A 5D universe is plausible given the data
If I told you that the universe had an additional dimension beyond the familiar three space and one time dimensions, the first thing you…
If I told you that the universe had an additional dimension beyond the familiar three space and one time dimensions, the first thing you might ask is: how do I see this dimension? Or why don’t I see it? Can I travel through it?
One of the main reasons we actually perceive the dimensions we do is not because they are dimensions but because the universe provides a mechanism to travel through each of them in a symmetric way. That is, each one behaves something like the other (with the exception of time which I’ll get to).
Einstein taught us that these three dimensions are related through a process called Lorentz transformation (named after the great physicist Alfred Lorentz). Lorentz transformation is simply the six possible ways that you can move in time and space: three kinds of rotation that you can think of as pitch, yaw, and roll and three kinds of acceleration (one for each spatial direction). It is the phenomenon of Lorentz symmetry (the equivalence of all states of motion related by Lorentz transformation) that allows us to perceive the three space and one time dimension, not their status as dimensions.
To see why, imagine a metal rod. You can turn it in three different rotational directions. You can also lob it across the room accelerating it at someone you dislike. A third option is to apply an electric current through it. Now, suppose my rod is connected to a circuit such that I can run an alternating current through it. This alternating current runs a motor. Now, one of the options I have in generating the current is to adjust its phase, that is, at what time the current has its most positive and most negative voltages. While adjusting the amplitude or frequency of the current would certainly alter the behavior of the motor, adjusting the phase would give me essentially identical motor performance. Thus, phase is a symmetry as far as my motor is concerned because changing it doesn’t change how the motor behaves.
Phase shift symmetry is also a symmetry of electromagnetic fields. Altering the phase of an electromagnetic field potential does not change any measurements. Thus, can we think of phase as another dimension? I argue yes, it is simply a dimension that is mediated by phase symmetry rather than Lorentz.
Indeed, rather than thinking of dimensions as fundamentally physical, they can be thought of as independent variables that are related by some transformation like rotation, acceleration, or phase shifting.
This may explain one reason why we do not perceive the 5th dimension. It is simply that we do but don’t think of it as a dimension.
But perhaps this definition of dimension is too broad. After all, from that perspective, anything that we can independently adjust is a dimension. That can’t be, fundamentally, a good way to define dimension unless you are a mathematician.
Fair point.
Suppose instead we define a dimension based on some geometric definition like that of Einstein’s general relativity. We can all agree that Einstein’s theory is all about time and space, so naturally, anything that we can call a dimension should be somehow related to time and space directly, as through the Einstein field equations that govern gravity and all the space and time that we normally perceive. Thus, phase shifting doesn’t count unless we can include it in a general relativity of a higher dimensional space and time.
This is exactly what Kaluza did in 1920. He defined a 5D spacetime using Einstein’s equations and showed that, if you made a few assumptions, electromagnetism and 4D general relativity pop out. Einstein was thrilled, but he didn’t know what to do with it and neither did Kaluza. People were just getting used to the idea of time and space being interchangeable. Now we had another dimension? It was all too much.
That didn’t stop 5D spacetime theory from marching right along through the 20th and into the 21st century, albeit with little fanfare. Every test that confirmed Einstein’s theory of relativity, 5D theory came up behind and whispered: “me too”.
So, now, where does 5D theory stand today? Well, it has stood the test of time, for sure. It has been battered and beaten (constrained) like all alternatives to Einstein’s which, so far, is still King despite quite a few unexplained phenomena.
Most of the validation of the 5D theory has been done by either the late Paul Wesson of Waterloo or James Overduin of Towson University with their various collaborators. These can be categorized into: classic tests of general relativity, cosmological tests, and the results of Gravity Probe B (which demonstrated some more advanced tests).
All of these tests put constraints on deviations from Einstein’s general relativity which are largely classed as post-Newtonian variants (variants that attempt to modify the corrections that Einstein’s theory makes to Newtonian theory in various ways). Post-post Newtonian variants include second order corrections and these are few and far between so small are the deviations from Newtonian theory itself. (These are so small that mountains on Earth have a stronger gravitational influence.)
It may surprise you to know that although Einstein’s relativity has stood the test of time it has largely been unconfirmed in its higher order predictions. General relativity is a highly nonlinear theory that has essentially infinite order corrections to Newton’s laws. (Newton’s can be thought of as the zeroth order gravitational theory for slow speeds and weak gravitational fields.) The order of a correction is a qualitative measure of its significance in the theory. First order corrections are most significant, but, by comparison, in the most precisely confirmed physical theory, Quantum Electrodynamics (QED), we have confirmations of results up to the fourth order.
This means there is a fair bit of wiggle room in General Relativity and the problem of how to construct a new theory has as much to do with the first principles motivation for it (why we are constructing the theory this way) as it does to making predictions.
This is partly why string theory is so popular. It has a strong motivation for creating a theory of all matter and forces from a single cause. Unfortunately, we have no predictions from it.
Classic Tests of General Relativity
Classic or Solar System tests of General Relativity are ones that look at how small bodies behave in the presence of the relatively banal celestial bodies of the Sun and Earth. In order to make predictions you need to understand what the geometry of space and time is and then calculate what a small, gravitationally negligible “test” body would experience in traveling through that space and time. That test body can either be a physical object like a spaceship or even a small planet. Or it could be a beam of light in which case we can take its wavelength into account as well as its path.
Classic tests assume that the spacetime geometry is non-rotating, spherically symmetric, electrically neutral and that doesn’t change with time. These are pretty good assumptions since neither the Earth nor Sun rotates close to the speed of light where it might be more important, both are electrically more or less neutral, and in the time frames that we conduct experiments they don’t change that much. (We can take other effects into account that violate these assumptions and these generate small corrections larger than second order but smaller than first.)
In 4D, there is only one solution to Einstein’s field equations that fits these assumptions. It is the same solution that works for planets, stars, and even black holes. Sometimes it is called the “Schwarzschild solution” because of the first person who discovered it in 1916.
In 5D, there is more than one such solution and so we have to pick one. The one that makes the most sense for us is the soliton solution. The soliton solution matches the Schwarzschild solution if you assume nothing changes in the fifth dimension. It also has the odd characteristic that it is a “vacuum” solution meaning that it generates curved spacetimes in 4D without having any matter in 5D. This is an important assumption in 5D spacetime that matter is simply higher dimensional geometry. In other words, ripples in the 5th dimension create matter.
A measure of the deviation of a 5D spacetime as above from 4D general relativity is given by a parameter, sometimes given the symbol b, that scales the ratio of the mass of your gravitating body and distance to the test body you are studying as an exponent from 0 (no effect on the 5th dimension) to +/- 1.15. This means that b is an exponent that indicates how much a mass actually curves the fifth dimension. The goal of experiment is to constrain this parameter.
Perihelion Advance
The very first prediction that Einstein made with his theory was the anomalous perihelion advance of the orbit of the planet Mercury. Essentially, all planets’ orbits are ellipses which rotate themselves. In Newtonian gravity, all of this rotation can be attributed to tugs from other planets. In Einsteinian gravity even in the absence of other planets, there is still some advance. This additional advance has been measured very accurately with radar to be about 43.11 +/- 0.21 arcsec per 100 years. The predicted value in GR is 42.98.
A 5D soliton theory can agree with this provided its deviations in the 5th dimension are fairly small.
In 2000, in a paper by Liu, it was suggested that the contributions from the oblateness of the Sun (it bulges in the middle from rotation) were significant enough to rule out general relativity and a 5D theory could explain the result. This turns out not to be the case. In 2011, the Messenger probe visited Mercury and confirmed that gravity due to solar oblateness is about 20 times less than Liu assumed from previous estimates. (It is not the first time that solar oblateness has been used to try to disprove General relativity.) In fact, the sun’s oblateness only contributes about 0.029 arcsec per 100 years to the result, which, if anything, improves the GR prediction slightly. While it does not rule out the 5D theory, it does constrain it such that 5th dimensional deviations from flatness are less significant than other sources of error.
Light Deflection
Once he had worked out an answer to the perihelion anomaly of Mercury, Einstein needed to make a prediction of some phenomenon that nobody had ever seen. A scientific theory that explains an existing anomaly is one thing. One that predicts something completely new is quite another as it is unlikely that a theory that is wrong would predict an unseen phenomenon correctly.
This phenomenon turned out to be the deflection of light by gravity. This is something that Newton’s laws could not grapple with correctly and worked flawlessly in Einstein’s theory.
In general relativity, light follows paths called null geodesics. These are paths that only particles with no mass at rest follow.
In a 5D spacetime, all particles follow null geodesics because, since matter comes from geometry, all particles are made of gravity and gravity is, in most theories, massless. The appearance of mass comes from our 4D way of looking at the universe. Therefore, anything that appears “lightlike” like light itself in 4D is simply following a null geodesic in both 5D and 4D, while anything that acts like a massive particle is following a null geodesic in 5D but not in 4D.
Deviations for light deflection by the Sun can be no more than 0.17% from the prediction of general relativity. While this is very small, it turns out that 5D solitons can meet this with deviations from flatness of b of no more than +/-0.07.
There are several other solar system tests such as time dilation and red shift, but none of these constrain it further.
Non-Soliton Solutions
It turns out that because there are other solutions to Einstein’s field equations in 5D, the point may be moot. While solitons predict deviations from 4D GR, another solution called the canonical solution predicts no deviations at all unless the bodies are spinning. Unfortunately, we don’t know which solution planets, stars, and other spherical bodies represent. If that is true, then all our comparisons of non-spinning solutions to predictions will come out the same in either theory. We have to take rotation into account.
Gravity Probe B
Gravity Probe B launched in 2004 measured precisely an effect of general relativity called the geodetic effect. This is the tendency for gyroscopes not to return to their original orientation when orbiting a body. Thus, they involve spinning objects.
In particular, Gravity Probe B contained four very precise gyroscopes that could measure deviations from their orientation upon orbiting the Earth. In these tests, the deviation from the gyroscope’s spin axes in a polar orbit at an altitude of 642 km was supposed to be about -6606.1 milliarcseconds/year.
All of the gyroscopes measured close to this value and are similar to classic tests for the soliton. For the canonical coordinates, constraints are reasonable.
One of the interesting predictions is that the geodetic precession in the 5D theory must be less than that of 4D GR. This means that if the probe found values that were more than GR’s prediction the 5D theory would be ruled out.
Conclusion
Given the data, and adding in cosmology (the universe as a whole), measurements of the equivalence principle (equivalence between gravity and acceleration), Lorentz invariance, and so on, a 5D theory is plausible as are many other theories. Nothing so far has indicated that Einstein’s General Relativity doesn’t explain everything we see, but if problems like dark energy, dark matter, and quantum gravity are any indication, it is not the end of the story. A 5D theory might get us one step closer.
Liu, Hongya, and James M. Overduin. “Solar system tests of higher dimensional gravity.” The Astrophysical Journal 538.1 (2000): 386.
Genova, Antonio, et al. “Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission.” Nature communications 9.1 (2018): 1–9.
Overduin, J. M., R. D. Everett, and P. S. Wesson. “Constraints on Kaluza–Klein gravity from Gravity Probe B.” General Relativity and Gravitation 45.9 (2013): 1723–1731.