# The Universe as a four-dimensional sphere?

I was chatting with my 12yo cousin yesterday and we got to the Universe, its size and stuff like that.

Then he came up with the idea (I'll rephrase it), that the Universe could basically be a 4d sphere which looks like 3d, but is warped into the 4th dimension, so it can form a 4d sphere - the same way as the Earth is 3d sphere which looks like 2d when you stand on it, but is warped ito the 3rd dimension, so it can form a sphere (hopefully the analogy makes sense as I seem to lack the proper wording). That would allow the Universe to be both "infinite" and have its size, also questions like what's beyond the edge are not meaningful anymore. So it all looks very logical to him (and I admit, to me too).

But I am pretty sure this is incorrect as I don't recall reading anything about 4th spatial dimension (apart from String theory hidden dimensions), but I wasn't able to give him any kind of proof or counterexample.

So my question is: Is there a (as simple as possible) explanation/measurement/demonstration I could give/show to him to disprove his idea?

Although it's obvious, I confess I am definitely no expert, so please be forgiving to my formulations and assumptions.

• What would differentiate a dimension we do not detect (which is what "being 4D but looking 3D" means to me) from a dimension that doesn't exist? That is to say - do not disprove it, just say that assuming it exists does not add anything of value. – ACuriousMind Sep 4 '14 at 14:40
• The word 'warped' seems very much out of place here. More appropriate would be 'extends'. This also shows why this idea has nothing to do with things being both finite and infinite... – Danu Sep 4 '14 at 15:06

The simple answer is that your cousin could be correct. If his theory is that:

1. the scale of the sphere is far larger than the observable universe

2. there's no way to detect the 4th (spatial) dimension

then no experiment we can do could prove him wrong. But then there's no experiment that we can do that could prove him right either, so as theories go it doesn't get us very far.

Now the tl;dr stuff:

Physics is a process of constructing theories to describe the universe, using those theories to make predictions, then doing the experiments to see if your predictions are correct. If two theories make exactly the same predictions there is no way to distinguish between them, in which case physicists (being a down to Earth bunch) tend to choose the simplest theory.

At the moment the generally accepted theory to describe the universe on the large scale is general relativity. This describes the universe as a four dimensional manifold equipped with a metric. We know there must be at least four dimensions because we need four numbers, e.g. $t$, $x$, $y$ and $z$, to uniquely identify a spacetime point. The metric is the equation that tells us the distance between points.

It's important to note that while there are one timelike and three spacelike dimensions, you cannot uniquely split spacetime into separate time and space parts. This is because different observers will disagree about what bits are space and what bits are time. A vector that looks like a purely time displacement to me may look like a combination of a time and space displacement to another observer. All observers will agree there is one time dimension and three space dimensions, but they won't all agree on how those dimensions are defined.

Anyhow, you've undoubtably heard that spacetime is curved, and you've probably seen the rubber sheet analogies for spacetime, so it's natural to ask if there is a fifth dimension (fourth spatial dimension) for the universe to curve into. In general relativity no extra dimension is required because the curvature in intrinsic not extrinsic. To learn more about what this means you might be interested in looking at the questions What is the universe 'expanding' into? and If the universe is expanding, what is it expanding into?.

So GR requires no extra dimensions to describe the universe, regardless of whether the universe is open or closed - current indications are that the universe is probably open. But there is one last point to mention. General Relativity is a local theory because it relates the local curvature to the local stress-energy tensor. It does not and cannot tell us anything about the global topology of the universe. The universe could be closed in the sense that if you travel far enough you return to your starting point, though if this is the case the scale of the universe must be larger than anything we can observe at the moment otherwise we'd be seeing repeated patterns in the CMB. However this type of closure doesn't require any extra dimensions either.

So the bottom line is that nothing in our current theories requires there to be an extra dimension and your cousin is introducing an unnecessary extra complication. If maybe one day an experiment at the LHC reveals that extra dimensions do exist your cousin should consider writing off for his Nobel prize. Until then the burden of proof rests with him to demonstrate that his extra dimension is required to give a better description of the universe than the one we are currently using.

• At the moment the generally accepted theory to describe the universe on the large scale is general relativity. Which is totally going to be supplanted by Shape Dynamics :D ;) – Kyle Kanos Sep 4 '14 at 15:55
• Awesome, this together with physics.stackexchange.com/a/82414/37033 helped me get at least some tiny portion of knowledge, so I'll be able to explain to my cousin he doesn't really need 4th spatial dimension in order to achieve what he missed in the 3d version. Thanks! – Erlik Sep 5 '14 at 8:25

Your 12-year-old cousin might be correct; it isn't yet known for sure. However, some existing experiments are pointing in the direction of your cousin being wrong.

What you're calling a "4D sphere" and a "3D sphere", a mathematician would call a "3-sphere" or a "2-sphere", respectively, because mathematically an "$n$-sphere" means something that's equivalent to just the surface of an $n+1$-dimensional ball.

If the universe has the shape of a 3-sphere, it should in principle be possible to detect that, even though the observable universe presumably isn't the entire universe, because the shape of the universe affects the curvature of space (or technically, the curvature of a spatial section of spacetime according to comoving coordinates).

On a 2-sphere (the usual kind of sphere), you can tell that the sphere isn't flat by just making measurements on a small portion of the sphere. Because the area of the region within a radius $r$ from a given point on the sphere is less than $\pi r^2$, you can tell that that region can't be a part of a flat surface, because the area of a circular part of a flat surface is given by $A=\pi r^2$. If the universe was in the shape of a 3-sphere, it should be possible to detect that, using similar kinds of measurements.

Experiments are indeed being done which take measurements that help determine what the shape of the universe is. One such experiment is the Wilkinson Microwave Anisotropy Probe (WMAP). However, according to data from WMAP, the universe appears to be flat, within a 0.4% margin of error. So the data isn't looking like the universe is in the shape of a 3-sphere. However, it might still be possible that the universe is in the shape of a 3-sphere, if the whole universe is sufficiently larger than the observable universe to make the observable universe just appear to be flat.

Your cousin is right. The Universe is a 4D sphere W = r + Ix + Jy + Kz = [f, V] The universe is defined by energy W = -vh/2pir + cP where -vh/2pir = -vp = -mv^2, a real number potential energy.

Newton found this in his theory of Gravity W=-mGM/r = -vh/2pir = -vp = - mv^2. Newton's energy is a real number or scalar 1 dimensional energy. Newton calculated the potential energy between m and M as if m is not moving. The fact is m is moving. The mass M creates a velocity field around it with speed v=(GM/r)^0.5 this field gives m a velocity V. Thus the mass m has a Momentum mV and vector and a vector energy cmV = cP the vector energy!

Physicists consider energy a scalar not a vector. The universe considers and most quantities a combination of a scalar and three vectors thus creating a 4d quantity called a Quaternion. William Rowan Hamilton developed Quaternions in 1843.

The vector energy is the so-called "Dark Energy'. This Dark Energy is hidden in plain sight. Every particle of matter that moves, m, creates Momentum P and Vector energy cP !

So the energy of the Universe is W = -vh/2pir + cP = [-vh/2pir, cP] 4D quantity.

Physics also does not have a 4D Derivative, I invented one X = d/dr + Del, the combination of Hamilton's Vector Derivative Del and a real scalar derivative d/dr=d/cdt.

With this we can find the real forces and they too are four dimensional;

Force = the first derivative of the energy W F = XW = [d/dr, Del] {-vh/2pir, cP] = [vp/r - cDel.P, cdP/dr + Del -vh/2pir + cDelxP] F = [vp/r - cp/r cos(P), -1P cp/r + 1R vp/r + 1L cp/r sin(P)]

F = cp/r[ v/c -cos(P), -1P + 1R v/c + 1L sin(P)]

The cp/r = cp/ct = p/t =mv/t = ma, Newton's famous F = ma.

The Quaternion has a scalar force cp =ma( v/c -cos(P)) the first term ma(v/c) is the Gravitational centripetal(cp) force the center seeking scalar force. This is the force that attracts the earth to the sun. The second term cf=ma cos(P) is the centrifugal (center fleeing) force is the force that keeps the earth from falling into the sun. When the two forces are equal the earth's orbit is stable and tis is called Continuity Condition. This is also the explanation of the red shift. The red shift is the condition v/c when the orbit is stable.

The vector force is three vectors: -1P cp/r is the tangent force; 1R vp/r is the Gradient and 1L cp/r sin(P) is the circulation or curl force. Here again your cousin is 4 D universe.

another finding is possible, The Universe is stable if the first derivative is zero.

this zero is possible when F = ma{v/c -cos(P), -1P + 1R v/c + 1L sin(P)] =0.

This force is zero when the Quater nion is zero meaning the v ector is zero and the scalar is zero. This happens when cDelxP= cpsin(P)=0 meaning that 1P is parallel to 1R and coa(P)=1. Then the quaternion is zero when v/c=1 or v=c ! The Boundary condition is when the velocity is the speed of ;light. the Unverse is bounded at v=c and this is the

Max energy W = -vp + cp = cp + cP = mc^2[-1,1P]

My estimate of the minimum size is 155E24 meters and Mass 2E53kg and Power 3645E49 watts.

The universe is larger when not at Maximum c^2= GM/r.

Your cousin is right the Universe is 4 D sphere.

Einstein's spacetime is not a 4D universe, Einstein and Minkowski could be called a 2D complex universe (x + y _+ z + + Ir) this is 3 scalars and 1 vector equal a scalar and 1 VECTOR i. THIS IS NOT A 4D UNIVERSE.