What Does a 4-Dimensional Sphere Look Like?

[...]

I wanted to take some time to reply to your questions about 4D volumes in Minkowski space:

1. Is the 4D-“volume” of a 4D-sphere finite or infinite?
2. Note that the 4D-sphere is clearly a bounded object, but how we compute the 4D-“volume” in such a geometry?
3. How we integrate when the metric is not the usual Euclidean one?
4. What is a 4D-cube in relativistic geometry? A tetrahedron?

Let me emphasize that I’m in no way an expert in this area, but from time to time I’ve dabbled in general relativity and related math topics a little bit. But I can at least try to shed some light on your questions.

As a natural consequence of the higher dimensional and more complex nature of 4D Minkowski space, there are a number of reasonable interpretations of your questions — ie, answering your question requires sharpening it, and it can be sharpened in a variety of ways. So rather than try to identify a single answer, my goal will be to clarify the issues. I believe that understanding this brief “tutorial” should enable you to pursue such questions as you wish. I’d appreciate knowing if this is helpful, and if you obtain any additional results.

I’ll start at square one with the bleeding obvious, because basic ideas have to be sharp to retain one’s footing in unfamiliar geometries. I’ll keep things as simple as possible and bring things up only if they’re really needed.

4-Rectangle

Let me start with (4), since the rectangle volume is straightforward. In volume spacetime integrals, the 4-volume element

dV4 = dt dx dy dz

has a standard orientation (just as does a 3D volume element), in which the time axis points to the future and the xyz axes are right-handed, or equivalently, the integral

     ⌠⌠⌠⌠
V4 = ││││ dt dx dy dz
     ⌡⌡⌡⌡

is positive.

Therefore, the theory of integration gives the same cartesian volume elements in 4D Minkowski space as it does for 4D Euclidean space,

     ⌠⌠⌠⌠
V4 = ││││ du dx dy dz
     ⌡⌡⌡⌡

despite the opposite sign of the fourth component of the cartesian metrics

(M4c) ds² = –du² + dx² + dy² + dz²

(E4c) ds² = du² + dx² + dy² + dz²

With the volume element in standard orientation, the 4-volume integral is simply the same as for (E4c),

∫∫∫∫ = ∫ dt ∫ dx ∫ dy ∫ dz = Δt · Δx · Δy · Δz.

So this 4-volume is finite (for finite deltas).

The above is stated explicitly in MTW p147, “Volume Integrals in Spacetime”. The same will be covered in any text such as tensor analysis that covers general curvilinear coordinates — look for things like

√(det(g))

where g is the metric matrix, and depending on the context g will have a minus sign or not, such that the standard orientation gives a positive integral.

Just as with integrals in Euclidean space, any spacetime integral can be integrated using the above by chopping up the integrand into rectilinear elements. But for dealing with spheres, the only sane thing to do is to use spherical coordinates.

Embedding Euclidean Spheres in Euclidean Space

For our purposes, understanding the basics of embeddings, mappings, and metrics are essential. Embeddings are great aids in reasoning about geometries. Mappings are the general class to which embeddings belong. Metrics are the way geometries are defined intrinsically (without reference to an embedding), and in the case of orthogonal coordinates, which we will always use, the side lengths whose product gives the volume element can be read off from the metric

Visualizations are another very helpful tool for understanding geometries. I’ll briefly touch on that the first time we have to consider a non-trivial higher dimensional space (Case 1).

First, imagine an ordinary sphere in 3D Euclidean space, ie, a 2D surface embedded in E3. You could call the integral of the surface a “surface integral” and the integral of the embedding space enclosed by the surface a “volume integral”, but such names just confuse things in higher dimensions, so I’ll use the term “d-volume” regardless of d.

Let’s first consider d-spheres in Euclidean space. First I’ll make a certain specific comparison to illustrate the main point, and then discuss the specifics in more general terms. Everything now is Euclidean.

The equation of a 2D sphere, which can be viewed as embedded in 3D euclidean space E3, is

(s2)       x² + y² + z² = r²

The equation of a 3D sphere, which can be viewed as embedded in 4D euclidean space E4, is

(s3)       u² + x² + y² + z² = r²

Now using spherical coordinates let r denote the radial variable and w₁, w₂, … the angle variables. E.g., a point on the surface of a 2D sphere of radius r is given by (w₁, w₂), and on a 3D sphere by (w₁,w₂,w₃). The surface of a sphere is covered by the range 0 to 2π for the single angle that appears only as its differential, and 0 to π for all others.

2-vol of the surface of a 2-sphere (ordinary 2D surface area)

(s2,2) = ∫∫ (r dw₁)(r sinw₁ dw₂)

       = ∫∫ r² sinw₁ dw₁ dw₂ = 4 π r²

 3-vol of the E3 embedding space enclosed by a 2-sphere (ordinary 3D sphere volume)

(s2,3) = ∫∫∫ (dr)(r dw₁)(r sinw₁ dw₂)

       = ∫∫∫ r² sinw₁ dr dw₁ dw₂ = 4/3 π r³

3-vol of the surface of a 3-sphere

(s3,3) = ∫∫∫ (r dw₁)(r sinw₁ dw₂)(r sinw₁ sinw₂ dw₃)

       = ∫∫∫ r³ sin²w₁ sinw₂ dw₁ dw₂ dw₃ = 2 π² r³

4-vol of the E4 embedding space enclosed by a 3-sphere

(s3,4) = ∫∫∫∫ (dr)(r dw₁)(r sinw₁ dw₂)(r sinw₁ sinw₂ dw₃)

       = ∫∫∫∫ r³ sin²w₁ sinw₂ dr dw₁ dw₂ dw₃ = 1/2 π² r⁴

The main point I want to make here is that both (s2,3) and (s3,3) are 3-volumes, or triple integrals — yet they have different values.

(s2,3) is an “extrinsic” 3-volume, since it involves the embedding space (ie not on the surface)

(s3,3) is an “intrinsic” 3-volume (ie on the surface)

Embeddings and Mappings

Now some observations and language about the above. Above 3 dimensions, intuition inevitably starts to fail abruptly and precipitously, so the more ways one has to reason about geometries the better.

The two integrals for s2, (s2,2) and (s2,3) (and likewise the two integrals for s3), differ by whether one integrates the surface or the embedding space enclosed by the surface, which in this case is whether dr is part of the volume element or r = fixed. The dimensionality of a volume element is how many coordinate differentials appear (eg, dr or not). [Footnote: The pairs of integrals also are related by differentiation / integration with respect to r. Integration/differentiation works only for fixed d-dim, it can’t change the dim (eg (s2,3) is not the derivative of (s3,3)). The reason for this is easy to see by looking at the formulas for the volume and surface of d-dim spheres given in Wolfram’s book Notes).]

The transform equations for cartesian-spherical coords in E3 space are

(TE3)

x = r sin w₁ cos w₂

y = r sin w₁ sin w₂

z = r cos w₁

 The fact that the transform equations (TE3) define an embedding of a 2D sphere in E3 is shown by verifying that (s2) is satisfied by the transform variables, by squaring the RHS’s of the transform equations and finding that their sum equals r².

Since an embedding increases the number of dimensions, then as an embedding the transform (TE3) is regarded as a mapping from 2 variables (w₁,w₂) to 3 variables (x,y,z). Given this view, r = fixed and dr = 0. Note that an embedding can increase the dimensionality by more than 1, e.g. a 1D curve in E3. An embedding is also called an immersion, or a subspace of the embedding space.

An embedding is also a parametric representation. The coordinates of the embedded surface, before they are transformed to the embedding space, are the parameters of the parametric representation of the embedded surface. For example, the parameters of (s2) are w₁ and w₂, and the transforms (TE3) yield a parametric representation of (s2) in E3. Another example is a 1D curve in E3 where the curve is represented parametrically by the single parameter s and the transform equations that map s into x,y,z.

We’ll use the term mapping to refer to transforms between any two sets of variables, with any number of variables or dimensions each. If the number of dimensions is regarded as different, we’ll usually call it an embedding. Rigorous definitions of mappings and embeddings involve various details (one-to-one, what to do with singular points, etc) but we won’t require that level of detail.

The transform (TE3) may also be regarded as a mapping between 3 variables (r,w₁,w₂) and 3 variables (x,y,z). In this view, r is a variable so the differentials contain the dr terms. Thus, the presence of a dr in a geometry simply means that r is a variable, regardless of whether the geometry is arrived at via a mapping or an embedding.

 Don’t read too much into these ideas — they are only intended as aids in reasoning about geometries.

Metrics

Metrics are defined as quadratic forms in the differentials. In any orthogonal coords, the cross-terms vanish. The cartesian metric of E3 is:

(E3c) ds² = dx² + dy² + dz²

This metric in spherical coords can be found by taking differentials of the transform equations (TE3), with the dr terms included since we want the 3-space (r,w₁,w₂), and plugging into (E3c), yielding

(E3s) ds² = dr² + r² (dw₁² + sin²w₁ dw₂²)

For our purposes, we’ll regard a metric, a coordinate system, a geometry, a space, and a surface as all synonymous. It’s useful to know that it’s not always easy to determine whether two metrics are equivalent. Even metrics for Euclidean space can very easily be made quite unrecognizable by nonlinear transforms. In fact, at least according to Wolfram’s book, above 4D the isometric equivalence problem is actually undecidable.

Sometimes I find it useful to think in terms of degrees of freedom (dof).

• For the equation defining a 2-sphere

(s2) x² + y² + z² = r²

a)       There are 3 cartesian variables, but the equation is a constraint that lowers the dof to 2.

b)       For each value of r, (s2) is satisfied by 2 dof points on the 2D spherical surface of radius r

c)       (s2) is embeddable in E3, which can use its extra dof to make excursions off of the surface.

• For the metric of E3

(E3c) ds² = dx² + dy² + dz²

(E3s) ds² = dr² + r² (dw₁² + sin²w₁ dw₂²)

the number of coordinate differentials is 3, or 3 dof. To move around in E3, one has to let r vary.

• Inequalities change the region within which the dof can vary. The inequality

x² + y² + z² <= r²

can be thought of as reducing E3 to the region enclosed by (s2). In this view, an equality constraint corresponds to reducing a region to the limit of reducing its dof.

A differential volume element is the product of the differential side lengths. For our orthogonal coord systems, side lengths may be simply read off from the metric as the square root of the coordinate differentials. 

For (E3c) this is just

dV3 = (dx)(dy)(dz)

For (E3s) the 3-volume element is that of (s2,3)

dV3 = (dr)(r dw₁)(r sinw₁ dw₂)

[Footnote: The volume element in spherical coordinates is not equal to the product of the dx dy dz differentials of the transform equations (TE3). This is easier to see in 2D polar coordinates r w,

x = r cos w

y = r sin w

dx = cosw dr – r sinw dw

dy = sinw dr + r cosw dw

The product

dx dy = (cosw dr – r sinw dw)(sinw dr + r cosw dw)

is not equal to the correct 2D volume element

r dr dw

To use the transform equations, one can either compute the determinant of the linear differential transform, or use the language of differential forms and compute the exterior or wedge product.

In 3D spherical coords this gives the 3-form,

dx ^ dy ^ dz = r² sinw₁ dr ^ dw₁ ^ dw₂

and thus the correct volume element

r² sinw₁ dr dw₁ dw₂

But for orthogonal coord systems, if one has a metric by far the simplest way is to simply read off the volume element from the metric].

The Cases

Now let’s look at 4D Minkowski space (M4) and your question re the “volume of a sphere in M4”. Case 1 seems closely related to your question, and its 4-volume is infinite. Then I point out a couple of ways that it could be modified to give cases with finite 4-volumes. Case 2 is a related possibility for which the 4-volume is finite. While Case 2 is not the 4-volume you ask about, I’m guessing that it is the source of your intuition that the volume should be finite. Case 3 completes and illuminates Cases 1 and 2. Cases 1 2 and 3 start by defining a (spatial) 3-surface, and add the fourth coordinate (time) independently to get the 4-volume. Case 4 starts with a 4-surface, and the first of the two sub-cases seems like the other most closely related possibility for interpreting your question. All four Cases also have physical relevance, which will be briefly described.

I have calculated some but not all of the 4-volumes of the various cases. Again, my intention is primarily to shed light on why the possibilities are so numerous and how they relate to each other. Understanding this tutorial should enable you to calculate any integrals that interest you.

Case 1

Your questions are about the volume associated with the equation of the 3-surface

(ps3) –u² + x² + y² + z² = r²

which is closely related to the Euclidean 3-sphere

(s3)  u² + x² + y² + z² = r²

The name (ps3) indicates “pseudo-sphere” (here a 3-surface). The term “pseudo” is used in the literature to refer to a space with exactly one dimension with opposite sign from the rest. Thus M4 space is also called pseudo-E4 (pE4 here) and the (ps3) space is called a pseudo-sphere. “Spherical” means “rotationally symmetric”, and can be made to include (ps3) via a more general definition of rotation, in which hyperbolic are allowed in addition to circular (ordinary trig) rotations. Re. relativity, Lorentz transforms are a certain hyperbolic rotation.

It turns out that a 3-pseudo-sphere (ps3) cannot be embedded in E4.The reason for this is described below. At this point, intuitions begin to fail abruptly — in particular, don’t (try to) picture (ps3) as a hyperbolic surface in E4, because such a thing does not exist.

However, (ps3) can be embedded in pE4 (M4), as a “hyperbolic surface in M4” — whatever that might look like.

Re. “whatever that might look like”, visualizations are another helpful tool for understanding geometries. This is typically done by taking some kind projection or slice of the space that suppresses one or more dimensions. Animation is another possibility, as your article enjoyably demonstrates.

Returning to the embedding of (ps3) in M4, the metric of M4 (pE4) is

(M4c) ds² = –du² + dx² + dy² + dz²

and in spherical coordinates of the “spatial” components,

(M4s) ds² = –du² + dr² + r² (dw₁² + sin²w₁ dw₂²)

Note

a)  4 coordinates (u,x,y,z) or (u,r,w₁,w₂)

b)  If one identifies u with t, this is the metric of “flat” spacetime, or special relativity, or Minkowski/Lorentz. One of the principles of general relativity is that the tangent spacetime at every event in spacetime is flat (M4). Just as any 2D spatial surface is everywhere locally Euclidean, spacetime is everywhere locally Lorentzian. Also, physical spacetime is a 4D surface that we are confined to deal with purely intrinsically, since we have no access to any supposed embedding space of it.

c)  In flat spacetime, space is flat

• The spatial part of (M4c) is the metric (E3c)
• The spatial part of (M4s) is the metric (E3s). The rotational functions are all circular and none hyperbolic as is required for spatial flatness.

d)  The hyperbolic nature of M4 thus necessarily involves the negative fourth component. This is reflected in the fact that Lorentz transformations are hyperbolic rotations with respect to velocities.

For comparison, the metric for E4 is

(E4c)  ds² = du² + dx² + dy² + dz²

(E4s)  ds² = dr² + r² [dw₁² + sin²w₁  (dw₂² + sin²w₂ dw₃³)]

It is useful to note that (E4s) is dr² plus the metric of the 3-sphere, and that this relationship holds in general for the metrics of EN and (N-1)-sphere.

Now, the metric of the 3-pseudo-sphere (ps3) is

(ps3s) ds² = r² [dw₁² + sinh²w₁ (dw₂² + sin2² dw₃²)]

Note

a)  Fixed r, 3 dof

b)  Differs from the 3-sphere part of (E4s) by the sinh(w₁) instead of sin(w₁). This will be shown to be the metric of a 3-hyperboloid (in M4) so the hyperbolic function is expected.

The embedding of (ps3) in M4 is achieved by the transform equations

(TM4)

x = r sinh w₁ sin w₂ cos w₃

y = r sinh w₁ sin w₂ sin w₃

z = r sinh w₁ cos w₂

u = r cosh w₁

The embedding of (ps3) in M4 by (TM4) is shown by taking the differentials of (TM4) — without dr’s since we’re regarding (TM4) as an embedding from 3 angles to 4 cartesians — and substituting into the metric (M4c) to obtain the metric (ps3s).

Squaring the (TM4) equations yields

        u² – x² – y² – z² = r²

which shows that

(ps3) –u² + x² + y² + z² = r²

is a (“two sheet”) 3-hyperboloid in M4.

The reason (ps3) is not embeddable in E4 (or higher) is that as the “radial coordinate” w₁ increases from 0 to inf, the circumference of the corresponding 2-spheres increase as sinh(w₁). Since sinh(w₁) > w₁ for all w₁, the circumferences increase faster than in fat space.

Since (ps3) is a 3-surface, the question remains what “the volume of (ps3)” means. For this Case 1, I presume you’re thinking of the volume of the embedding space M4, enclosed by he surface of (ps3), in analogy with the embedding volume of E3 enclosed by the surface of a 2D Euclidean sphere. However,

• The 3-surface (ps3) is open and doesn’t enclose a region of M4
• The 3-volume of the entire 3-surface (ps3) is infinite

First let’s examine the 3-volume of the entire surface (ps3)

V3 = ∫∫∫ (r dw₁) (r sinh w₁ dw₂) (r sinh w₁ sin w₂ dw₃)

   = ∫∫∫ r³ sinh²w₁ sin w₂ dw₁ dw₂ dw₃

   = ∫∫ r³ sin w₂ dw₂ dw₃ ∫ sinh²w₁ dw₁

   = r² ∫[0,π]dw₃ ∫[0,2π]dw₂ ∫[0,inf]sinh²w₁ dw₁

and

∫[0,inf]sinh²w₁ dw₁ = inf

The corresponding (spacetime) 4-volume

V4 = V3 ∫ dt

is therefore also infinite for any t.

It’s not surprising that the 3-hyperbola given by equation (ps3) is unbounded and hence has infinite 3-volume. The equation doesn’t bound or limit the surface in exactly the same way that the 1D hyperbola curve

y² – x² = r²

doesn’t bound or limit the curve. Unlike the 1D circle, the equation doesn’t define a curve of finite length, nor does it enclose a finite region of E2.

We have embedded (ps3) as a 3-hyperboloid in M4, ie we found the locus of points in M4 that satisfy (ps3), took the 3-integral over the region of M4 defined by (ps3), and added the fourth component independently to get the 4-volume. The crucial point is that the constraint (ps3) doesn’t restrict the surface from being unbounded.

Comparing this integral to the finite volume 4-rectangle of M4 in the first section: A cube is a closed surface that encloses a region of its embedding space. A sphere is rotationally symmetric, and is open or closed depending on whether or not a hyperbolic rotation is involved.

Case 1 Finite Variants

One could alter the (ps3) geometry to have finite 3-volume in at least a couple different ways:

1. Add additional constraints that define a finite portion of (ps3). Note that the condition
   
   –u² + x² + y² + z² <= r²
   
   does restrict the space, but not to a finite portion of it (one could think of it as the set of all such 3-hyperbolas up to r², each of which is unbounded). So it seems that any additional constraint that specifies a finite portion of (ps3) would no longer be the pseudo-sphere but rather only a portion of it.
   
   
2. Add a global topology to the 3-space that would make the entire surface closed and hence finite. Metrics are local and intrinsic, and say nothing about the global topology. The prototypical idea for this is flat E2 rolled into a cylinder in one (both) directions making a cylinder (torus), thus making x (and y) cyclic. The extrinsic (3D) surface is no longer flat, but it is still intrinsically flat (a 2D inhabitant detects no curvature anywhere, but can recognize global repetition). This surface has the combination of being flat, having no boundaries, and yet being closed.

Global topological closure is fundamentally very different from selecting a portion of a surface. A global topology is required in addition to a metric to fully specify a space. The implicit or default specification of an open global topology that is presumed in the absence of an explicit specification is in reality an arbitrary choice.

Physics

4D geometries can be interpreted purely mathematically or also physically, by either regarding the fourth coordinate simply as an additional mathematical index or by identifying it with time.

Cases 1 2 and 3 form a natural family in terms of both mathematics and physics (which is relativistic). Re. the latter, they are the 3 variants of the “isotropic and homogenous” cosmology, with constant spatial curvature. Case 1 is the open universe of negative constant curvature, Case 2 is the closed universe of positive constant curvature, and Case 3 is the flat universe that divides them. These 3 cosmologies are all of the “expanding universe” type, with the famous analogy of “pennies on a balloon”. The pennies represent objects of galactic scale which themselves do not expand (the length of a meter stick remains fixed within them), but the space between them expands as the balloon is blown up. The cosmology of Case 4 is described there.

The spatial 3-surfaces of constant curvature are “hypersurfaces of homogeneity”, with one such surface for each moment of time. The full spacetime metrics for the three cases may be written with the same form

(STI) dst² = -dt² + a²(t) ds²

           = -dt² + a²(t) [ dw₁² + F²(dw₂² + sin²w₂ dw₃²) ]

where a(t) is the “relative expansion rate” that varies with t, and “STI” stands for “space-time isotropic” (isotropy implies homogeneity but not the converse), and ds² is the spatial component which is determined by the factor F,

F = sinh w₁ for negative constant curvature – Case 1

F =      w₁ for zero     constant curvature - Case 3

F = sin  w₁ for positive constant curvature – Case 2

In each case, the 4-volume is

V4 = ∫∫∫∫ = V3 ∫ dt

Thus the full spacetime metric of Case 1, with (ps3) 3-space, is given by

F    = sinh w₁

a(t) = r

so that

dst² = -dt² + r² [ dw₁² + sinh²w₁ (dw₂² + sin²w₂ dw₃²) ]

Cases 1 2 and 3 are covered in most any book on general relativity, but MTW p725f is exceptionally clear. Case 4 is compactly described in MTW p745, and in wonderful detail in Rindler p185f.

Case 2

This is not the case you ask about, but I suspect your intuition of finite 4-volume was based on this case.

This is like Case 1, but with the 3-surface having positive constant curvature instead of negative, and it is embeddable in E4 rather than M4. Instead of the Case 1 pseudo-sphere (ps3), the surface is a 3-sphere

(s3) u² + x² + y² + z² =  r²

The metric of the 3-sphere is

(s3s) ds² = r² [dw₁² + sin²w₁ (dw₂² + sin²w₂ dw₃³)]

The metric of E4 from above

(E4c)  ds² = du² + dx² + dy² + dz²

The 3-surface (s3) is embeddable in E4 via

(TE4)

x = r sin w₁ sin w₂ cos w₃

y = r sin w₁ sin w₂ sin w₃

z = r sin w₁ cos w₂

u = r cos w₁

which differs from (TM4) of Case 1 by having all circular and no hyperbolic functions.

The embedding is shown by differentiating (TE4) and plugging into (E4c) to obtain (s3s). Squaring (TE4) and plugging into (s3) yields r², showing that the surface (s3) embedded in E4 is a 3-sphere.

The 3-volume of the 3-sphere is finite (for finite r), since this 3-surface is closed. It is (s3,3) from the first section

V3 = (s3,3)

   = ∫∫∫ (r dw₁)(r sinw₁ dw₂)(r sinw₁ sinw₂ dw₃)

   = ∫∫∫ r³ sin²w₁ sinw₂ dw₁ dw₂ dw₃

   = 2 π² r³

The 4-volume with this spatial 3-surface is

V4 = V3 ∫ dt

   = 2 π² r³ t

which is finite for finite r and t.

Re. physics, the full spacetime metric

      dst² = -dt² + r²    [dw₁² + sin²w₁ (dw₂² + sin²w₂ dw₃²)]

is obtained from our cosmological metric

(STI) dst² = -dt² + a²(t) ds²

           = -dt² + a²(t) [dw₁² +     F² (dw₂² + sin²w₂ dw₃²)]

by the substitutions

F    = sin w₁

a(t) = r

Case 3

This is the zero curvature case that completes the family with Cases 1 and 2 and is interesting in that it yields M4 itself as the 4-metric.

The zero curvature 3-surface is simply E3 space

(E3s) dr² + r²(dw₁² + sin²w₁ dw₂²)

The transform equations is the mapping (not embedding) between cartesian and spherical

(TE3)

x = r sin w₁ cos w₂

y = r sin w₁ sin w₂

z = r cos w₁

which simply show that (E3s) is E3.

The 3-volume of the entire surface is

V3 = ∫∫∫ (dr)(r sin w₁ dw₂)(r dw₁)

   = ∫∫∫ (r² sin w₁ dw₂ dw₁) ∫[0,inf]dr

   = inf

which is the integral of all of E3 space. Like Case 1 it is unbounded and hence infinite. The infinity comes from the [0, inf] range of the radial variable, r here and w₁ in Case 1.

Thus of Cases 1 2 and 3 only the positive curvature Case 2 has finite 3-volume of the entire 3-surface.

The 4-volume with this 3-volume is the 4-volume of all of M4

V4 = V3 ∫ dt = inf

which is infinite for any t.

Re. physics, the spacetime metric for this case is that of M4 itself

(M4s) dst² = -dt² + dr² + r²(dw₂² + sin²w₂ dw₃²)

(where the two angles end up being named w₂ and w₃)

which is obtained from the cosmological metric

(STI) dst² = -dt² + a²(t)[dw₁² + F²(dw₂² + sin²w₂ dw₃²)]

by the substitutions

F       = w₁

a(t) w₁ = r

As in Case 1, the 3-volume of the entire 3-surface is infinite. However, since cartesian coordinates are natural in flat space, the toroidal topological closure described above is done very easily here by matching boundaries to create the identifications

x = x+Δx,  y = y+Δy, z = z+Δz

Thus the integral of the entire surface can thereby made finite,

∫∫∫ = Δx · Δy · Δz

which is the same as the corresponding cube volume.

Case 4

In this case we begin with a 4D surface, embeddable in 5D, so that the entire 4-volume integral may be performed on the 4D surface. There are two cases, one with negative constant curvature and the other with positive constant curvature.

Re. physics, these 4-surfaces are isometric to the de Sitter metric, with positive “cosmological constant” for the negative curvature case and negative cosmological constant for the positive curvature case. These cases are called the de Sitter and the anti-de Sitter universes (I think I may have seen the terms de Sitter / anti-de Sitter reversed in some of the literature). The de Sitter cosmology is covered in virtually any book on general relativity (although not necessarily under that name), but the embedding of each case is spelled out in Rindler in unusual detail.

The first case is for a 4-pseudo-sphere, which has negative constant curvature,

(ps4)       –  u² +  w² +  x² +  y² +  z² = r²

The surface (ps4) is embeddable in M5 (or pE5)

(M5)  ds² = – du² + dw² + dx² + dy² + dz²

The pseudo-sphere (ps4) seems like a natural possibility for the definition of a 4D-sphere in a space related to M4. I haven’t carried out the integral, but I would expect the 4-volume of this entire surface to be infinite, since (ps4) defines an unbounded surface in a directly analogous manner to Case 1. But such spaces are full of surprises…

The second case is for a 4-pseudo-pseudo-sphere, which has positive constant curvature,

(pps4)       –  u² –  w² +  x² +  y² +  z² = r²

The surface (pps4) is embeddable in pseudo-M5 (or ppE5)

(pM5)  ds² = – du² – dw² + dx² + dy² + dz²

For the embedding transform equations, see Rindler.

Again I haven’t carried out the integral, but I would expect the 4-volume of this entire surface to be finite. But this positive curvature case (pps4) is more remote than this negative curvature case (ps4) from the sphere equation you cite (just as the positive curvature Case 2 (s3) is more remote than the negative curvature Case 1 (ps3)).

References

MTW:  Gravitation, by Charles Misner, Kip Thorne, and John Archibald Wheeler.

W.H.Freeman and Company, 1973.

Written by major contributors to the field. Affectionately known as “the Bible” or “the phone book” due to its size. Encyclopedic, conversational, filled with pictures, metaphors, stories, history, etc. In differential geometry, an increasingly abstract series of mathematical concepts and notations have developed, and rather than choosing one, MTW both explains these languages and discusses most topics in term of all of them in parallel, as well as graphically in pictures.

Rindler: Essential Relativity: Special, General, and Cosmological, by Wolfgang Rindler.

Springer, Revised Second Edition, 1977.

An expository gem. Apparently not widely read, but I can’t imagine why.

I own only a few introductory GR books in addition to the above.

A First Course in General Relativity, by Bernard F. Schutz.

Cambridge University Press, 1990.

Highly recommended, along with Rindler.

The Geometry of Spacetime: An Introduction to Special and General Relativity, by James J Callahan

Springer, 2000

This more recent intro lays out the full mathematical framework of parameterizations, mappings, and so on, which I’ve found sometimes useful in straightening out my thoughts. Otherwise not a favorite.