Four-vector

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In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (1/2,1/2) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame).

Four-vectors describe, for instance, position x^μ in spacetime modeled as Minkowski space, a particle's four-momentum p^μ, the amplitude of the electromagnetic four-potential A^μ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra.

The Lorentz group may be represented by 4×4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by

$${\displaystyle X^{\prime }=\mathrm{\Lambda }X,}$$

(matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors x_μ, p_μ and A_μ(x). These transform according to the rule

$${\displaystyle X^{\prime }={\left({\mathrm{\Lambda }}^{-1}\right)}^{\text{T}}X,}$$

where ^T denotes the matrix transpose. This rule is different from the above rule. It corresponds to the dual representation of the standard representation. However, for the Lorentz group the dual of any representation is equivalent to the original representation. Thus the objects with covariant indices are four-vectors as well.

For an example of a well-behaved four-component object in special relativity that is not a four-vector, see bispinor. It is similarly defined, the difference being that the transformation rule under Lorentz transformations is given by a representation other than the standard representation. In this case, the rule reads X′ = Π(Λ)X, where Π(Λ) is a 4×4 matrix other than Λ. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars, spinors, tensors and spinor-tensors.

The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.

Notation

The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors, capital bold for four dimensional vectors (except for the four-gradient), and tensor index notation.

Four-vector algebra

Four-vectors in a real-valued basis

A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:

$${\displaystyle \begin{array}{rl}\mathbf{A}& =\left(A^0,A^1,A^2,A^3\right)\\ & =A^0{\mathbf{E}}_0+A^1{\mathbf{E}}_1+A^2{\mathbf{E}}_2+A^3{\mathbf{E}}_3\\ & =A^0{\mathbf{E}}_0+A^i{\mathbf{E}}_i\\ & =A^{\alpha }{\mathbf{E}}_{\alpha }\\ & =A^{\mu }\end{array}}$$

where in the last form the magnitude component and basis vector have been combined to a single element.

The upper indices indicate contravariant components. Here the standard convention is that Latin indices take values for spatial components, so that i = 1, 2, 3, and Greek indices take values for space and time components, so α = 0, 1, 2, 3, used with the summation convention. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or raising and lowering indices.

In special relativity, the spacelike basis E₁, E₂, E₃ and components A¹, A², A³ are often Cartesian basis and components:

$${\displaystyle \begin{array}{rl}\mathbf{A}& =\left(A_t,A_x,A_y,A_z\right)\\ & =A_t{\mathbf{E}}_t+A_x{\mathbf{E}}_x+A_y{\mathbf{E}}_y+A_z{\mathbf{E}}_z\end{array}}$$

although, of course, any other basis and components may be used, such as spherical polar coordinates

$${\displaystyle \begin{array}{rl}\mathbf{A}& =\left(A_t,A_r,A_{\theta },A_{\varphi }\right)\\ & =A_t{\mathbf{E}}_t+A_r{\mathbf{E}}_r+A_{\theta }{\mathbf{E}}_{\theta }+A_{\varphi }{\mathbf{E}}_{\varphi }\end{array}}$$

or cylindrical polar coordinates,

$${\displaystyle \begin{array}{rl}\mathbf{A}& =(A_t,A_r,A_{\theta },A_z)\\ & =A_t{\mathbf{E}}_t+A_r{\mathbf{E}}_r+A_{\theta }{\mathbf{E}}_{\theta }+A_z{\mathbf{E}}_z\end{array}}$$

or any other orthogonal coordinates, or even general curvilinear coordinates. Note the coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of Minkowski diagram (also called spacetime diagram). In this article, four-vectors will be referred to simply as vectors.

It is also customary to represent the bases by column vectors:

$${\displaystyle {\mathbf{E}}_0=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),{\mathbf{E}}_1=\left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right),{\mathbf{E}}_2=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),{\mathbf{E}}_3=\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right)}$$

so that:

$${\displaystyle \mathbf{A}=\left(\begin{array}{c}{A}^0\\ A^1\\ A^2\\ A^3\end{array}\right)}$$

The relation between the covariant and contravariant coordinates is through the Minkowski metric tensor (referred to as the metric), η which raises and lowers indices as follows:

$${\displaystyle A_{\mu }={\eta }_{\mu \nu }{A}^{\nu },}$$

and in various equivalent notations the covariant components are:

$${\displaystyle \begin{array}{rl}\mathbf{A}& =(A_0,A_1,A_2,A_3)\\ & =A_0{\mathbf{E}}^0+A_1{\mathbf{E}}^1+A_2{\mathbf{E}}^2+A_3{\mathbf{E}}^3\\ & =A_0{\mathbf{E}}^0+A_i{\mathbf{E}}^i\\ & =A_{\alpha }{\mathbf{E}}^{\alpha }\end{array}}$$

where the lowered index indicates it to be covariant. Often the metric is diagonal, as is the case for orthogonal coordinates (see line element), but not in general curvilinear coordinates.

The bases can be represented by row vectors:

$${\displaystyle {\mathbf{E}}^0=\left(\begin{array}{cccc}1& 0& 0& 0\end{array}\right),{\mathbf{E}}^1=\left(\begin{array}{cccc}0& 1& 0& 0\end{array}\right),{\mathbf{E}}^2=\left(\begin{array}{cccc}0& 0& 1& 0\end{array}\right),{\mathbf{E}}^3=\left(\begin{array}{cccc}0& 0& 0& 1\end{array}\right)}$$

so that:

$${\displaystyle \mathbf{A}=\left(\begin{array}{cccc}{A}_0& A_1& A_2& A_3\end{array}\right)}$$

The motivation for the above conventions are that the inner product is a scalar, see below for details.

Lorentz transformation

Main article: Lorentz transformation

Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ:

$${\displaystyle {\mathbf{A}}^{\prime }=\mathbf{\Lambda }\mathbf{A}}$$

In index notation, the contravariant and covariant components transform according to, respectively:

$${\displaystyle {A^{\prime }}^{\mu }={\mathrm{\Lambda }}^{\mu }{}_{\nu }{A}^{\nu },{A^{\prime }}_{\mu }={\mathrm{\Lambda }}_{\mu }{}^{\nu }{A}_{\nu }}$$

in which the matrix Λ has components Λ^μ_ν in row μ and column ν, and the inverse matrix Λ⁻¹ has components Λ_μ^ν in row μ and column ν.

For background on the nature of this transformation definition, see tensor. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity.

Pure rotations about an arbitrary axis

For two frames rotated by a fixed angle θ about an axis defined by the unit vector:

$${\displaystyle \hat{\mathbf{n}}=\left({\hat{n}}_1,{\hat{n}}_2,{\hat{n}}_3\right),}$$

without any boosts, the matrix Λ has components given by:

$${\displaystyle \begin{array}{rl}{\mathrm{\Lambda }}_{00}& =1\\ {\mathrm{\Lambda }}_{0i}={\mathrm{\Lambda }}_{i0}& =0\\ {\mathrm{\Lambda }}_{ij}& =\left({\delta }_{ij}-{\hat{n}}_i{\hat{n}}_j\right)\cos \theta -{\epsilon }_{ijk}{\hat{n}}_k\sin \theta +{\hat{n}}_i{\hat{n}}_j\end{array}}$$

where δ_ij is the Kronecker delta, and ε_ijk is the three-dimensional Levi-Civita symbol. The spacelike components of four-vectors are rotated, while the timelike components remain unchanged.

For the case of rotations about the z-axis only, the spacelike part of the Lorentz matrix reduces to the rotation matrix about the z-axis:

$${\displaystyle \left(\begin{array}{c}{A^{\prime }}^0\\ {A^{\prime }}^1\\ {A^{\prime }}^2\\ {A^{\prime }}^3\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}{A}^0\\ A^1\\ A^2\\ A^3\end{array}\right).}$$

Pure boosts in an arbitrary direction

Standard configuration of coordinate systems for a Lorentz boost in the x-direction.

For two frames moving at constant relative three-velocity v (not four-velocity, see below), it is convenient to denote and define the relative velocity in units of c by:

$${\displaystyle \mathit{\beta }=({\beta }_1,{\beta }_2,{\beta }_3)=\frac{1}{c}(v_1,v_2,v_3)=\frac{1}{c}\mathbf{v}.}$$

Then without rotations, the matrix Λ has components given by:

$${\displaystyle \begin{array}{rl}{\mathrm{\Lambda }}_{00}& =\gamma ,\\ {\mathrm{\Lambda }}_{0i}={\mathrm{\Lambda }}_{i0}& =-\gamma {\beta }_i,\\ {\mathrm{\Lambda }}_{ij}={\mathrm{\Lambda }}_{ji}& =(\gamma -1)\frac{{\beta }_i{\beta }_j}{{\beta }^2}+{\delta }_{ij}=(\gamma -1)\frac{v_i{v}_j}{v^2}+{\delta }_{ij},\end{array}}$$

where the Lorentz factor is defined by:

$${\displaystyle \gamma =\frac{1}{\sqrt{1-\mathit{\beta }\cdot \mathit{\beta }}},}$$

and δ_ij is the Kronecker delta. Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.

For the case of a boost in the x-direction only, the matrix reduces to

$${\displaystyle \left(\begin{array}{c}{A}^{\prime 0}\\ A^{\prime 1}\\ A^{\prime 2}\\ A^{\prime 3}\end{array}\right)=\left(\begin{array}{cccc}\cosh \varphi & -\sinh \varphi & 0& 0\\ -\sinh \varphi & \cosh \varphi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}{A}^0\\ A^1\\ A^2\\ A^3\end{array}\right)}$$

Where the rapidity ϕ expression has been used, written in terms of the hyperbolic functions:

$${\displaystyle \gamma =\cosh \varphi }$$

This Lorentz matrix illustrates the boost to be a hyperbolic rotation in four dimensional spacetime, analogous to the circular rotation above in three-dimensional space.

Properties

Linearity

Four-vectors have the same linearity properties as Euclidean vectors in three dimensions. They can be added in the usual entrywise way:

$${\displaystyle \mathbf{A}+\mathbf{B}=\left(A^0,A^1,A^2,A^3\right)+\left(B^0,B^1,B^2,B^3\right)=\left(A^0+B^0,A^1+B^1,A^2+B^2,A^3+B^3\right)}$$

and similarly scalar multiplication by a scalar λ is defined entrywise by:

$${\displaystyle \lambda \mathbf{A}=\lambda \left(A^0,A^1,A^2,A^3\right)=\left(\lambda A^0,\lambda A^1,\lambda A^2,\lambda A^3\right)}$$

Then subtraction is the inverse operation of addition, defined entrywise by:

$${\displaystyle \mathbf{A}+(-1)\mathbf{B}=\left(A^0,A^1,A^2,A^3\right)+(-1)\left(B^0,B^1,B^2,B^3\right)=\left(A^0-B^0,A^1-B^1,A^2-B^2,A^3-B^3\right)}$$

Minkowski tensor

See also: spacetime interval

Applying the Minkowski tensor η_μν to two four-vectors A and B, writing the result in dot product notation, we have, using Einstein notation:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A^{\mu }{\eta }_{\mu \nu }{B}^{\nu }}$$

It is convenient to rewrite the definition in matrix form:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=\left(\begin{array}{cccc}{A}^0& A^1& A^2& A^3\end{array}\right)\left(\begin{array}{cccc}{\eta }_{00}& {\eta }_{01}& {\eta }_{02}& {\eta }_{03}\\ {\eta }_{10}& {\eta }_{11}& {\eta }_{12}& {\eta }_{13}\\ {\eta }_{20}& {\eta }_{21}& {\eta }_{22}& {\eta }_{23}\\ {\eta }_{30}& {\eta }_{31}& {\eta }_{32}& {\eta }_{33}\end{array}\right)\left(\begin{array}{c}{B}^0\\ B^1\\ B^2\\ B^3\end{array}\right)}$$

in which case η_μν above is the entry in row μ and column ν of the Minkowski metric as a square matrix. The Minkowski metric is not a Euclidean metric, because it is indefinite (see metric signature). A number of other expressions can be used because the metric tensor can raise and lower the components of A or B. For contra/co-variant components of A and co/contra-variant components of B, we have:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A^{\mu }{\eta }_{\mu \nu }{B}^{\nu }=A_{\nu }{B}^{\nu }=A^{\mu }{B}_{\mu }}$$

so in the matrix notation:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=\left(\begin{array}{cccc}{A}_0& A_1& A_2& A_3\end{array}\right)\left(\begin{array}{c}{B}^0\\ B^1\\ B^2\\ B^3\end{array}\right)=\left(\begin{array}{cccc}{B}_0& B_1& B_2& B_3\end{array}\right)\left(\begin{array}{c}{A}^0\\ A^1\\ A^2\\ A^3\end{array}\right)}$$

while for A and B each in covariant components:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A_{\mu }{\eta }^{\mu \nu }{B}_{\nu }}$$

with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector A with itself we get:

$${\displaystyle \mathbf{A}\cdot \mathbf{A}=A^{\mu }{\eta }_{\mu \nu }{A}^{\nu }}$$

which, depending on the case, may be considered the square, or its negative, of the length of the vector.

Following are two common choices for the metric tensor in the standard basis (essentially Cartesian coordinates). If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.

Standard basis, (+−−−) signature

In the (+−−−) metric signature, evaluating the summation over indices gives:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A^0{B}^0-A^1{B}^1-A^2{B}^2-A^3{B}^3}$$

while in matrix form:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=\left(\begin{array}{cccc}{A}^0& A^1& A^2& A^3\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)\left(\begin{array}{c}{B}^0\\ B^1\\ B^2\\ B^3\end{array}\right)}$$

It is a recurring theme in special relativity to take the expression

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A^0{B}^0-A^1{B}^1-A^2{B}^2-A^3{B}^3=C}$$

in one reference frame, where C is the value of the inner product in this frame, and:

$${\displaystyle {\mathbf{A}}^{\prime }\cdot {\mathbf{B}}^{\prime }={A^{\prime }}^0{B^{\prime }}^0-{A^{\prime }}^1{B^{\prime }}^1-{A^{\prime }}^2{B^{\prime }}^2-{A^{\prime }}^3{B^{\prime }}^3=C^{\prime }}$$

in another frame, in which C′ is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}={\mathbf{A}}^{\prime }\cdot {\mathbf{B}}^{\prime }}$$

that is:

$${\displaystyle C=A^0{B}^0-A^1{B}^1-A^2{B}^2-A^3{B}^3={A^{\prime }}^0{B^{\prime }}^0-{A^{\prime }}^1{B^{\prime }}^1-{A^{\prime }}^2{B^{\prime }}^2-{A^{\prime }}^3{B^{\prime }}^3}$$

Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "conservation law", but there is no "conservation" involved. The primary significance of the Minkowski inner product is that for any two four-vectors, its value is invariant for all observers; a change of coordinates does not result in a change in value of the inner product. The components of the four-vectors change from one frame to another; A and A′ are connected by a Lorentz transformation, and similarly for B and B′, although the inner products are the same in all frames. Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations. A particular example is with energy and momentum in the energy-momentum relation derived from the four-momentum vector (see also below).

In this signature we have:

$${\displaystyle \mathbf{A}\cdot \mathbf{A}={\left(A^0\right)}^2-{\left(A^1\right)}^2-{\left(A^2\right)}^2-{\left(A^3\right)}^2}$$

With the signature (+−−−), four-vectors may be classified as either spacelike if ${\displaystyle \mathbf{A}\cdot \mathbf{A}<0}$, timelike if ${\displaystyle \mathbf{A}\cdot \mathbf{A}>0}$, and null vectors if ${\displaystyle \mathbf{A}\cdot \mathbf{A}=0}$.

Standard basis, (−+++) signature

Some authors define η with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=-A^0{B}^0+A^1{B}^1+A^2{B}^2+A^3{B}^3}$$

while the matrix form is:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=\left(\begin{array}{cccc}{A}^0& A^1& A^2& A^3\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}{B}^0\\ B^1\\ B^2\\ B^3\end{array}\right)}$$

Note that in this case, in one frame:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=-A^0{B}^0+A^1{B}^1+A^2{B}^2+A^3{B}^3=-C}$$

while in another:

$${\displaystyle {\mathbf{A}}^{\prime }\cdot {\mathbf{B}}^{\prime }=-{A^{\prime }}^0{B^{\prime }}^0+{A^{\prime }}^1{B^{\prime }}^1+{A^{\prime }}^2{B^{\prime }}^2+{A^{\prime }}^3{B^{\prime }}^3=-C^{\prime }}$$

so that:

$${\displaystyle -C=-A^0{B}^0+A^1{B}^1+A^2{B}^2+A^3{B}^3=-{A^{\prime }}^0{B^{\prime }}^0+{A^{\prime }}^1{B^{\prime }}^1+{A^{\prime }}^2{B^{\prime }}^2+{A^{\prime }}^3{B^{\prime }}^3}$$

which is equivalent to the above expression for C in terms of A and B. Either convention will work. With the Minkowski metric defined in the two ways above, the only difference between covariant and contravariant four-vector components are signs, therefore the signs depend on which sign convention is used.

We have:

$${\displaystyle \mathbf{A}\cdot \mathbf{A}=-{\left(A^0\right)}^2+{\left(A^1\right)}^2+{\left(A^2\right)}^2+{\left(A^3\right)}^2}$$

With the signature (−+++), four-vectors may be classified as either spacelike if ${\displaystyle \mathbf{A}\cdot \mathbf{A}>0}$, timelike if ${\displaystyle \mathbf{A}\cdot \mathbf{A}<0}$, and null if ${\displaystyle \mathbf{A}\cdot \mathbf{A}=0}$.

Dual vectors

Applying the Minkowski tensor is often expressed as the effect of the dual vector of one vector on the other:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A^{\ast }(\mathbf{B})=A{}_{\nu }{B}^{\nu }.}$$

Here the A_νs are the components of the dual vector A* of A in the dual basis and called the covariant coordinates of A, while the original A^ν components are called the contravariant coordinates.

Four-vector calculus

Derivatives and differentials

In special relativity (but not general relativity), the derivative of a four-vector with respect to a scalar λ (invariant) is itself a four-vector. It is also useful to take the differential of the four-vector, dA and divide it by the differential of the scalar, dλ:

$${\displaystyle \underset{\text{differential}}{d\mathbf{A}}=\underset{\text{derivative}}{\frac{d\mathbf{A}}{d\lambda }}\underset{\text{differential}}{d\lambda }}$$

where the contravariant components are:

$${\displaystyle d\mathbf{A}=\left(d{A}^0,d{A}^1,d{A}^2,d{A}^3\right)}$$

while the covariant components are:

$${\displaystyle d\mathbf{A}=\left(d{A}_0,d{A}_1,d{A}_2,d{A}_3\right)}$$

In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in proper time (see below).

Fundamental four-vectors

Four-position

A point in Minkowski space is a time and spatial position, called an "event", or sometimes the position four-vector or four-position or 4-position, described in some reference frame by a set of four coordinates:

$${\displaystyle \mathbf{R}=\left(ct,\mathbf{r}\right)}$$

where r is the three-dimensional space position vector. If r is a function of coordinate time t in the same frame, i.e. r = r(t), this corresponds to a sequence of events as t varies. The definition R⁰ = ct ensures that all the coordinates have the same units (of distance). These coordinates are the components of the position four-vector for the event.

The displacement four-vector is defined to be an "arrow" linking two events:

$${\displaystyle \mathrm{\Delta }\mathbf{R}=\left(c\mathrm{\Delta }t,\mathrm{\Delta }\mathbf{r}\right)}$$

For the differential four-position on a world line we have, using a norm notation:

$${\displaystyle \Vert d\mathbf{R}{\Vert }^2=\mathbf{d}\mathbf{R}\cdot \mathbf{d}\mathbf{R}=d{R}^{\mu }d{R}_{\mu }=c^{2}d{\tau }^2=d{s}^2,}$$

defining the differential line element ds and differential proper time increment dτ, but this "norm" is also:

$${\displaystyle \Vert d\mathbf{R}{\Vert }^2=(cdt{)}^2-d\mathbf{r}\cdot d\mathbf{r},}$$

so that:

$${\displaystyle (cd\tau {)}^2=(cdt{)}^2-d\mathbf{r}\cdot d\mathbf{r}.}$$

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time ${\displaystyle \tau }$. As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the coordinate time t of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (cdt)² to obtain:

$${\displaystyle {\left(\frac{cd\tau }{cdt}\right)}^2=1-\left(\frac{d\mathbf{r}}{cdt}\cdot \frac{d\mathbf{r}}{cdt}\right)=1-\frac{\mathbf{u}\cdot \mathbf{u}}{c^2}=\frac{1}{\gamma (\mathbf{u}{)}^2},}$$

where u = dr/dt is the coordinate 3-velocity of an object measured in the same frame as the coordinates x, y, z, and coordinate time t, and

$${\displaystyle \gamma (\mathbf{u})=\frac{1}{\sqrt{1-\frac{\mathbf{u}\cdot \mathbf{u}}{c^2}}}}$$

is the Lorentz factor. This provides a useful relation between the differentials in coordinate time and proper time:

$${\displaystyle dt=\gamma (\mathbf{u})d\tau .}$$

This relation can also be found from the time transformation in the Lorentz transformations.

Important four-vectors in relativity theory can be defined by applying this differential ${\displaystyle \frac{d}{d\tau }}$.

Four-gradient

Considering that partial derivatives are linear operators, one can form a four-gradient from the partial time derivative ∂/∂t and the spatial gradient ∇. Using the standard basis, in index and abbreviated notations, the contravariant components are:

$${\displaystyle \begin{array}{rl}\mathbf{\partial }& =\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_0},-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2},-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3}\right)\\ & =({\mathrm{\partial }}^0,-{\mathrm{\partial }}^1,-{\mathrm{\partial }}^2,-{\mathrm{\partial }}^3)\\ & ={\mathbf{E}}_0{\mathrm{\partial }}^0-{\mathbf{E}}_1{\mathrm{\partial }}^1-{\mathbf{E}}_2{\mathrm{\partial }}^2-{\mathbf{E}}_3{\mathrm{\partial }}^3\\ & ={\mathbf{E}}_0{\mathrm{\partial }}^0-{\mathbf{E}}_i{\mathrm{\partial }}^i\\ & ={\mathbf{E}}_{\alpha }{\mathrm{\partial }}^{\alpha }\\ & =\left(\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t},-\mathrm{\nabla }\right)\\ & =\left(\frac{{\mathrm{\partial }}_t}{c},-\mathrm{\nabla }\right)\\ & ={\mathbf{E}}_0\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\mathrm{\nabla }\end{array}}$$

Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector. The covariant components are:

$${\displaystyle \begin{array}{rl}\mathbf{\partial }& =\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^0},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^1},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^2},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^3}\right)\\ & =({\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)\\ & ={\mathbf{E}}^0{\mathrm{\partial }}_0+{\mathbf{E}}^1{\mathrm{\partial }}_1+{\mathbf{E}}^2{\mathrm{\partial }}_2+{\mathbf{E}}^3{\mathrm{\partial }}_3\\ & ={\mathbf{E}}^0{\mathrm{\partial }}_0+{\mathbf{E}}^i{\mathrm{\partial }}_i\\ & ={\mathbf{E}}^{\alpha }{\mathrm{\partial }}_{\alpha }\\ & =\left(\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t},\mathrm{\nabla }\right)\\ & =\left(\frac{{\mathrm{\partial }}_t}{c},\mathrm{\nabla }\right)\\ & ={\mathbf{E}}^0\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathrm{\nabla }\end{array}}$$

Since this is an operator, it doesn't have a "length", but evaluating the inner product of the operator with itself gives another operator:

$${\displaystyle {\mathrm{\partial }}^{\mu }{\mathrm{\partial }}_{\mu }=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2=\frac{{{\mathrm{\partial }}_t}^2}{c^2}-{\mathrm{\nabla }}^2}$$

called the D'Alembert operator.

Kinematics

Four-velocity

Main article: Four-velocity

The four-velocity of a particle is defined by:

$${\displaystyle \mathbf{U}=\frac{d\mathbf{X}}{d\tau }=\frac{d\mathbf{X}}{dt}\frac{dt}{d\tau }=\gamma (\mathbf{u})\left(c,\mathbf{u}\right),}$$

Geometrically, U is a normalized vector tangent to the world line of the particle. Using the differential of the four-position, the magnitude of the four-velocity can be obtained:

$${\displaystyle \Vert \mathbf{U}{\Vert }^2=U^{\mu }{U}_{\mu }=\frac{d{X}^{\mu }}{d\tau }\frac{d{X}_{\mu }}{d\tau }=\frac{d{X}^{\mu }d{X}_{\mu }}{d{\tau }^2}=c^2,}$$

in short, the magnitude of the four-velocity for any object is always a fixed constant:

$${\displaystyle \Vert \mathbf{U}{\Vert }^2=c^2}$$

The norm is also:

$${\displaystyle \Vert \mathbf{U}{\Vert }^2={\gamma (\mathbf{u})}^2\left(c^2-\mathbf{u}\cdot \mathbf{u}\right),}$$

so that:

$${\displaystyle c^2={\gamma (\mathbf{u})}^2\left(c^2-\mathbf{u}\cdot \mathbf{u}\right),}$$

which reduces to the definition of the Lorentz factor.

Units of four-velocity are m/s in SI and 1 in the geometrized unit system. Four-velocity is a contravariant vector.

Four-acceleration

The four-acceleration is given by:

$${\displaystyle \mathbf{A}=\frac{d\mathbf{U}}{d\tau }=\gamma (\mathbf{u})\left(\frac{d\gamma (\mathbf{u})}{dt}c,\frac{d\gamma (\mathbf{u})}{dt}\mathbf{u}+\gamma (\mathbf{u})\mathbf{a}\right).}$$

where a = du/dt is the coordinate 3-acceleration. Since the magnitude of U is a constant, the four acceleration is orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

$${\displaystyle \mathbf{A}\cdot \mathbf{U}=A^{\mu }{U}_{\mu }=\frac{d{U}^{\mu }}{d\tau }{U}_{\mu }=\frac{1}{2}\frac{d}{d\tau }\left(U^{\mu }{U}_{\mu }\right)=0}$$

which is true for all world lines. The geometric meaning of four-acceleration is the curvature vector of the world line in Minkowski space.

Dynamics

Four-momentum

For a massive particle of rest mass (or invariant mass) m₀, the four-momentum is given by:

$${\displaystyle \mathbf{P}=m_0\mathbf{U}=m_0\gamma (\mathbf{u})(c,\mathbf{u})=\left(\frac{E}{c},\mathbf{p}\right)}$$

where the total energy of the moving particle is:

$${\displaystyle E=\gamma (\mathbf{u})m_0{c}^2}$$

and the total relativistic momentum is:

$${\displaystyle \mathbf{p}=\gamma (\mathbf{u})m_0\mathbf{u}}$$

Taking the inner product of the four-momentum with itself:

$${\displaystyle \Vert \mathbf{P}{\Vert }^2=P^{\mu }{P}_{\mu }=m_0^2{U}^{\mu }{U}_{\mu }=m_0^2{c}^2}$$

and also:

$${\displaystyle \Vert \mathbf{P}{\Vert }^2=\frac{E^2}{c^2}-\mathbf{p}\cdot \mathbf{p}}$$

which leads to the energy–momentum relation:

$${\displaystyle E^2=c^2\mathbf{p}\cdot \mathbf{p}+{\left(m_0{c}^2\right)}^2.}$$

This last relation is useful relativistic mechanics, essential in relativistic quantum mechanics and relativistic quantum field theory, all with applications to particle physics.

Four-force

The four-force acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum in Newton's second law:

$${\displaystyle \mathbf{F}=\frac{d\mathbf{P}}{d\tau }=\gamma (\mathbf{u})\left(\frac{1}{c}\frac{dE}{dt},\frac{d\mathbf{p}}{dt}\right)=\gamma (\mathbf{u})\left(\frac{P}{c},\mathbf{f}\right)}$$

where P is the power transferred to move the particle, and f is the 3-force acting on the particle. For a particle of constant invariant mass m₀, this is equivalent to

$${\displaystyle \mathbf{F}=m_0\mathbf{A}=m_0\gamma (\mathbf{u})\left(\frac{d\gamma (\mathbf{u})}{dt}c,\left(\frac{d\gamma (\mathbf{u})}{dt}\mathbf{u}+\gamma (\mathbf{u})\mathbf{a}\right)\right)}$$

An invariant derived from the four-force is:

$${\displaystyle \mathbf{F}\cdot \mathbf{U}=F^{\mu }{U}_{\mu }=m_0{A}^{\mu }{U}_{\mu }=0}$$

from the above result.

Thermodynamics

See also: Relativistic heat conduction

Four-heat flux

The four-heat flux vector field, is essentially similar to the 3d heat flux vector field q, in the local frame of the fluid:

$${\displaystyle \mathbf{Q}=-k\mathbf{\partial }T=-k\left(\frac{1}{c}\frac{\mathrm{\partial }T}{\mathrm{\partial }t},\mathrm{\nabla }T\right)}$$

where T is absolute temperature and k is thermal conductivity.

Four-baryon number flux

The flux of baryons is:

$${\displaystyle \mathbf{S}=n\mathbf{U}}$$

where n is the number density of baryons in the local rest frame of the baryon fluid (positive values for baryons, negative for antibaryons), and U the four-velocity field (of the fluid) as above.

Four-entropy

The four-entropy vector is defined by:

$${\displaystyle \mathbf{s}=s\mathbf{S}+\frac{\mathbf{Q}}{T}}$$

where s is the entropy per baryon, and T the absolute temperature, in the local rest frame of the fluid.

Electromagnetism

Examples of four-vectors in electromagnetism include the following.

Four-current

The electromagnetic four-current (or more correctly a four-current density) is defined by

$${\displaystyle \mathbf{J}=\left(\rho c,\mathbf{j}\right)}$$

formed from the current density j and charge density ρ.

Four-potential

The electromagnetic four-potential (or more correctly a four-EM vector potential) defined by

$${\displaystyle \mathbf{A}=\left(\frac{\varphi }{c},\mathbf{a}\right)}$$

formed from the vector potential a and the scalar potential ϕ.

The four-potential is not uniquely determined, because it depends on a choice of gauge.

In the wave equation for the electromagnetic field:

• In vacuum,
  $${\displaystyle (\mathbf{\partial }\cdot \mathbf{\partial })\mathbf{A}=0}$$

  
• With a four-current source and using the Lorenz gauge condition ${\displaystyle (\mathbf{\partial }\cdot \mathbf{A})=0}$,
  $${\displaystyle (\mathbf{\partial }\cdot \mathbf{\partial })\mathbf{A}={\mu }_0\mathbf{J}}$$

  

Waves

Four-frequency

A photonic plane wave can be described by the four-frequency defined as

$${\displaystyle \mathbf{N}=\nu \left(1,\hat{\mathbf{n}}\right)}$$

where ν is the frequency of the wave and ${\displaystyle \hat{\mathbf{n}}}$ is a unit vector in the travel direction of the wave. Now:

$${\displaystyle \Vert \mathbf{N}\Vert =N^{\mu }{N}_{\mu }={\nu }^2\left(1-\hat{\mathbf{n}}\cdot \hat{\mathbf{n}}\right)=0}$$

so the four-frequency of a photon is always a null vector.

Four-wavevector

See also: De Broglie relation

The quantities reciprocal to time t and space r are the angular frequency ω and angular wave vector k, respectively. They form the components of the four-wavevector or wave four-vector:

$${\displaystyle \mathbf{K}=\left(\frac{\omega }{c},\overrightarrow{\mathbf{k}}\right)=\left(\frac{\omega }{c},\frac{\omega }{v_p}\hat{\mathbf{n}}\right).}$$

A wave packet of nearly monochromatic light can be described by:

$${\displaystyle \mathbf{K}=\frac{2\pi }{c}\mathbf{N}=\frac{2\pi }{c}\nu \left(1,\hat{\mathbf{n}}\right)=\frac{\omega }{c}\left(1,\hat{\mathbf{n}}\right).}$$

The de Broglie relations then showed that four-wavevector applied to matter waves as well as to light waves:

$${\displaystyle \mathbf{P}=\hslash \mathbf{K}=\left(\frac{E}{c},\overrightarrow{p}\right)=\hslash \left(\frac{\omega }{c},\overrightarrow{k}\right).}$$

yielding ${\displaystyle E=\hslash \omega }$ and ${\displaystyle \overrightarrow{p}=\hslash \overrightarrow{k}}$, where ħ is the Planck constant divided by 2π .

The square of the norm is:

$${\displaystyle \Vert \mathbf{K}{\Vert }^2=K^{\mu }{K}_{\mu }={\left(\frac{\omega }{c}\right)}^2-\mathbf{k}\cdot \mathbf{k},}$$

and by the de Broglie relation:

$${\displaystyle \Vert \mathbf{K}{\Vert }^2=\frac{1}{{\hslash }^2}\Vert \mathbf{P}{\Vert }^2={\left(\frac{m_{0}c}{\hslash }\right)}^2,}$$

we have the matter wave analogue of the energy–momentum relation:

$${\displaystyle {\left(\frac{\omega }{c}\right)}^2-\mathbf{k}\cdot \mathbf{k}={\left(\frac{m_{0}c}{\hslash }\right)}^2.}$$

Note that for massless particles, in which case m₀ = 0, we have:

$${\displaystyle {\left(\frac{\omega }{c}\right)}^2=\mathbf{k}\cdot \mathbf{k},}$$

or ‖k‖ = ω/c . Note this is consistent with the above case; for photons with a 3-wavevector of modulus ω / c , in the direction of wave propagation defined by the unit vector ${\displaystyle \hat{\mathbf{n}}.}$

Quantum theory

Four-probability current

In quantum mechanics, the four-probability current or probability four-current is analogous to the electromagnetic four-current:

$${\displaystyle \mathbf{J}=(\rho c,\mathbf{j})}$$

where ρ is the probability density function corresponding to the time component, and j is the probability current vector. In non-relativistic quantum mechanics, this current is always well defined because the expressions for density and current are positive definite and can admit a probability interpretation. In relativistic quantum mechanics and quantum field theory, it is not always possible to find a current, particularly when interactions are involved.

Replacing the energy by the energy operator and the momentum by the momentum operator in the four-momentum, one obtains the four-momentum operator, used in relativistic wave equations.

Four-spin

The four-spin of a particle is defined in the rest frame of a particle to be

$${\displaystyle \mathbf{S}=(0,\mathbf{s})}$$

where s is the spin pseudovector. In quantum mechanics, not all three components of this vector are simultaneously measurable, only one component is. The timelike component is zero in the particle's rest frame, but not in any other frame. This component can be found from an appropriate Lorentz transformation.

The norm squared is the (negative of the) magnitude squared of the spin, and according to quantum mechanics we have

$${\displaystyle \Vert \mathbf{S}{\Vert }^2=-|\mathbf{s}{|}^2=-{\hslash }^{2}s(s+1)}$$

This value is observable and quantized, with s the spin quantum number (not the magnitude of the spin vector).

Other formulations

Four-vectors in the algebra of physical space

A four-vector A can also be defined in using the Pauli matrices as a basis, again in various equivalent notations:

$${\displaystyle \begin{array}{rl}\mathbf{A}& =\left(A^0,A^1,A^2,A^3\right)\\ & =A^0{\mathit{\sigma }}_0+A^1{\mathit{\sigma }}_1+A^2{\mathit{\sigma }}_2+A^3{\mathit{\sigma }}_3\\ & =A^0{\mathit{\sigma }}_0+A^i{\mathit{\sigma }}_i\\ & =A^{\alpha }{\mathit{\sigma }}_{\alpha }\end{array}}$$

or explicitly:

$${\displaystyle \begin{array}{rl}\mathbf{A}& =A^0\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+A^1\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)+A^2\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)+A^3\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\\ & =\left(\begin{array}{cc}{A}^0+A^3& A^1-i{A}^2\\ A^1+i{A}^2& A^0-A^3\end{array}\right)\end{array}}$$

and in this formulation, the four-vector is represented as a Hermitian matrix (the matrix transpose and complex conjugate of the matrix leaves it unchanged), rather than a real-valued column or row vector. The determinant of the matrix is the modulus of the four-vector, so the determinant is an invariant:

$${\displaystyle \begin{array}{rl}|\mathbf{A}|& =\left|\begin{array}{cc}{A}^0+A^3& A^1-i{A}^2\\ A^1+i{A}^2& A^0-A^3\end{array}\right|\\ & =\left(A^0+A^3\right)\left(A^0-A^3\right)-\left(A^1-i{A}^2\right)\left(A^1+i{A}^2\right)\\ & ={\left(A^0\right)}^2-{\left(A^1\right)}^2-{\left(A^2\right)}^2-{\left(A^3\right)}^2\end{array}}$$

This idea of using the Pauli matrices as basis vectors is employed in the algebra of physical space, an example of a Clifford algebra.

Four-vectors in spacetime algebra

In spacetime algebra, another example of Clifford algebra, the gamma matrices can also form a basis. (They are also called the Dirac matrices, owing to their appearance in the Dirac equation). There is more than one way to express the gamma matrices, detailed in that main article.

The Feynman slash notation is a shorthand for a four-vector A contracted with the gamma matrices:

$${\displaystyle \mathbf{A}/=A_{\alpha }{\gamma }^{\alpha }=A_0{\gamma }^0+A_1{\gamma }^1+A_2{\gamma }^2+A_3{\gamma }^3}$$

The four-momentum contracted with the gamma matrices is an important case in relativistic quantum mechanics and relativistic quantum field theory. In the Dirac equation and other relativistic wave equations, terms of the form:

$${\displaystyle \mathbf{P}/=P_{\alpha }{\gamma }^{\alpha }=P_0{\gamma }^0+P_1{\gamma }^1+P_2{\gamma }^2+P_3{\gamma }^3={\displaystyle \frac{E}{c}}{\gamma }^0-p_x{\gamma }^1-p_y{\gamma }^2-p_z{\gamma }^3}$$

appear, in which the energy E and momentum components (p_x, p_y, p_z) are replaced by their respective operators.

Shape of the universe

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Shape_of_the_universe

The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as a continuous object. The spatial curvature is described by general relativity, which describes how spacetime is curved due to the effect of gravity. The spatial topology cannot be determined from its curvature, due to the fact that there exist locally indistinguishable spaces that may be endowed with different topological invariants.

Cosmologists distinguish between the observable universe and the entire universe, the former being a ball-shaped portion of the latter that can, in principle, be accessible by astronomical observations. Assuming the cosmological principle, the observable universe is similar from all contemporary vantage points, which allows cosmologists to discuss properties of the entire universe with only information from studying their observable universe. The main discussion in this context is whether the universe is finite, like the observable universe, or infinite.

Several potential topological and geometric properties of the universe need to be identified. Its topological characterization remains an open problem. Some of these properties are:

1. Boundedness (whether the universe is finite or infinite)
2. Flatness (zero curvature), hyperbolic (negative curvature), or spherical (positive curvature)
3. Connectivity: how the universe is put together as a manifold, i.e., a simply connected space or a multiply connected space.

There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite. Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one. For example, a multiply connected space may be flat and finite, as illustrated by the three-torus. Yet, in the case of simply connected spaces, flatness implies infinitude.

To this day, the exact shape of the universe remains a matter of debate in physical cosmology. In this regard, experimental data from various independent sources (WMAP, BOOMERanG, and Planck for example) confirm that the universe is flat with only a 0.4% margin of error. Yet, the issue of simple versus multiple connectivity has not yet been decided based on astronomical observation. On the other hand, any non-zero curvature is possible for a sufficiently large curved universe (analogously to how a small portion of a sphere can look flat). Theorists have been trying to construct a formal mathematical model of the shape of the universe relating connectivity, curvature and boundedness. In formal terms, this is a 3-manifold model corresponding to the spatial section (in comoving coordinates) of the four-dimensional spacetime of the universe. The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat, but the data is also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space, the multiply connected three-torus, and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by a 2-dimensional lattice).

Physical cosmology is based on the theory of General Relativity, a physical picture cast in terms of differential equations. Therefore, only the local geometric properties of the universe become theoretically accessible. Thus, Einstein's field equations determine only the local geometry but have absolutely no saying on the topology of the universe. At present, the only possibility to elucidate such global properties relies on observational data, especially the fluctuations (anisotropies) of the temperature gradient field of the Cosmic Microwave Background (CMB).

Shape of the observable universe

Main article: Observable universe

See also: Distance measures (cosmology)

As stated in the introduction, there are two aspects to consider:

1. its local geometry, which predominantly concerns the curvature of the universe, particularly the observable universe, and
2. its global geometry, which concerns the topology of the universe as a whole.

The observable universe can be thought of as a sphere that extends outwards from any observation point for 46.5 billion light-years, going farther back in time and more redshifted the more distant away one looks. Ideally, one can continue to look back all the way to the Big Bang; in practice, however, the farthest away one can look using light and other electromagnetic radiation is the cosmic microwave background (CMB), as anything past that is opaque. Experimental investigations show that the observable universe is very close to isotropic and homogeneous.

If the observable universe encompasses the entire universe, it may be possible to determine the structure of the entire universe by observation. However, if the observable universe is smaller than the entire universe, our observations will be limited to only a part of the whole, and we may not be able to determine its global geometry through measurement. From experiments, it is possible to construct different mathematical models of the global geometry of the entire universe, all of which are consistent with current observational data; thus it is currently unknown whether the observable universe is identical to the global universe, or is instead many orders of magnitude smaller. The universe may be small in some dimensions and not in others (analogous to the way a cuboid is longer in the dimension of length than it is in the dimensions of width and depth). To test whether a given mathematical model describes the universe accurately, scientists look for the model's novel implications—phenomena in the universe that have not yet been observed, but that must exist if the model is correct—and they devise experiments to test whether those phenomena occur or not. For example, if the universe is a small closed loop, one would expect to see multiple images of an object in the sky, although not necessarily images of the same age.

Cosmologists normally work with a given space-like slice of spacetime called the comoving coordinates, the existence of a preferred set of which is possible and widely accepted in present-day physical cosmology. The section of spacetime that can be observed is the backward light cone (all points within the cosmic light horizon, given time to reach a given observer), while the related term Hubble volume can be used to describe either the past light cone or comoving space up to the surface of last scattering. To speak of "the shape of the universe (at a point in time)" is ontologically naive from the point of view of special relativity alone: due to the relativity of simultaneity, different points in space cannot be said to exist "at the same point in time" nor, therefore, of "the shape of the universe at a point in time". However, the comoving coordinates (if well-defined) provide a strict sense to those by using the time since the Big Bang (measured in the reference of CMB) as a distinguished universal time.

Curvature of the universe

Further information: Curvature § Space

The curvature is a quantity describing how the geometry of a space differs locally from the one of the flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases:

1. Zero curvature (flat); a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E³.
2. Positive curvature; a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S³.
3. Negative curvature; a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H³.

Curved geometries are in the domain of Non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.

The local geometry of the universe is determined by whether the density parameter Ω is greater than, less than, or equal to 1.

From top to bottom: a spherical universe with Ω > 1, a hyperbolic universe with Ω < 1, and a flat universe with Ω = 1. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.

General relativity explains that mass and energy bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called the density parameter, represented with Omega (Ω). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. Put another way,

• If Ω = 1, the universe is flat.
• If Ω > 1, there is positive curvature.
• If Ω < 1 there is negative curvature.

One can experimentally calculate this Ω to determine the curvature two ways. One is to count up all the mass-energy in the universe and take its average density then divide that average by the critical energy density. Data from Wilkinson Microwave Anisotropy Probe (WMAP) as well as the Planck spacecraft give values for the three constituents of all the mass-energy in the universe – normal mass (baryonic matter and dark matter), relativistic particles (photons and neutrinos), and dark energy or the cosmological constant:

Ω_mass ≈ 0.315±0.018

Ω_relativistic ≈ 9.24×10⁻⁵

Ω_Λ ≈ 0.6817±0.0018

Ω_total = Ω_mass + Ω_relativistic + Ω_Λ = 1.00±0.02

The actual value for critical density value is measured as ρ_critical = 9.47×10⁻²⁷ kg m⁻³. From these values, within experimental error, the universe seems to be flat.

Another way to measure Ω is to do so geometrically by measuring an angle across the observable universe. We can do this by using the CMB and measuring the power spectrum and temperature anisotropy. For instance, one can imagine finding a gas cloud that is not in thermal equilibrium due to being so large that light speed cannot propagate the thermal information. Knowing this propagation speed, we then know the size of the gas cloud as well as the distance to the gas cloud, we then have two sides of a triangle and can then determine the angles. Using a method similar to this, the BOOMERanG experiment has determined that the sum of the angles to 180° within experimental error, corresponding to an Ω_total ≈ 1.00±0.12.

These and other astronomical measurements constrain the spatial curvature to be very close to zero, although they do not constrain its sign. This means that although the local geometries of spacetime are generated by the theory of relativity based on spacetime intervals, we can approximate 3-space by the familiar Euclidean geometry.

The Friedmann–Lemaître–Robertson–Walker (FLRW) model using Friedmann equations is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of fluid dynamics, that is, modeling the matter within the universe as a perfect fluid. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe. Another way of saying this is that if all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed (rather than the distortions caused by 'dense' objects such as galaxies). This assumption is justified by the observations that, while the universe is "weakly" inhomogeneous and anisotropic (see the large-scale structure of the cosmos), it is on average homogeneous and isotropic.

Global universe structure

Global structure covers the geometry and the topology of the whole universe—both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. The universe is often taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably. A global geometry is a local geometry plus a topology. It follows that a topology alone does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

As stated in the introduction, investigations within the study of the global structure of the universe include:

• whether the universe is infinite or finite in extent,
• whether the geometry of the global universe is flat, positively curved, or negatively curved, and,
• whether the topology is simply connected like a sphere or multiply connected, like a torus.

Infinite or finite

One of the presently unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".

With or without boundary

Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.

If space were infinite (flat, simply connected), perturbations in the temperature of the CMB radiation would exist on all scales. If, however, space is finite, then there are those wavelengths missing that are larger than the size of the space. Maps of the CMB perturbation spectrum made with satellites like NASA's WMAP and the ESA's Planck have shown a striking amount of missing perturbations at large scales. The properties of the observed fluctuations of the CMB show a 'missing power' on scales beyond the size of the universe. That would imply that our universe is multiply-connected and finite. The spectrum of the CMB fits much better with the universe as a gigantic three-torus, a cosmos connected to itself in all three dimensions.

Curvature

The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected, and infinite, but there are tori which are flat, multiply connected, finite, and compact (see flat torus).

In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.

The latest research shows that even the most powerful future experiments (like the SKA) will not be able to distinguish between flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10⁻⁴. If the true value of the cosmological curvature parameter is larger than 10⁻³ we will be able to distinguish between these three models even now.

Final results of the Planck mission, released in 2018 show the cosmological curvature parameter, 1 − Ω = Ω_K = −Kc²/a²H², to be 0.0007±0.0019, consistent with a flat universe. (i.e. positive curvature: K = +1, Ω_K < 0, Ω > 1, negative curvature: K = −1, Ω_K > 0, Ω < 1, zero curvature: K = 0, Ω_K = 0, Ω = 1).

Universe with zero curvature

In a universe with zero curvature, the local geometry is flat. The most obvious global structure is that of Euclidean space, which is infinite in extent. Flat universes that are finite in extent include the torus and Klein bottle. Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. These are the Bieberbach manifolds. The most familiar is the aforementioned 3-torus universe.

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe.

A flat universe can have zero total energy.

Universe with positive curvature

A positively curved universe is described by elliptic geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere.

Poincaré dodecahedral space is a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003 and an optimal orientation on the sky for the model was estimated in 2008.

Universe with negative curvature

A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood. Those of finite volume can be understood via the Mostow rigidity theorem. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called "horn topologies", so called because of the shape of the pseudosphere, a canonical model of hyperbolic geometry. An example is the Picard horn, a negatively curved space, colloquially described as "funnel-shaped".

Curvature: open or closed

When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively. These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in topological spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a closed manifold (i.e., compact without boundary) and open manifold (i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.

Milne model (hyperbolic expanding)

Main article: Milne model

If one applies Minkowski space-based special relativity to expansion of the universe, without resorting to the concept of a curved spacetime, then one obtains the Milne model. Any spatial section of the universe of a constant age (the proper time elapsed from the Big Bang) will have a negative curvature; this is merely a pseudo-Euclidean geometric fact analogous to one that concentric spheres in the flat Euclidean space are nevertheless curved. Spatial geometry of this model is an unbounded hyperbolic space. The entire universe in this model can be modelled by embedding it in Minkowski spacetime, in which case the universe is included inside a future light cone of a Minkowski spacetime. The Milne model in this case is the future interior of the light cone and the light cone itself is the Big Bang.

For any given moment t > 0 of coordinate time within the Milne model (assuming the Big Bang has t = 0), any cross-section of the universe at constant t' in the Minkowski spacetime is bounded by a sphere of radius ct = ct'. The apparent paradox of an infinite universe "contained" within a sphere is an effect of the mismatch between coordinate systems of the Milne model and the Minkowski spacetime in which it is embedded.

This model is essentially a degenerate FLRW for Ω = 0. It is incompatible with observations that definitely rule out such a large negative spatial curvature. However, as a background in which gravitational fields (or gravitons) can operate, due to diffeomorphism invariance, the space on the macroscopic scale, is equivalent to any other (open) solution of Einstein's field equations.