The Causal Diagram of the Black Hole

A week ago, I explained the idea of causal diagrams, or Penrose-Carter diagrams, and we discussed the diagram for the static black hole metric shown below.

As I pointed out, though a solution of Einstein's field equations, this diagram does not actually describe a situation we find in reality. The black hole shown in this diagram is accompanied by a white hole, and both have existed since forever, and will continue to exist, unchanging, until eternity. Today, I thus want to discuss the metric for a realistic black hole, a black hole formed from collapse of matter. I will also briefly touch on the evaporation but, as you know if you've been around for a while, the exact way the evaporation proceeds, in particular the final stage, is still under debate.

To obtain the causal diagram of the black hole, recall that Einstein's field equations are local and the black hole solution is a vacuum solution. Yes, that is right. This means that in General Relativity empty space is not necessarily flat. (Flat meaning the curvature tensor vanishes identically. Empty space however has a property called "Ricci-flatness.") If we want to describe collapsing matter, we thus know that outside of that matter the previously found solution, depicted above, still holds. So, what we do is drawing into the diagram the surface of the collapsing matter, and keep the part that is outside that matter. This is shown below.



Now the blue shaded part is the one that no longer correctly describes the black hole that forms from collapse and has to be discarded. This means in particular that the white hole as well as the second asymptotically flat regions are both gone and do not exist in real world situations (addressing a concern that Andrew brought up in the previous post).

What we do then is to attach an interior solution that does not describe vacuum. In some simplified cases this can be done explicitly. For example if the collapsing density is homogeneous (which would be a piece of a FRW-metric), or if it is null dust (described by the Vaidya-metric). Then, one can calculate the interior solution and use a matching condition to join both parts together. For our purposes however, we don't have to bother with the details since we just want to capture the causal structure. For what the causal structure is concerned, the inside solution is rather dull. There is nothing specific going on. The radius just shrinks until it falls below the Schwarzschild radius associated to its total mass. Then the horizon forms, and the matter collapses to a singular point. This is shown in the diagram below.



Note that there is no particular meaning to curves that are exactly horizontal or vertical, we are thus free to deform them, which has been done to make the r=0 curve vertical. This is fine as long as we make sure that the null curves on 45° angles remain the same, and thus spacelike remains spacelike and timelike remains timelike.

As pointed out in the previous post, the use of radial coordinates means that ingoing curves look as if they are reflected at r=0 when they actually go through. The lightray marked v₀ in the above figure is the last light ray that just manages to escape the forming horizon. It is in this background, not the static background, that Hawking did his calculation which showed that black holes do emit radiation.

Knowing the black hole, once formed, emits radiation of course brings up the next question: how do we incorporate the evaporation into the diagram? One can add the evaporation of the black hole by using another non-vacuum patch that describes outgoing radiation which leads to a decreasing of the mass. The Schwarzschild-radius of the black hole then gets closer to the singularity until both, the horizon and the singularity, vanish in the endpoint of evaporation. In this process, the event horizon remains lightlike. What changes for the observer at scri minus is the mass associated to the black hole. When the black hole is completely evaporated, we are left with a spacetime filled with very dilute radiation. This spacetime is to good precision flat and described by another piece of Minkowski metric. If you patch the pieces together you get the diagram below.

If you followed me so far, then we are now in an excellent shape to discuss the black hole information loss problem, which can basically be read off the causal diagram, and the possible solutions Lee and I classified in our recent paper. Let me know in the comments if you're interested in another post on that.

With kisses from Germany

I'm presently in Germany, about to give yet another talk. Yesterday, I had the great idea to get a flu shot with the result that today I feel pretty shitty, like, as if I'm getting the flu. Best conditions to give a seminar. On the upside, I finally managed to get my BlackBerry desktop manager to download the photos I've been taking, so here's some quick blogging.

Yesterday, Stefan and I walked by booths selling Christmas candy and gloves and toys and that stuff. The below photo shows the sign on one of them. It reads "Negerküsse," literally "Negro's kisses." It's a sort of candy, a soft fluffy cream covered by chocolate. They are meanwhile mostly called "chocolate kisses," but occasionally, as you see here, the older word creeps up.


And here's another photo from my BlackBerry. That coffee looked better than it was:

And another image that I found worth taking. This is nearby the parking lot behind the AlbaNova University building

And finally a random shot somewhere in Stockholm during the summer:

Swedish Research Council requires Open Access

The Swedish Research Council decided some weeks ago that research supported by a grant from their agency has to be made publicly available within 6 months. From the press release:

"To obtain a research grant, the Swedish Research Council now requires researchers to publish their material so as to make it available to all. The public and other researchers should have free access to all material financed by public means.

The thought behind so-called Open Access is that everyone should have free and unrestricted access to scientifically assessed articles. The Research Council has now determined that researchers granted funds by the Authority should publish their scientifically assessed texts in journals and from conferences in this manner.... Researchers will have to guarantee that publications are available according to Open Access within a maximum period of six months."

This requirement is even stronger than that of the US National Institute of Health (NIH), which demands the public has access to the published results of NIH funded research no later than 12 months after publication.

Seems like the Open Access movement has a reason to celebrate :-)

Causal Diagrams

I once witnessed a physicist explain the universe to an artist. The artist had approached the physicist to learn how to understand extra dimensions, a concept, so he explained, that would undoubtedly enhance the depth of his artwork, and be of great inspirational value for his quest to capture the contextuality of essence. Or maybe essence of contextuality. Or something like that. Either way, the physicist took a piece of chalk and drew a line on the blackboard. "That is our universe," he said. It took several minutes before the artist stopped laughing and said "Now THAT is what I'd call an abstraction."

You see, the fact that our universe is at least 4-dimensional and infinitely large (or damned close to that) creates some problem with visualization. The average blackboard is 2-dimensional, somewhat smaller than infinite, and my female brain already finds 3d plots messy and confusing. Add to this that most physicists aren't particularly great in drawing the universe.

Thus arises the need to picture 4 dimensions in an intuitive and illuminating way. Penrose-Carter diagrams, also called "causal diagrams," do exactly that. Though they do not work for the most general space-times, but only when additional symmetries simplify the scenario, they capture the essence of a 4-dimensional space-time. Or maybe the essence of 4-dimensional contextuality.

Understanding causal diagrams is one of the most basic skills you need if you want to work in General Relativity.

It works like this.

First, we have the problem of getting 4 dimensions down to 2, where one of the 4 dimensions is time. That's not so complicated. We will assume that space is spherically symmetric, such that when you sit in one point, all directions from that point look similar. This would be the case for example if you sat in the middle of a ball or, to reasonably good precision, if you sat in the middle of the Earth. The only interesting information is then in the change of scenery as a function of the distance from you, who you are sitting in the center of symmetry. We can thus capture the full 3 space dimensions by just considering what happens with the distance to the center of symmetry. This distance is of course just the radial coordinate r. Besides that, we will draw the time-coordinate t, which is usually depicted vertically, whereas r is horizontally. This is shown in the picture below, left. You've seen that before.

An infinitely flat 4-dimensional space-time is then just a half-plane. Note that a flat space is spherically symmetric around every point. (If you want to nitpick, what I mean with "flat" is that the curvature tensor identically vanishes.)

Next thing we do is to notice that if we had a particle moving towards the center of symmetry at r=0, passing through it, and moving away from it again, it would look on the half-plane like a reflection instead. Sometimes we thus mirror the half-plane to the other side, such that the curves of particles just go through. Keep in mind though that r increases in both directions. The world-lines of particles with a fixed velocity move on straight lines in that plane. Don't try to draw curves for particles that do not approach the center radially because the symmetry doesn't allow it. We now adopt the first convention for causal diagrams:

Light moves on 45° angles.

Curves on which light moves are called "lightlike," or, due to their property of having zero length in a Minkowski-metric, "null curves."

The next step is more tricky, because now we have to deal with the infinitely large space. How do we get it to fit on a blackboard? If you have ever done perspective drawing, you know the answer already. The "horizon line" and the "vanishing points" depict the infinite distance on a finite sheet of paper. The price to pay is that what is equally spaced far away, moves closer and closer together on the 2-dimensional picture. An example is the photo with railroad tracks to the right.

To draw a picture of an infinite space-time, we do exactly the same: we make infinity finite by squeezing together what is far away. Since the space-time is infinite in more than one direction an additional assumption is that we

Squeeze infinity equally in all directions.

The resulting squeeze is also called a "conformal transformation," and has the merit of preserving angles, such that most importantly null curves still move on 45°, no matter which such transformation you used. There are many different squeezes, though qualitatively they look all similar. An example for an often used squeeze is the tangent function in the interval [-π/2,π/2], shown to the left. If you take equal spaces on the vertical axis, the corresponding values on the horizontal axis produce a no longer evenly spaced representation of that infinite vertical axis.

If we now go and squeeze our flat space-time what we get is a diamond.


In this diagram, spacelike curves always have angles less than 45°, and timelike curves on which particles can move have angles more than 45° (in every point). All spacelike curves come from and end in the side corners, called "spacelike infinity," whereas timelike curves all come from the bottom corner and end in the upper corner, called "past timelime infinity" and "future timelike infinity," rspt. Light comes from the lower V-shaped boundary and end at the upper Λ-shaped boundary, called "past null infinity" and "future null infinity." The null infinities are usually denoted with an I in a script font, and are thus for short often called "scri minus" for past null infinity and "scri plus" for future null infinity.

So far so good, but flat Minkowski space is admittedly somewhat boring. Let us thus look at something more interesting. The causal diagram of the maximally analytically extended Schwarzschild-solution, describing a static black hole. You have seen it thousands of times in the header of this website.


It is futile trying to explain how to obtain the diagram without telling you what a metric is and what to do with it, but the big advantage of these diagrams is exactly that you can learn something about the space-time properties without bothering with tensor equations, so let's see.

The first thing you will notice is that the diagram contains regions (A and B) that cannot be connected by any lightlike or timelike curve. This means there is no way to send information from one to the other, and A and B are thus causally disconnected. You will also see that there are two spacelike boundaries on the bottom and top where time- and lightlike curves end without having reached infinity. The spacetime is thus geodesically incomplete or, equivalently, has singularities. The maybe most important property to identify is the boundary of the region from which lightlike curves can reach future null infinity. If not at an infinite distance, this boundary it is called a future event horizon. Similarly, the boundary of the region to which light can be send from past null infinity is a past event horizon. These horizons are always lightlike surfaces.

When you're done thinking, take time to see how pretty it is.

This Schwarzschild-metric does not only depict a black hole in the upper part, which contains a region where no information can ever come out to future infinity, but also a region in the lower part where no information can ever get in from past infinity. That second region is called a white hole. It is however a mathematical artifact since this diagram describes an unrealistic situation: a black hole that has been there since forever and will be there until eternity. In reality, black holes are formed from collapsing matter and later evaporate. We will discuss the more realistic diagram in another post, so stay tuned.

Finally, upon Googling for images I found that somebody else had used the same motivation from perspective drawing that I came up with. Well. If one thousand monkeys hit they keyboard for long enough, they will eventually type the complete Misner, Thorne, Wheeler. Not only once, but an infinite amount of time.

If you arrived here by just scrolling down, shame on you. The minimum amount of information you should take home is that Penrose-Carter diagrams, aka "causal diagrams," are used to depict the causal properties of 4-dimensional space-times with additional symmetries.

Thoughts and Experiments

The other day I got in an argument over what constitutes a "scientific question," after I invoked thought experiments in an attempt to illuminate some features of a model. Is a question that cannot be tested even in principle a scientific question? And thus, if it isn't, should scientists think about it at all?

Forced to take a point of view, I want to offer the possibly only intelligent words the former German chancellor Helmut Kohl ever uttered "Entscheidend ist, was hinten rauskommt." - "What matters is what comes out in the end."

I don't care very much about somebody's personal philosophy or what -ism they classify themselves under, I care whether it's an approach promising to lead to progress. Discarding a question as unscientific when it concerns extreme, and potentially untestable, limits extends the requirement on a theory to be scientific to the method how to get to a good theory. And with that cuts off frequently used methods, not only thought experiments, but also demanding mathematical consistency in regimes we have not tested. I simply think limiting allowable questions to those practically testable constrains our imagination and thus our opportunity to find answers.

One can certainly take such though experiments and the questions they raise for more important than they are. Steve Giddings for example claims that our failure to solve the black hole information loss paradox is akin to the failure of classical electrodynamics to explain the stability of atoms (see this talk or this paper). Well, the only difference is that Bohr's very existence was evidence for the stability of atoms, whereas we have yet to see a single black hole evaporate. Nevertheless, the information loss problem indicates clearly a lack in our understanding of Nature, and, wanting to understand, physicists have turned it inside-out and upside-down for decades. However, whether that eventually leads to something besides loads of papers yet has to be decided.

A more fruitful though experiment was of course Einstein's chase after a photon. And then one should not neglect that thought experiments are not only inexpensive but also fun.

HIT, the Heidelberg Ion-Beam Therapy Center

Last Monday morning, more cars than usually were crowding the road through the Heidelberg "Neuenheimer Feld" Campus. A parking lot had been reserved for guests, and signs all over the Campus showed the way to a "Press Conference". Later I learned that all this gathering was for the opening ceremony of the Heidelberg Ion-Beam Therapy Center (HIT).

With a background in heavy-ion physics from Frankfurt University, it would have been hard for me not to have heard of this project before: Kind of a large spin-off of the GSI facility near Darmstadt, the Heidelberg Ion-Beam Therapy Center is a dedicated heavy-ion accelerator for deployment in radiotherapy to treat tumours. It is the first medical heavy-ion machine in Europe.

The HIT building in Heidelberg. Source: Photo Gallery of the HIT.
The building, half-buried in the ground to minimise radiation exposure for the environment, houses an ion source, a linear accelerator (LINAC) and a synchrotron with a circumference of 65 metres, which can accelerate protons (hydrogen nuclei), alpha particles (helium nuclei), or nuclei of carbon and oxygen to final energies of 50 to 430 MeV/nucleon. For carbon nuclei with 12 nucleons, this means a maximal energy of 5.16 GeV. This energy corresponds to a bit less than half the rest mass of the carbon nucleus, meaning a gamma factor of 1.45, or motion of the nucleus at 73 percent of the speed of light.

HIT accelerator layout. Source: T. Winkelmann et al., Proceedings of ECRIS08 (PDF).
The HIT facility. Source: Photo Gallery of the HIT.
The heavy-ion beam, which is focussed to a diameter of about a millimetre, is steered by the high energy beam transport (HEBT) system to one of two treatment places, or into a big, pivotable installation of bending magnets, the so-called gantry. The gantry allows the beam to be directed from any direction of a vertical plane into one point.

The HIT gantry. Source: Photo Gallery of the HIT.
At the treatment places, the beam, which is focussed to a diameter of about a millimetre, enters the body of patients, and deposits its energy in tumour cells, thereby corrupting the DNA of the tumour cells and stopping their runaway replication.

Treatment place at the focus of the HIT Gantry. Source: Photo Gallery of the HIT.
What is so special about heavy ions for the treatment of cancer that justifies the construction of a large, highly specialised 120 million Euro facility?

It's an effect discovered in 1904 by the Australian physicists William H. Bragg and Richard D. Kleeman, who studied energy energy deposition of alpha particles from radioactive decays when penetrating matter. To their surprise, and different from gamma rays or X rays, alpha particles deposit their energy predominantly around the end point of their track.

The Bragg peak for carbon ions. From Fokas et al.: "Ion beam radiobiology and cancer: Time to update ourselves", Biochimica et Biophysica Acta 1796 (2009) 216–229.)
On a plot showing energy deposition along the path, this pattern shows up as a curve strongly peaked at the end point, in the so-called Bragg peak. (The Bragg peak should not be confused with the Bragg reflections in X ray scattering, which were discovered by William H. Bragg and his son, William Lawrence, in 1913, winning them the 1915 Nobel Prize in physics.)

This strongly localised energy deposition, along with the sharp focus of the beam, makes heavy ions such as carbon an ideal tool to attack tumour cells while doing as less harm as possible to the surrounding tissue. The penetration depth can be controlled by the ion beam energy. Thus, the ion beam hits its target precisely and transfers an exact dosage of energy to the tumour.

HIT will be used to treat tumours which are deeply situated in the body and can hardly be reached by conventional radiation treatment. Tests at GSI so far have been very promising, and there are good chances that a large part of the about 1300 patients per year who will be treated at HIT eventually can be cured.

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• Web page of the Heidelberg Ion-Beam Therapy Center (HIT)
• Photo Gallery of the HIT
• HIT Brochure as PDF file
• For technical details about the accelerator facility, check out T. Winkelmann et al, "Experience at the Ion Therapy Center (HIT) with two years of continuous ECR ion source operation", Proceedings of ECRIS08, Chicago, IL USA (PDF file).
• About the costs of HIT, check out e.g O. Jäkel et al.: "On the cost-effectiveness of Carbon ion radiation therapy for skull base chordoma.", Radiotherapy and Oncology 83(2) (2007) 133-8.

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Urban Physics Myths

Stefan and I, we had a good laugh at the LHC baguette. I've been wondering whether the PR department made it up to entertain us while waiting. For the next filler, how about the technician who hung his coat on the regulator for the server-room AC and caused a total system breakdown?

The baguette also brought to mind incidents of beer bottles that occasionally appear in beam pipes, and made me scratch my head about other frequently told physics stories. You know, stories of the sort that typically happened to a friend of a friend.

There is for example the story about the postdoc who got scurvy by living from Snickers and Coke out of vending machines for several months. In some cases, it's pizza and Coke instead. In one version said postdoc was located at Fermilab, in another version at Brookhaven. Nobody ever met that postdoc.

Another story that I've heard in several versions circulates around the organizer of an Italian summer school who we don't want to name for his alleged mafia connections. You see, as the story goes, one of the speakers had his bag stolen at the airport. Mentioning this to the organizer, the bag promptly reappeared the next day. In other versions it's been several pieces of baggage, a purse, or a car. The summer school and the organizer remained the same.

Then there's the story of the student who, in a case of utter frustration, adds a sentence to his thesis offering whoever reads this a beer. Needless to say, the thesis gets accepted and printed with the beer-offer. The conditional statement "if and only if Mike's dog really ate his frog," that Eric Weinstein mentioned is a variation on that theme of the not-even-reading advisor.

Add your story in the comments!

Experimental Search for Quantum Gravity 2010

I'm organizing a workshop!

Experimental Search for Quantum Gravity 2010

will take place July 12-16 at Nordita, in the top intelligent city of the world, beautiful Stockholm, Sweden. This is the 2010 installation of our 2007 PI workshop of which you can find my summary here.

We meanwhile have the website up, and a preliminary list of participants.

The purpose of the workshop is to bring together people who study various possibilities to experimentally test quantum gravity in order to assess these possibilities and encourage discussions. Some topics are:

• Predictions for existent and planned earth based experiments (accelerators, high-precision measurements), possibly in scenarios with a lowered Planck scale.
• Cosmological measurements: signatures from the universe's early quantum phase (in the cosmic microwave/neutrino/graviton background, large and small scale structure)
• Astrophysical measurements: cosmic rays, gamma-ray bursts, supernovae.
• Quantum effects caused by space-time fuzziness (decoherence), status and proposals for experiments.
• Miscellaneous and other (emergent gravity, non-locality, etc.).

Yesterday, I submitted an application for conference support to the Swedish Research Council. (Many thanks to Thomas for helping with a Swedish title!) If this application goes through, most of the grant is meant to support students or postdocs in the early career stages, who have little or no travel grants available. I would really like to give these young researchers the opportunity to participate and learn something about what is still a very young field. So, if you feel addressed, mark the week July 12-16, and send me a note to be put on the waiting list (hossi at nordita dot org).

Since the topic touches on many different fields the workshop is bound to be interesting, and I'm very much looking forward to it.

The Waterloo Institute for Complexity & Innovation

Waterloo, Ontario, once named the Top Intelligent Community of the world, home of the Perimeter Institute for Theoretical Physics and the Center Centre for International Governance Innovation, makes another shot at solving all problems of mankind: The Waterloo Institute for Complexity & Innovation (WICI).

Presently in its planning phase, the WICI's is an interdisciplinary effort, located at the University of Waterloo. According to the website, its goals are to

• Develop a common, transdisciplinary language and methodology and an integrated, coherent theory for the study and pedagogy of complex adaptive systems; and,


• Apply these tools to stimulate rapid and beneficial innovation that will increase the resilience of complex adaptive systems worldwide – including social, political, economic, and ecological systems – that are currently under threat.

So far, the activities consist of a seminar series, which, innovation etc, has the talks recorded and uploaded to Google videos. You find there, among others, Stuart Kauffman speaking on "The Evolution of Economic Wealth and Innovation," Thomas Homer-Dixon on "Ingenuity Theory: Adaptation Failure and Societal Crisis," and if you have been waiting for a summary of Lee Smolin's last year's q-fin paper, here's his talk on "Symmetries in Economic Models and their Consequences".

Unfortunately, they have forgotten about another complex system currently under threat, the academic system. Which plays a central role for my virtual institute, and that for a good reason.

The top intelligent community of the year 2009 btw is Stockholm.

Einstein's Summer House in Caputh

On our way back from Potsdam last week, Stefan insisted we absolutely should stop to see Einstein's summer house in Caputh. Caputh is a small place South-West of Potsdam, located idyllically at the slope of a hill on the lakefront of Schwielowsee ("See" = "Lake"). Thanks to unexpected one-way traffic, an abundance of construction sites, and a malfunctioning railway crossing gate, it took just an hour drive for the five kilometers from Telegraphenberg to Caputh.


Einstein's summer house in Caputh.
In the summer of 1929, Einstein invested his savings in the construction of a small house to spend the summer there, away from busy Berlin. The site, at the border of a forest, offers a great view over the lake and the Brandenburg landscape. Unfortunately, last week this view was hidden in foggy haze. The house is built out of wood, which is uncommon in Germany, and looks very elegant and modern.

The summer house in Caputh is the only of the places where Einstein has lived in Germany that is left. It has been restored a few years ago, and is used today as a location for a variety of lectures and cultural events.

Einstein and his wife spent four summers in Caputh, before they left Germany in December 1932 never to come back. Einstein later said that he never felt more comfortable and at ease than when in Caputh. TIME magazine told its readers in its "People" column of August 31, 1931, that "last week he was vacationing at Caputh near Potsdam, wearing white linen pajamas, no socks, no shoes."


Unfortunately, last week was a bit to chilly to dispense with socks and shoes.




(If you think the photo looks like I have to use a bathroom really urgently, that's exactly correct.)

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See also:

Website of the Einstein Forum / Einsteinhaus

Website of the Initiativkreis Albert-Einstein-Haus Caputh e.V.

Are you what you are or what?

“I'm not aware of too many things
I know what I know, if you know what I mean”

~ Edie Brickell

Some days ago, I received a complimentary issue of PhysicsToday which contained a letter I wrote in reply to an article by David Mermin “What’s Bad About This Habit” (PhysicsToday, May 2009, page 8). Since the article is subscription only, let me briefly summarize what Mermin wrote.

Mermin comments on the “bad habit of physicists to take their most successful abstractions to be real properties of our world.” He starts with commenting on the reality of the quantum state:

[T]he recognition that quantum states are calculational devices and not real properties of a system forces one to formulate the sources of that discomfort in more nuanced, less sensational terms. Taking that view of quantum states can diminish the motivation for theoretical or experimental searches for a “mechanism” underlying “spooky actions at a distance” or the “collapse of the wavefunction”—searches that make life harder than it needs to be.

He then goes on to distinguish between the real and the abstract on the example of quantum field theory

I hope you will agree that you are not a continuous field of operators on an infinite-dimensional Hilbert space. Nor, for that matter, is the page you are reading or the chair you are sitting in. Quantum fields are useful mathematical tools. They enable us to calculate things.

and the spacetime continuum

[S]pacetime is an abstract four-dimensional mathematical continuum of points that approximately represent phenomena whose spatial and temporal extension we find it useful or necessary to ignore. The device of spacetime has been so powerful that we often reify that abstract bookkeeping structure, saying that we inhabit a world that is such a four- (or, for some of us, ten-) dimensional continuum.

He warns of the consequences of mistaking abstractions for reality

So when I hear that spacetime becomes a foam at the Planck scale, I don’t reach for my gun. (I haven’t any.) But I do wonder what that foam has to do with the macroscopic events that spacetime was constructed to represent and the macroscopic means we use to locate events.

(Referring to Stephen Hawking's remark “When I hear of Schrödinger's cat I reach for my gun.”) Mermin concludes with

Quantum mechanics has brought home to us the necessity of separating that irreducibly real experience from the remarkable, beautiful, and highly abstract superstructure we have found to tie it all together.

I completely agree with Mermin. One shouldn't mistake mathematical tools for reality, and mixing up both leads to confusions. Our task as physicists is to explain observations and make predictions for experiments, not to unravel the fundamental nature of reality (wink, wink). However, one should not throw out the baby with the bath water. We have a clear goal, but no map telling us how to get there. And while some might find the philosophy of science a waste of time, and others might say taking abstractions too seriously only creates artificial problems, these considerations could contain the clue, or the inspiration, necessary for progress.

Thus, while I personally am not too enchanted by taking maths to be reality, I think one should not simply dismiss these studies on the basis of gut-feeling. I was therefore put off, not by the actual opinion Mermin expressed, but by it being uninsightful, and - in its polemic way - potentially counterproductive by encouraging shallow argumentations.

So I wrote the following letter:

“I hope you will agree,” David Mermin writes, “that you are not a continuous field of operators on an infinite-dimensional Hilbert space. Nor, for that matter, is the page you are reading or the chair you are sitting in.” His comment is a nice example of the logical fallacy known as “appeal to belief”: Most people believe X is true, so X is true. That many people believe they are not operators in Hilbert spaces, believe they do have free will, or do or don’t believe in global warming makes no difference as to whether a statement is true or false. I have no basis on which to decide what I “really” am. And though I personally think any such argument is a waste of time because it can never be decided anyway, and though I am sympathetic to the opinion Mermin expresses, his article dismisses the relevance of both quantum foundations and the philosophy of science out of hand in a rather polemic and not very insightful way.

Sabine Hossenfelder
Waterloo, Ontario, Canada

To which Mermin replies

Sabine Hossenfelder takes my rhetorical flourish as an attempt to argue, fallaciously, for the truth of that proposition. That was not my intent any more than I intended, by calling attention to the agreement among most of us that the ether is not real, to establish thereby its unreality. Although Hossenfelder takes my column as a shallow, polemical dismissal of both philosophy of science and quantum foundations, I had viewed it as an amateurish attempt to contribute to both disciplines.

It leaves me to wonder though why Physics Today prints such amateurish attempts. It's like opening a journal on medicine and reading a column proclaiming talking is a bad habit since, I hope you will agree, the human body is not made of words. And then find it explained as being an amateurish contribution to psychology.

In any case, I will now go act on some wave-functions.

“Philosophy is the talk on a cereal box
Religion is the smile on a dog
I'm not aware of too many things
I know what I know, if you know what I mean

Choke me in the shallow waters
Before I get too deep

What I am is what I am
Are you what you are or what?

~Edie Brickel, What I am