# Is it reasonable to visualise an additional dimension of time as a part of 4th dimensional spacetime

[EDIT] In the spirit of asking a "good question", this question is considerably more refined that the one I decided to blurt out earlier thanks to a little additional research and nudges from comments.

I'm not a physicist, but love to let ideas roll around in my head - so please excuse any crossed boundaries. I'm not equipped with the mathematics required to examine or perhaps even pose this question in a way many of you are probably used to.

So, my question:

Spacetime as we know it consists of 3 spatial dimensions + 1 temporal dimension. Is it reasonable to visualise spacetime in our next-highest dimension as consisting of 4 spatial + 2 temporal dimensions?

From another point of view:

If we view our 3D space as a submanifold of a 4D manifold, is it reasonable to suppose that time as we experience it may be a similar submanifold of time in a higher dimension?

Or:

Adding and subtracting spatial dimensions is the easy bit (an axis perpendicular to all other axes in that dimension) ... do we also get to add and subtract temporal dimensions from spacetime as we journey from one dimension to the next?

(For posterity, my original question was phrased as "Does the 4th dimension include imaginary time as part of its fabric of spacetime?" - referring to the concept of imaginary time as popularised by Hawking. In case it helps, this annoying little thought experiment arose from musings of the possible mechanics behind quantum entanglement)

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References I've found helpful so far:

"Survey of two time physics": http://inspirehep.net/record/532282?ln=en

"Dual field theories in (d-1)+1 emergent spacetimes from A unifying field theory in d+2 spacetime": http://inspirehep.net/record/750980

• If our universe is somehow curved into a fourth dimension The Universe in not embedded in an additional dimension, Google intrinsic curvature. I think imaginary time is just a trick when there is no curvature (so you don't need to use a matrix doing a scalar product, instead you use imaginary numbers), but I don't know if Wick rotation is useful in curved space-time. In general, it's better to use real time and forget about imaginary numbers. Jun 21 '14 at 12:12
• @jinawee as a matter of fact virtually all calculations in QFT in curved spacetime are carried out in Euclidean signature, so the Wick rotation is incredibly useful there too. Some would argue that there is some physical content in this trick too, given how well it works...
– Danu
Jun 21 '14 at 12:31
• Thanks for the pointer - I found a great definition of intrinsic and extrinsic curvature at Wolfram Mathworld. I think what I'm visualizing when I say "our universe curved into a fourth dimension" is extrinsic curvature. As in, our 3 dimensions are a submanifold of a 4 dimensional manifold ... I'm wondering if time as we experience it may be a similar "submanifold" of a higher dimension of time. (It was musing about the mechanism behind quantum entanglement which gave rise to this annoying thought experiment). Jun 21 '14 at 12:34
• The 2-Time Physics seems cutting edge research, so don't expect it's right. The first edit seems off-topic, since it's pure especulation. Btw, do you mean imaginary in the senso of complex number with no real part? Jun 21 '14 at 14:36
• I was using the only words I knew an hour ago to explain the concept of 4+2 dimensions - borrowed of course from Hawking. en.wikipedia.org/wiki/Imaginary_time Jun 21 '14 at 14:39

Two or more timelike dimensions is a situation that is difficult if not impossible to reconcile with the notion of causality.

Suppose you want to think of a five dimensional universe with three spatial and two time dimensions. What you mean then is the metric has a $(2,\,3)$ signature, which means that at each point Riemann normal co-ordinates centered at that point have the metric tensor at that point:

$$g = \left(\begin{array}{cc|ccc}1&0&0&0&0\\0&1&0&0&0&\\\hline0&0&-1&0&0\\0&0&0&-1&0\\0&0&0&0&-1\end{array}\right)$$

So, suppose we confine ourselves to isometries that conserve the form $\mathrm{d}t_1^2 + \mathrm{d}t_2^2 - \mathrm{d}x_1^2- \mathrm{d}x_2^2- \mathrm{d}x_3^2$ but which are confined to the time subspace. This means we are conserving the form $\mathrm{d}t_1^2 + \mathrm{d}t_2^2$; but this means we are talking about a true, real-angle rotation (i.e not a boost thought of as an imaginary-angle rotation), a transformation from $SO(2)$; therefore, the sense of any vector pointing to the future in time can be rotated so that it points backwards in at least one of the time co-ordinates and we can choose which time direction has its sense inverted. Therefore, there is always a generalized Lorentz transformation that will invert the time ordering of any process for the inertial observer whose frame the transformation transforms to.

Thus we see that physical reasonable theories must either

1. Have a metric where the sign of the time component in the signature is the opposite to the signs of the spatial components in the signature. This situation allows the generalized Lorentz transformation to always conserve the time ordering of any process as long as we postulate an impossibility of faster-than-light relative motion between inertial observers; Or
2. Make some further postulates as to why the time reversing rotations described above are not physical, or at least some explanation of how causality can still be manifest.
• and what about a secondary time dimension at the infinitesimal small level, impacting only some behaviors at this scale ? ( it's not a point of view, just a question about this proof )
– user46925
Aug 11 '16 at 4:34
• @igael That may be a valid point: as I said, "difficult if not impossible". One would need to make postulates or arguments to reason why rotations as I outline could not happen and I believe they do play with ideas like yours. I've changed the last paragraph accordingly. Aug 11 '16 at 4:51

Time is the perception of the order of change in 3 dimensions. Time does not exists, only 3 dimensions of space exists in reality. Time is not a 4th dimension.

I think your question highlights the damage done to innocent brains in high schools. Small wonder we do not have enough smart physics students, their brains get destroyed with false assumptions forced upon them in primary and high school.

The similarity of time and space is limited to Lorentz symmetry. Beyond Lorentz symmetry, the time dimension cannot be assimilated to space dimensions.

Within spacetime, time is not intrinsically curved: Any observed time corresponds to the proper time of a clock, and the clock is always counting straightforward, even if according to our coordinates we may observe time dilation.

With view to these facts, any presumed extrinsic curvature seems to be highly speculative and unnatural.

It is not reasonable to view time as imaginary in the four dimensional space.

Proper 4 D space includes r + Ix + Jy + Kz = r + V = ct + V, where time is real dimension and x,y and z are the imaginary or vector dimensions. This is called Quaternions. The Cosmos seems to consists of Quaternions. William Rowan Hamilton, discovered Quaternions in 1843.

The Cosmos and laws of mathematics confirm Quaternion space as reasonable and real.

• "The Cosmos seems to consists of Quaternions". What? I don't really see a reason to assume that points of spacetime need to form a division algebra. Jun 21 '14 at 14:32