide between the Us and the Uk . In the Us I think polls are around 85-95% of the population believe in god. A recent poll in the Uk found only 17% of the populaiton belived that god created the unvierse. Churches in the Uk are turned into aprtments and nightclubs.
I can understand a science educator in the USA wanting to promote Gould's NOMA for diplomatic/tactical reasons. If you are going to admit there's a conflict between science and religion and 90% of the poulatin is religious, you are likely to lose.
Not so in the UK.
The question that follows then is forgetting tactical/diplomatic reaosn, is NOMA valid? I think the problem is Gould was just wrong to presume they don't overlap. Religions are extremley interested in origins and any science that deals with origins is going to arouse the interest of religion. In some context some of the religious wing will deny the science ie evolution and sometimes climate change . In others they will hijack it and pretend it proves god ie mono theists with cosmology or new age mystics with QM. The truth is NOMA doesn't work becuase the magisterium do overlap. If atheists cosmologists like Krauss don't talk about comsology you can be damm sure theolgians and mystics will.
 
  • #25
skydivephil said:
To be fair to Krauss he's not arguing anything more than the idea is plausible, not correct.

Right. From the end of Chapter 10, 'A Universe From Nothing':

Does this prove that our universe arose from nothing? Of course not. But it does take us one rather large step closer to the plausibility of such a scenario.
 
  • #26
Chalnoth said:
Counter-productive to what, precisely?
When project funding is cut or denied due to political friction, I would call that counter-productive.
 
  • #27
skydivephil said:
Having thought about it a bit more, I wonder if the reason LQC bounce scenarios are "boring" is that they are less of an ultimate answer. Why do you think? Suppose we found the universe was eternal say like in the Caroll/Chen model or something like that or suppose we found it did come from nothing. Now that's a lot closer to some ultimate answer than our universe bounced from another one that collapsed, where did that unvierse come from? Answer : no idea.
The latter I personally find v exciting becuase it means there's more to learn. But perhaps others dont... But whilst LQC may be less satisfying it seems more liklely to get into a testable format and let's hope someone should be making more noise about that.
...

Perceptive comment. It reminds me that a group sense of TIMING and gradualism is so important to how science works. A delicate partial consensus concerning which questions are ripe to be asked and investigated---and which ones we are not yet ready to tackle. A sense that human concepts, language, math tools, models are part of an evolutionary process where something analogous to natural selection deciding which models work best and which concepts to replicate: a kind of empirical selection of the fittest and best, by scientific communities. Empiricism has a subjective element and requires people to discuss and act in good faith (a delicate business!)

There is a kind of co-evolution of the scientific community and its conceptual tools---and a selective pressure which seems to be towards simpler&more reliable explanations---which probably can't be rushed.

The probing into QG (more generally into quantum geometry and its interaction with matter) is an example. Maybe we are not yet ready to address what seem like "ultimate theory" questions, but should be more patient and try to understand geometry better. How did our particular "big bang" work? Since we see lots of information in the sky resulting from it, let's try to understand this.

Your mentioning religion gave me a chuckle. I tried to imagine a church with a sign out in front of it saying "We don't know the answer yet, but we're working on it." Just think if the Vatican would issue a statement like that: human knowledge evolves--human language evolves--no final concepts.

Perhaps this is the asymmetry that makes it hard to use NOMA. In one magisterium you want immediate answers to the ultimate questions (like why does existence exist? what's it mean and what's the right thing to do?)
In the other magisterium there is a sense of time and evolution: we know this picture isn't quite right, but it is the most reliable picture we can make for now, and stay tuned for further changes...

skydivephil said:
I think there is a great divide between the Us and the Uk . In the Us I think polls are around 85-95% of the population believe in god. A recent poll in the Uk found only 17% of the populaiton belived that god created the unvierse. Churches in the Uk are turned into aprtments and nightclubs.
I can understand a science educator in the USA wanting to promote Gould's NOMA for diplomatic/tactical reasons. If you are going to admit there's a conflict between science and religion and 90% of the populatin is religious, you are likely to lose.
Not so in the UK.
The question that follows then is forgetting tactical/diplomatic reaosn, is NOMA valid? I think the problem is Gould was just wrong to presume they don't overlap. Religions are extremley interested in origins and any science that deals with origins is going to arouse the interest of religion. In some context some of the religious wing will deny the science ie evolution and sometimes climate change . In others they will hijack it and pretend it proves god ie mono theists with cosmology or new age mystics with QM. The truth is NOMA doesn't work becuase the magisterium do overlap. If atheists cosmologists like Krauss don't talk about comsology you can be damm sure theolgians and mystics will.

I am uncomfortable with Krauss' book (of which I've only read the preface and gotten a sense of from his video and the discussion about the book) because it seems to me to be doing the same thing that dogmatic religions do which is to inflame the human appetite for immediate answers to ultimate questions, and pander to that appetite.

I may be wrong: I've only read part of the book and really shouldn't comment on it directly. Skydive you said:
"Religions are extremley interested in origins and any science that deals with origins is going to arouse the interest"
Especially any science that appears to deal with ultimate origins. It seems to me that religions are primarily interested in "first" causes (though this may be an oversimplification.) By contrast, cosmology at this stage is merely interested in understanding the big bang*. For someone fascinated by ultimate questions that could well be boring.
But how can one ask "what came before what came before?" until one has a good understanding of the big bang?

*plus various other things including what goes on in black holes and expectations about the longrange future of what we see and live in.
 
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  • #28
Chronos said:
When project funding is cut or denied due to political friction, I would call that counter-productive.
And I would call it counter-productive to pollute science with money from religious institutions, such as the Templeton Foundation.
 
  • #29
Mark M said:
A universe with the topology of a 3-torus would have Euclidean geometry, but no boundary.

Essentially, a 2-torus is the familiar surface of a doughnut. This may seem to curved, but it is still Euclidean. Parallel lines stay parallel, the angles of a triangle added up to 180 degrees, etc.

A 3-torus would be the higher dimensional generalization.
Great! I hadn't thought of that. Certainly the flat torus is a compact 2D space without boundary. I suppose the flat 3D torus is what you get when you take a cube and identify the three pairs of opposing faces, with the correct sense. That would certainly meet the requirements of a completely flat space that is compact.

Are you sure about the regular torus being flat though? The standard coordinate system is orthogonal but that's not enough for flatness is it? It's homeomorphic and diffeomorphic to the flat torus, but there's no isometric mapping. I would have thought that it would follow from that that there is no mapping from one to the other that preserves the metric tensor everywhere, and that it would follow in turn that one could construct a triangle on a regular torus whose angles don't add to 180 degrees.

I know I'm being lazy here. I should sit down and work this out for myself. But maybe somebody can point me to somewhere that it's already been worked out (I googled a bit but had no luck).

I suppose what I'm asking is whether there exists a diffeomorphism [itex]\pi[/itex] from the flat torus F to the regular torus R for which the induced metric [itex]g^*_R[/itex] on R defined by [itex]g^*_R(\vec{u},\vec{v}):=g_F(d\pi^{-1}(\vec{u}),d\pi^{-1}(\vec{v}))[/itex], where [itex]d\pi[/itex] is the differential of [itex]\pi[/itex], satisfies:
[itex]g^*_R(\vec{u},\vec{v})=g_R(\vec{u},\vec{v})[/itex] everywhere on R.
 
  • #30
andrewkirk said:
Are you sure about the regular torus being flat though?
I believe you have to embed the 2D torus in four dimensions in order for it to be flat everywhere (note: you don't have to embed the torus at all). In three dimensions, there is some negative curvature on the inner side, and some positive curvature on the outer side.
 
  • #31
This thread is closed until I have the time to make major edits. Blatant violations of Physics Forums Rules, to which everyone agrees when they register, can result in infractions being issued.

Greg Bernhardt said:
Religious Discussion: Discussions that assert the a priori truth or falsity of religious dogmas and belief systems, or value judgments stemming from such religious belief systems, will not be tolerated.
 

1. What are Lawrence Krauss' theories in physics?

Lawrence Krauss is a theoretical physicist who has proposed various theories in cosmology, particle physics, and astrophysics. Some of his notable theories include the "anthropic principle" and the "vacuum energy hypothesis."

2. Are Lawrence Krauss' theories widely accepted in the physics community?

While some of Krauss' theories have gained traction and are widely accepted, others have been met with criticism and skepticism within the physics community. The acceptance of his theories varies depending on the evidence and support for them.

3. What evidence supports Lawrence Krauss' theories?

Krauss' theories are typically based on mathematical models and observations from experiments and observations. Some of his theories, such as the "inflationary universe" model, have also received support from other physicists and researchers in the field.

4. Have any of Lawrence Krauss' theories been proven or disproven?

The scientific process involves constant testing and refinement of theories, so it is not uncommon for theories to be modified or even disproven as new evidence and data emerge. Some of Krauss' theories have been supported by evidence, while others have been challenged or modified based on new research.

5. How does Lawrence Krauss' work contribute to the field of physics?

Krauss' work has made significant contributions to our understanding of the universe and has sparked important discussions and debates within the physics community. His research has also inspired new experiments and observations that have furthered our understanding of the cosmos.

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