- #1
mollwollfumble
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Not sure whether this should be posted under GR, QM, Beyond the standard model, or Topology and analysis. I understand GR best so am posting here. I'm having trouble visualising the "instanton". My limited understanding is that the instanton appears both in GR and QM, and has a role in the recent ER = EPR paper by Maldacena and Susskind. Also, from A. Zee's book I understand that an instanton is a topological defect similar to a vortex and monopole.
I find the standard visualisation of an instanton, as a cuplike depression between a singularity and the outer universe, completely bewildering because it directly contradicts the classical picture of the Schwarzschild black hole in unmodified space-time coordinates, where an infalling particle heads towards infinite positive time at the event horizon and then reappears at an infinite negative time just inside the event horizon to move forward in time to hit the singularity a small positive time after approaching the BH. There is nothing even remotely like a cup-shape depression in the region between the outer universe and the singularity.
I also find the instanton illustrated as ring around the tightest part of a wormhole unconvincing, because the wormhole throat shrinks to zero when anything enters, so all that illustration shows is a point.
As a person with a PhD in classical fluid mechanics, I came up with the following possible visualisation of an instanton. The topological defect called the monopole/hedgehog/source/sink can be illustrated by "particle paths" = "flow lines" = "streamlines" in potential flow radiating from a central point singularity. The topological defect called a "vortex" can be illustrated by "particle paths" = "flow lines" = "streamlines" in potential flow circling a central point singularity.
Perhaps, if I understand it correctly, an instanton can be seen as a "source and sink" = "black hole and white hole" = "equal positive and negative charges" in extremely close proximity. In potential flow we call that the "doublet". One feature that the doublet has in common with the instanton is that the integral of particle speed at infinity for the doublet and instanton is zero for both. This integral is not zero for either the sink or the vortex.
Typical illustrations are: http://nptel.ac.in/courses/112104118/lecture-21/images/fig21.5.gif , http://nptel.ac.in/courses/101103004/module3/lec7/images/3.png , http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec1011/img79.gif.
Is my idea reasonable? or daft?
I find the standard visualisation of an instanton, as a cuplike depression between a singularity and the outer universe, completely bewildering because it directly contradicts the classical picture of the Schwarzschild black hole in unmodified space-time coordinates, where an infalling particle heads towards infinite positive time at the event horizon and then reappears at an infinite negative time just inside the event horizon to move forward in time to hit the singularity a small positive time after approaching the BH. There is nothing even remotely like a cup-shape depression in the region between the outer universe and the singularity.
I also find the instanton illustrated as ring around the tightest part of a wormhole unconvincing, because the wormhole throat shrinks to zero when anything enters, so all that illustration shows is a point.
As a person with a PhD in classical fluid mechanics, I came up with the following possible visualisation of an instanton. The topological defect called the monopole/hedgehog/source/sink can be illustrated by "particle paths" = "flow lines" = "streamlines" in potential flow radiating from a central point singularity. The topological defect called a "vortex" can be illustrated by "particle paths" = "flow lines" = "streamlines" in potential flow circling a central point singularity.
Perhaps, if I understand it correctly, an instanton can be seen as a "source and sink" = "black hole and white hole" = "equal positive and negative charges" in extremely close proximity. In potential flow we call that the "doublet". One feature that the doublet has in common with the instanton is that the integral of particle speed at infinity for the doublet and instanton is zero for both. This integral is not zero for either the sink or the vortex.
Typical illustrations are: http://nptel.ac.in/courses/112104118/lecture-21/images/fig21.5.gif , http://nptel.ac.in/courses/101103004/module3/lec7/images/3.png , http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec1011/img79.gif.
Is my idea reasonable? or daft?