# String Topology and Time

1. Sep 17, 2011

### Lost in Space

If a point particle is a string viewed end on in three dimensions, or is a zero brane, isn't this merely a perspective? Are open strings really more like ribbons given the dimension of time and are closed strings more like tubes seen end on? If so, could this mean that the string's length is actually determined by temporal properties in four dimensions as a point particle is a line if the dimension of time is included? I've read that strings can't be any smaller than the Planck length but that some may have expanded during the inflationary period to become 'cosmic strings'. But surely these same factors would apply whatever the size of the string because of the temporal aspect?

Does particle decay really mean that strings lose energy over time because of quantum effects? The vibrational energy of a string must surely be confined within the string in order to maintain its energy state so isn't any possible energy loss due to something similar to Hawking radiation? If a string made of energy 'vibrates' with energy doesn't the vibrational energy of a string consist of a different type of energy from the string's constituence? Are these oscillations 'bouncing' from one end of the string to the other, and are these confined merely to the 'end' of the string that we perceive from our dimensional perspective or could they go beyond, i.e. from past to future and vice versa if time is included as length?

2. Sep 17, 2011

### mitchell porter

Inflation is supposed to happen because there's a field, the "inflaton" field, with a very high energy density. In string theory, there are at least three ways to get a field: from the strings themselves, from branes, and from the geometry of space. At some high level of understanding, these should all be somewhat interchangeable, but I'm not at that level. So let me just talk in terms of strings, branes, and space as three separate things. You have ten-dimensional space-time, you have branes taking up space in it, and then you have strings either attached to the branes or moving freely through the space between the branes.

What is a field? It is a distinct type of physical quantity which has a value at every point in space (though the value can be zero). In classical field theory, you have differential equations which describe how waves spread through a field - or how waves spread through several interacting fields in a correlated way. In quantum field theory, these waves become probabilistic and also have energy levels, and so you get "particles". The inflaton field of cosmology is one of these quantum fields, with a dual particle/field nature.

So now let's examine how to get a field from strings, from branes, and from space itself. The string is the easiest example, because it's so much like a particle: a minute self-contained entity that flies through space, vibrating with a fixed energy and spinning with a fixed angular momentum. If you ignore the internal details of the string and just focus on the energy and angular momentum that it carries, you get a particle description. If you try to describe the interactions of strings but only at this particle level, you'll get quantum fields. So that's one way to get fields out of a string theory.

Next, branes. A brane is a blob in space, which can come in various numbers of dimensions, and which can emit, absorb, or hold on to a string. (The strings which are attached to branes can be compared to lines of flux attached to a charge.) A brane can vibrate in space, just like a string. So for each point on the brane, you can ask how far it is displaced from an average position - again, just like the waves on a string. These vibrations of a brane are another way to get a field: in braneworld models, there are branes which stretch out along all the large directions of space, and so there is an amount of brane-vibration at each point in space, and I already mentioned that a field has a value at each point in space. The brane is something that the strings can interact with, and in fact the vibrations of an individual brane are a type of string, a string attached at both ends to that same brane.

Finally, space itself. In string theory you can have the famous extra dimensions that are closed, and form a geometry of their own, like a Calabi-Yau space. These extra dimensions can have sizes - how far around it is, in each new direction of space, until you return to where you started. And these sizes can change - the curvature in one extra dimension might be increasing, or it might be decreasing in another direction. We assume that in the present these properties are more or less stable, because they affect observable physics in a very fundamental way. Anyway, again we have here quantitative properties defined at every point in space (from the perspective of three large space dimensions), and so these are also fields, "moduli fields". They interact with the strings and branes according to the rules of general relativity, which say how space curves in response to energy density.

So in a full working model of string theory, you will have strings flying around, branes vibrating, and space itself curving and even changing topology; and the modes of strings, the vibrations of branes, and the geometric moduli of the extra dimensions all interact. They are all places where energy can be exchanged.

The reason I went into these details is to begin to explain how to get inflation in string theory. In the original string theory models that got people excited in the 1980s; space is flat forever, there is no inflation, the geometry is stable. The string/brane/moduli interactions are in equilibrium. To get inflation in string theory, you need to have a setup that is out of equilibrium for a while. You have to have something to play the role of the inflaton field, and there is nothing to tell you in advance whether the inflaton is a string mode, a brane property, or a geometric modulus.

In your questions, you seem to speculate about whether time itself could somehow inflate the string - whether the string length scale could somehow be set by the future lifetime of the string. The closest thing to this that I can think of, would be an approach to inflation which somehow obtains it from the global structure of space. There are various speculations about how to do string theory in de Sitter space which seem to be aiming for this. But it's one of the problem of such approaches that in the real world, inflation isn't just a ubiquitous process that happens everywhere and everywhen. If it did happen (the evidence for inflation is at best indirect), it stopped at a definite time, at least in our part of the universe. And you can explain that in terms of the usual concept of inflation: the inflaton field here ran out of energy, that's why inflation stopped. For example, if it was a particular brane vibration that was the driver of inflation, what happened is that the energy of those brane vibrations was all transferred elsewhere - into particular species of string, for example - and that was the end of inflation.

So I think the sensible attitude towards inflation in string theory is that it came about from a very specific interaction, under very special circumstances. I don't entirely discount the possibility that there might be an explanation for inflation, in terms of the deeper level of string theory (where strings, branes, and the geometry of space are all unified), which ends up sounding more fundamental and less arbitrary, but good luck trying to guess it without making a more down-to-earth model first. :-)

3. Sep 19, 2011

### Lost in Space

A very lucid explanation, Mitchell, thanks. But does any of it figure without time? Talk of inflation, vibrations, quantum fluctuations etc, is all well and good but surely they are all dependent on a primal temporal dimension in order to be? To my mind it seems that time needs to be, if not a causal factor, at least a primary factor to allow such things to occur. As for Calabi-Yau manifolds, I can appreciate that six of the ten dimensions can be 'tied up' as such, but the dimension of time is surely not one that is?

4. Sep 20, 2011

### mitchell porter

Change of any sort is impossible without time, and fortunately enough, string theory does contain time, in the same way that all theories since relativity contain time - as a direction in "space-time" that is distinguished from the others by being "timelike". In relativistic theories, there is a unified mathematical recipe for calculating http://en.wikipedia.org/wiki/Space-time_interval#Spacetime_intervals", if it's timelike separation, then the square of the interval is negative, and this will remain so in all valid coordinate systems (validity means that the laws of physics still look the same in that coordinate system; the definition sounds circular, but the content lies in the specific coordinate transformations which are allowed; the symmetry principles of modern physics, about which you hear so much, can all be expressed in the form that "you can relabel coordinates - or other physical properties - in a specified way, without having to rewrite the laws of physics for the new coordinates").

The relativistic concept of time is interesting, and it acquires even more flexibility in general relativity (Einstein's theory of gravity); how this physical concept of time relates to the subjective experience of time is also interesting and problematic. But the concept of time in string theory is essentially the same as the concept of time in special and general relativity (with an extra complication or two that come from quantum mechanics). Time mixes with space in a relativistic transformation, but timelike directions are timelike in all valid coordinate systems, and always distinct from spacelike directions. The complete interchangeability of length and time, which your original comment hints at, doesn't work because length is spacelike and time is timelike, and those are coordinate-independent properties.

There are innumerable features of physics which involve time, and not just in the banal sense with which I started (that you need time for events to happen in). The mixing of time with space is part of larger symmetries like supersymmetry and diffeomorphism invariance; diffeomorphism invariance of the string worldsheet has implications for how you quantize it; the technicalities go on. So time definitely matters.

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