# B Is the Hawking-Hartle Euclidean approach confirmed by CMBR data?

#### Cerenkov

Summary
Stephen Hawking and Jim Hartle pioneered the Euclidean approach to early universe cosmology. This involves the use of a technique called Imaginary Time. According to Hawking this technique was used at the 1982 Nuffield workshop.
http://inspirehep.net/search?p=773__w%3AC82/06/21.2+or+773__w%3AC82-06-21.2+and+980__a%3AConferencePaper
I am interested in the CMBR-related predictions that resulted from that workshop.
Hello.

I have three questions about a claim made by Stephen Hawking in his book, 'My Brief History' and I would be grateful to receive some help concerning it please. Here is a .pdf version of it.
Here is the relevant section, from chapter 12, Imaginary Time.

I HAD been working mainly on black holes, but my interest in cosmology was renewed by the suggestion that the early universe had gone through a period of inflationary expansion. Its size would have grown at an ever-increasing rate, just as prices go up in the shops. In 1982, using Euclidean methods, I showed that such a universe would become slightly non-uniform. Similar results were obtained by the Russian scientist Viatcheslav Mukhanov about the same time, but that only became known later in the West.

These non-uniformities can be regarded as arising from thermal fluctuations due to the effective temperature in an inflationary universe that Gary Gibbons and I had discovered eight years earlier. Several other people later made similar predictions. I held a workshop in Cambridge, attended by all the major players in the field, and at this meeting we established most of our present picture of inflation, including the all-important density fluctuations that give rise to galaxy formation, and so to our existence.

This was ten years before the Cosmic Background Explorer (COBE) satellite recorded differences in the microwave background in different directions produced by the density fluctuations. So again, in the study of gravity, theory was ahead of experiment. These fluctuations were later confirmed by the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite, and were found to agree exactly with predictions.

Now to my questions.

1.
Firstly, I read Hawking's words to mean that the fluctuations observed by WMAP and Planck were first predicted by him and Hartle, by their usage of the Euclidean approach and the Imaginary Time technique. Have I drawn a false conclusion, here?

2.
If that is so, then which fluctuations are these? These? https://physics.stackexchange.com/questions/155508/angular-power-spectrum-of-cmb Or some other?

3.
Or, if I have misread what Hawking is saying, could I please be informed as to what he actually means about the relationship between the Euclidean approach and the CMBR?

Any help given at my basic level would be much appreciated. Thank you.

Cerenkov.

#### PAllen

On your question 1, that appears to be what Hawking is claiming. Alan Guth would present a very different history and priority. I think most cosmologists would say Hawking is taking excessive credit here. This same point has been criticized in other popular books by Hawking.

#### Cerenkov

Ah thanks PAllen. I suspected that might be the case. Thanks for confirming my suspicions.

If I may ask you one further, related question. In the same chapter Hawking writes...

"Our calculation [Hartle and I] made use of the concept of imaginary time, which can be regarded as a direction of time at right angles to ordinary real time. When I returned to Cambridge I developed this idea further with two of my former research students, Gary Gibbons and Malcolm Perry. We replaced ordinary time with imaginary time. This is called the Euclidean approach, because it makes time become a fourth direction of space. It met with a lot of resistance at first but is now generally accepted as the best way to study quantum gravity."

So, is the Euclidean approach now generally accepted as being the best way of studying quantum gravity?

Thanks again, Cerenkov.

#### kimbyd

Gold Member
2018 Award
Ah thanks PAllen. I suspected that might be the case. Thanks for confirming my suspicions.

If I may ask you one further, related question. In the same chapter Hawking writes...

"Our calculation [Hartle and I] made use of the concept of imaginary time, which can be regarded as a direction of time at right angles to ordinary real time. When I returned to Cambridge I developed this idea further with two of my former research students, Gary Gibbons and Malcolm Perry. We replaced ordinary time with imaginary time. This is called the Euclidean approach, because it makes time become a fourth direction of space. It met with a lot of resistance at first but is now generally accepted as the best way to study quantum gravity."

So, is the Euclidean approach now generally accepted as being the best way of studying quantum gravity?

Thanks again, Cerenkov.
On your last question, I don't think so. And I don't think this description of "imaginary time" is at all accurate. As I understand it, the imaginary time notation comes from the fact that the space-time distance element can (in flat space-time) be defined as:
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$
In General Relativity, you can use much more complex formulations for the length element, but the fact that time comes in with a different sign from space is a fundamental property.

You can, if you wish, make the equations look more beautiful by taking time to be an imaginary number: $t' = it$. Then, if you square the time length element, you'll get a negative number. So you can write down the length element more simply:
$$ds^2 = dt'^2 + dx^2 + dy^2 + dz^2$$
The time element no longer looks weird. It's just another dimension. But when you do calculations, because $t'$ is imaginary, it comes in with a negative sign instead of a positive one.

I don't think this concept of "imaginary time" means anything special. It's just one way of representing the fact that when measuring space-time distances, the contribution from time is opposite the contribution from space. Perhaps this representation makes some equations in quantum gravity look simpler. I'm not sufficiently familiar to say. But I don't think it's a radical concept either way.

#### PAllen

No, @kimbyd , I think Hawking is referring to Wick rotation, not the imaginary time metric convention.

#### PeterDonis

Mentor
In General Relativity, you can use much more complex formulations for the length element, but the fact that time comes in with a different sign from space is a fundamental property.
Only in a very restricted sense: that at any event you can always find a coordinate chart in which the metric takes its Minkowski form. But that, in itself, is not enough to make the "ict" metric convention valid in a general curved spacetime, once you go beyond restricting attention to physics in a single small patch that can be considered flat. That is why, for example, MTW stresses that the "ict" approach is not valid in GR, and therefore they don't use it even in their presentation of SR.

I think Hawking is referring to Wick rotation, not the imaginary time metric convention.
To be fair, I think Hawking has referred to both and wasn't always clear which. But either way, the issue I raised above applies. Wick rotation has the same issues in a general curved spacetime that the "ict" metric convention does.

#### kimbyd

Gold Member
2018 Award
No, @kimbyd , I think Hawking is referring to Wick rotation, not the imaginary time metric convention.
True, but Wick rotation is a concept which derives from the "ict" metric composition, as described here:
https://en.wikipedia.org/wiki/Wick_rotation

My understanding was that for the subset of cases where "ict" works, so does Wick rotation, and each produces results which are identical to the real time solution.
Only in a very restricted sense: that at any event you can always find a coordinate chart in which the metric takes its Minkowski form. But that, in itself, is not enough to make the "ict" metric convention valid in a general curved spacetime, once you go beyond restricting attention to physics in a single small patch that can be considered flat. That is why, for example, MTW stresses that the "ict" approach is not valid in GR, and therefore they don't use it even in their presentation of SR.
One fundamental issue with the "ict" convention in General Relativity is that the coordinate labeled "t" is not always the one that has a negative sign. For instance, consider the Schwarzschild metric:
$$ds^2 = -\left(1 - {r_s \over r}\right) c^2 dt^2 + \left(1 - {r_s \over r}\right)^{-1}dr^2 + r^2 d\Omega^2$$
where $d\Omega^2 = d\theta^2 + sin^2\theta d\phi^2$.

For locations inside the event horizon, where $r < r_s$, the term in front of $dt^2$ becomes positive, while the term in front of the $dr^2$ becomes negative. If you were to simply replace $t$ with $ict$, you'd end up with two time-like coordinates, which is obviously invalid.

What I'm curious about is: is this the only situation in which the $ict$ convention breaks in GR?

#### PAllen

True, but Wick rotation is a concept which derives from the "ict" metric composition, as described here:
https://en.wikipedia.org/wiki/Wick_rotation

My understanding was that for the subset of cases where "ict" works, so does Wick rotation, and each produces results which are identical to the real time solution.
Ok, that makes sense.
One fundamental issue with the "ict" convention in General Relativity is that the coordinate labeled "t" is not always the one that has a negative sign. For instance, consider the Schwarzschild metric:
$$ds^2 = -\left(1 - {r_s \over r}\right) c^2 dt^2 + \left(1 - {r_s \over r}\right)^{-1}dr^2 + r^2 d\Omega^2$$
where $d\Omega^2 = d\theta^2 + sin^2\theta d\phi^2$.

For locations inside the event horizon, where $r < r_s$, the term in front of $dt^2$ becomes positive, while the term in front of the $dr^2$ becomes negative. If you were to simply replace $t$ with $ict$, you'd end up with two time-like coordinates, which is obviously invalid.

What I'm curious about is: is this the only situation in which the $ict$ convention breaks in GR?
It doesn't break down in the Schwarzschild interior at all, at least not for this reason. In the interior metric, you should replace r with t, and t with z, since the former is the timelike coordinate and the latter is an axial coordinate along S2XR1 spacelike hypersurfaces foliating the interior. Then, the 'correct' t can be locally identified with ict producing a 'formally' positive definite metric.

More generally, in any pseudo-Riemannian manifold, you can always make the metric 'nearly Minkowski' near a given event by suitable coordinate choice (e.g. Riemann Normal or Fermi-Normal coordinates).

The relevant question is then whether Wick rotation makes sense if it is only locally possible rather than globally possible. I confess I don't know the answer to this. My understanding was that for FLRW cosmologies it could be done globally, not just locally.

#### PeterDonis

Mentor
One fundamental issue with the "ict" convention in General Relativity is that the coordinate labeled "t" is not always the one that has a negative sign.
It's worse than that. You don't even have to have a timelike coordinate at all. You can have null coordinates. You can even have a chart where all four coordinates are spacelike (e.g., Painleve coordinates in the interior of a black hole). Or you can have multiple timelike coordinates (see below).

If you were to simply replace $t$ with $ict$, you'd end up with two time-like coordinates, which is obviously invalid.
No, it's not. You can have two timelike vectors that are linearly independent. They won't be orthogonal, but nothing requires coordinate basis vectors to be orthogonal. They just have to be linearly independent.

is this the only situation in which the $ict$ convention breaks in GR?
The $ict$ convention breaks in lots of ways in GR (see above for some examples). That's why textbooks like MTW don't use it.

#### kimbyd

Gold Member
2018 Award
It's worse than that. You don't even have to have a timelike coordinate at all. You can have null coordinates. You can even have a chart where all four coordinates are spacelike (e.g., Painleve coordinates in the interior of a black hole). Or you can have multiple timelike coordinates (see below).
Interesting. The Painlevé coordinates also highlight another problem: the possibility of off-diagonal metric elements. The metric for a black hole in these coordinates includes the term $2\sqrt{2M/r} dt_r dr$. If we replace $t_r$ with $it_r$ (using units where $c=1$), then we'd have a metric which is no longer real-valued. Maybe there's a way to make sense of a metric which is a complex number, but it certainly seems like a serious problem.

#### PAllen

I don’t think failure in some coordinates is a valid argument against ict convention. For example, the following metric is flat Minkowski space in a particular set of all light like coordinates - but this doesn’t preclude using ict in Minkowski space, just not with these coordinates:

ds2 = du dv + du dw + du db + dv dw + dv db + dw db

where u,v,w,b are the lightlike coordinates.

Conversely, as I already mentioned, any pseudo-riemannian manifold can by coordinatized in patches that are small perturbations from the standard Minkowski metric.

The real question is, under what circumstances can Wick rotation methods be used in GR? Clearly, Hawking does this for cosmology. I don't have any knowledge of this question (the criteria for validity of Wick rotation methods in GR).

#### PeterDonis

Mentor
this doesn’t preclude using ict in Minkowski space,
In Minkowski space, yes. But we are talking about using the ict convention in a general curved spacetime. That's what is not, in general, possible.

The real question is, under what circumstances can Wick rotation methods be used in GR? Clearly, Hawking does this for cosmology.
My understanding is that Hawking's use of it only works for Hawking's particular "no boundary" cosmological model, in which the geometry of spacetime at the "beginning" is an actual 4-dimensional sphere. I am not aware of it working for a standard FRW geometry, or any other general curved spacetime. But I am not very familiar with the literature in this area.

#### kimbyd

Gold Member
2018 Award
In Minkowski space, yes. But we are talking about using the ict convention in a general curved spacetime. That's what is not, in general, possible.
I'm pretty sure you can use it on a local coordinate patch in any space-time, for the trivial reason that every location in space-time is locally-Minkowski. I think the problem stems from the fact that it only works for a subset of possible coordinate choices for that region of space-time.

As anything that's coordinate-specific is problematic in General Relativity, and this seems to add very weird restrictions to that specificity, I can see how it would be just too complicated to be useful in GR.

Edit: Or, to put it another way, making use of General Relativity from the point of view of a particular coordinate system has always been a source of potential errors in GR calculations. As the ict convention not only requires doing calculations in a specific coordinate system, but also adds entirely new ways in which that coordinate system can break things, it makes sense that it would be unpopular.

#### PeterDonis

Mentor
I'm pretty sure you can use it on a local coordinate patch in any space-time
Sure, but everything breaks down when you try to extend beyond that single local coordinate patch. Or when you try to match things up between two different local coordinate patches.

"Is the Hawking-Hartle Euclidean approach confirmed by CMBR data?"

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