# Space-time in 2 + 1 Dimensions

*Most recent answer: 10/22/2007*

Q:

Hello and thank you for taking the time to answer my question. My name is Marc, and I am currently a graduate student in philosophy, and have maintained a non-physicist’s interest in questions concerning the "nature" of spacetime.

My question is:

Using the analogue of a three-dimensional spacetime (as opposed to our 4-dimensional one), would is be possible, from a conceptual and physical standpoint, to represent objects on a two-dimensional spatial manifold, and represent time as a third spatial dimension, albeit one not spatially accessible to entities living on a two-dimensional manifold? What I am driving at is whether it is possible to reduce the physical concept of time to description in terms of higher-dimensional spatial coordinates; in our case, time would thus be defined as a fourth spatial dimension. In the example above, it would represent a third spatial dimension for two-dimensional entities.

Is there anything in extant physical theory explicitly to rule out this possibility? Also, if you could recommend any further reading on this topic for the non-physicist, I would really appreciate it.

Thanks again!

Marc

- Marc L. (age 25)

Hasbrouck Heights, NJ, USA

My question is:

Using the analogue of a three-dimensional spacetime (as opposed to our 4-dimensional one), would is be possible, from a conceptual and physical standpoint, to represent objects on a two-dimensional spatial manifold, and represent time as a third spatial dimension, albeit one not spatially accessible to entities living on a two-dimensional manifold? What I am driving at is whether it is possible to reduce the physical concept of time to description in terms of higher-dimensional spatial coordinates; in our case, time would thus be defined as a fourth spatial dimension. In the example above, it would represent a third spatial dimension for two-dimensional entities.

Is there anything in extant physical theory explicitly to rule out this possibility? Also, if you could recommend any further reading on this topic for the non-physicist, I would really appreciate it.

Thanks again!

Marc

- Marc L. (age 25)

Hasbrouck Heights, NJ, USA

A:

Our ordinary space-time theory deals with what are called 3+1 dimensions; three spatial and one time. It agrees with experimental observations. However, there are perfectly good mathematical theories that work in 2 + 1 dimensions. If you Google "2 + 1 dimensions" you will get about a 1000 hits. Unfortunately none of the sites I looked at are for the casual reader; pretty far out mathematically.

If you really want to blow your mind you might consider a 10 dimensional theory as proposed by ’String Theorists’. At least in 10 dimensions there is a readable exposition on Wikipedia:

enjoy,

LeeH

I think I’m picking up a little different question than the one Lee answered. The time dimension is mathematically not like the space dimensions. Its coordinates enter into a ’metric’ with the opposite sign of the space coordinates. In the Special Relativistic approximation, (which is much easier to follow than General Relativity), the transformations from one coordinate system to another mix the time and space coordinates, but the time coordinate mixes in a different way than the space coordinates.

Mike W.

If you really want to blow your mind you might consider a 10 dimensional theory as proposed by ’String Theorists’. At least in 10 dimensions there is a readable exposition on Wikipedia:

enjoy,

LeeH

I think I’m picking up a little different question than the one Lee answered. The time dimension is mathematically not like the space dimensions. Its coordinates enter into a ’metric’ with the opposite sign of the space coordinates. In the Special Relativistic approximation, (which is much easier to follow than General Relativity), the transformations from one coordinate system to another mix the time and space coordinates, but the time coordinate mixes in a different way than the space coordinates.

Mike W.

*(published on 10/22/2007)*