A simplified presentation of Einstein's unified field equations
Date: 2026-04-02T00:00:00+00:00
Keywords: Gravitation; Einstein; Albert; 1879-1955. Zur einheitlichen Feldtheorie; Electromagnetism

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Title: A simplified presentation of Einstein's unified field equations

Author: Tullio Levi-Civita

Translator: John Dougall

Release date: April 2, 2026 [eBook #78343]

Language: English

Original publication: London: Blackie & Son Limited, 1929

Other information and formats: www.gutenberg.org/ebooks/78343

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*** START OF THE PROJECT GUTENBERG EBOOK A SIMPLIFIED PRESENTATION OF EINSTEIN'S UNIFIED FIELD EQUATIONS ***

A SIMPLIFIED PRESENTATION
OF
EINSTEIN'S
UNIFIED FIELD EQUATIONS

By
TULLIO LEVI-CIVITA

Professor of Rational Mechanics in the University of Rome
Fellow of R. Accademia Nazionale del Lincei

Authorized Translation by
JOHN DOUGALL, M.A., D.Sc.

BLACKIE & SON LIMITED
LONDON AND GLASGOW
1929

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A SIMPLIFIED PRESENTATION OF EINSTEIN'S UNIFIED FIELD EQUATIONS

In his recent paper, "Zur einheitlichen Feldtheorie",[1] Einstein made use of the fundamental idea that it is both possible and useful to give a geometrical interpretation of the complete system of the sixteen field equations (consisting of Einstein's celebrated gravitational equations and Maxwell's equations) in such a way as to include the definition (and the definition only) of an orthogonal quadruplet[2] embedded in the space-time world.

Conversely, the sixteen parameters determining a quadruplet are to give a complete definition not only of the Riemannian metric of space (as is well known, this takes place automatically), but of the phenomena of electromagnetism as well.

For this purpose the eminent author introduced covariant derivatives with respect to the quadruplet, and suggested relationships between them which to a first approximation lead to the required co-ordination of gravitational and electromagnetic phenomena.

It appears to me, however, that the root problem raised by Einstein can be solved in a simpler and more general way by making use of perfectly familiar methods of the absolute differential calculus on the one hand, while, on the other hand, retaining unaltered all results previously obtained.

1. Geometrical and formal preliminaries.[3]

Let be general co-ordinates of a Riemannian space , and the parameters of congruences, which define a lattice of lines in and an -uplet[4] at every point.

Following Einstein's example I shall use Greek letters for co-ordinate indices (such as ), and Roman letters, on the other hand, for indices referring to the -uplet (such as ). I shall leave out signs of summation with respect to Greek indices (provided they occur once above and once below), but other 's will be retained.

As usual, let the quantities be the elements reciprocal to (normalized cofactors). For every they form a covariant system (moments of the -uplet in question). By composition with the quantities , we obtain, from every mixed tensor of rank with the components an "-uplet tensor",[5] the components of which are defined by the formulæ and conversely, since these formulæ can be solved for the co-ordinate components in the form

The components of the -uplet tensor are pure invariants with respect to transformations of co-ordinates; they essentially depend on the -uplet considered, but, as is easily verified, they also behave like a tensor when the quantities and are simultaneously subjected to orthogonal transformations.

If we put a definite metric (for real values of the quantities involved) is introduced into in such a way that our -uplet turns out orthogonal. Later (§3) I shall give the (unimportant) modifications required to transfer the -uplet theory, avoiding any appearance of imaginaries, to an indefinite metric (with a given index of inertia).

Meanwhile I suppose that the covariant derivatives of the moments have been introduced, and, following Ricci, I take the coefficients of rotation

In virtue of the identities (which result from the relationships between parameters and moments), Ricci's quantities form invariants with respect to transformations of co-ordinates, which of course essentially depend on the given -uplet and necessarily include all its geometrical differential properties of the first order. With respect to orthogonal transformations with constant coefficients the quantities behave like a tensor of the third rank. In order to emphasize the limitation to transformations with constant coefficients I shall call such systems local -uplet tensors. True -uplet tensors behave as invariants with respect to all orthogonal transformations whose coefficients can vary in any way with the quantities .

Perhaps it is not superfluous to remark that the explicit expressions for the coefficients of rotation, , can also be obtained directly by ordinary differentiation without making use of the covariant derivatives of the quantities .

In order to do this, we have to introduce either the Pfaffian expressions or the operators (derivatives of a function in the direction of the lines of the congruences), and then to form the corresponding bilinear covariants or Poisson brackets. We can, however, attain the desired result even more rapidly by using (4) and noticing that, according to the definition of covariant differentiation, we have the identity We thus obtain and all the quantities are uniquely determined by these equations together with (5).

Equations (4) can be solved for the quantities , giving from which we obtain the conditions of integrability of by repeated covariant differentiation and formation of differences. For this we require the commutation-formula where denotes the Riemannian tensor. In this way we obtain where for brevity we write From (7) we conclude that the 4-index symbols, , form a (true) -uplet tensor. In virtue of the well-known identities satisfied by the Riemannian symbols the formulæ (7) lead to similar identities for the 4-index symbols, , namely

Now for the Einstein tensor

Its components , with respect to the two members , of the -uplet are expressed, by (1), by whence, by (7),

The linear (co-ordinate and -uplet) invariant consequently takes the form

In conclusion, I shall emphasize one other fact, namely that contraction of two indices in an -uplet tensor leads to a reduced tensor—of the th rank if the original tensor is of the th rank.

As we have already seen, the quantities form a local -uplet tensor of the third rank, which in virtue of (5) is skew-symmetrical with respect to the two first indices , . The same is true for the differences , which for are called anormalities (i.e. quantities which vanish when the th congruence of the -uplet is normal).

If we apply the differential operator to the elements (where stands for ) of a local or true -uplet tensor, we obtain a new local -uplet tensor , the rank of which exceeds that of the original tensor by unity. In particular, we obtain in this way the local -uplet tensor of the fourth rank which is skew-symmetrical with respect to and . By contraction we obtain so that we have obviously formed a skew-symmetrical local -uplet tensor of the second rank. Its covariant and contravariant components are respectively

We may mention in addition that the quantities may be interpreted as mean curvatures of the -fold sections, drawn orthogonally to the lines of the -uplet. By what we have said above, they are line-components of a local -uplet vector. From the tensor of the third rank, , and this vector we obtain by contraction a new local -uplet tensor of the second rank, namely which is also skew-symmetrical.

2. Formation of divergences. The special case .

If are the contravariant components of a vector , its divergence is defined by the invariant where, as usual, denotes the determinant and is written (instead of simply ) because the formula is then valid as it stands even for an indefinite .

For the divergence of a tensor of the second rank with the contravariant components we obtain a vector with the contravariant components

Following von Laue,[6] we shall write simply

If we here replace the covariant derivatives by their explicit values, we obtain in the case of a skew-symmetrical tensor ; hence, by (16),

Owing to the skew-symmetry of the quantities , the right-hand side vanishes identically.

Thus if we again make use of covariant derivatives, we obtain the identity or finally, in tensor notation,

That is, in an arbitrary Riemannian space the divergence of the divergence of a skew-symmetrical tensor of the second rank is identically zero.

In order to express the right-hand sides of (16) and (17) in -uplet tensor components, it is sufficient to apply the operator to the formulæ of definition

By replacing ordinary differentiation by covariant differentiation on the right-hand side (which is permissible, as we are dealing with invariants), we obtain whence, by , (16), and (17), which give the divergences and of -uplet tensors (of the first or second rank) directly by means of -uplet components and -uplet operations.

For we have an elementary tensor of the fourth rank at our disposal, namely the well-known Riccian -system, the covariant and contravariant components of which, , respectively, are equal to zero if the four indices are not all different. The other components have the respective values , , the upper or lower sign being taken according as the permutation is even or odd with respect to (0123).

Let again be a skew-symmetrical tensor of the second rank with the contravariant components . If we put which means the same as in von Laue's notation, we are justified in calling the vector with the above covariant and contravariant components the Pfaffian divergence of , because the 's vanish identically if, and only if, the 's coincide with the coefficients of the bilinear covariants of a Pfaffian expression . This is most easily proved by replacing the covariant derivatives in by their explicit values and noting that, owing to the skew-symmetry of the quantities , all that we have left is The right-hand sides obviously vanish if the quantities are replaced by the differences .

By substituting the expression for the 's in the second form (16) of the divergence of a vector we immediately obtain , which, bearing in mind, may be written that is, the divergence of the Pfaffian divergence of a skew-symmetrical tensor of the second rank in vanishes identically.

Further, we shall proceed to represent the vector (the Pfaffian divergence) directly in terms of the -uplet components of the given tensor. Here it suggests itself to start from the solved form of the equations which we have just written down, namely and to calculate the quantities by covariant differentiation of the right-hand side.

From and we obtain hence, by , where for brevity we have put Thus these quantities are equal to zero if two of the four indices are equal. If, on the other hand, is a permutation of the numbers 0123, has the value , according as the class of the substitution is even or odd. We accordingly see that in the expression which we have just obtained for the 's the two last terms are equal to each other, so that we finally obtain

3. Transformations for an indefinite metric.

According to Eisenhart[7] all the formulæ of the -uplet theory can be transferred in a readily intelligible way to indefinite metrics, without leaving the real region even temporarily.

If we consider an indefinite we (as is well known) call a (real) direction time-like or space-like, according as the corresponding turns out greater or less than zero; null directions are those directions, in number, for which .

In any case we call the ratios parameters of a proper (i.e. non-null) direction.

Hence we have if we henceforth denote positive or negative unity by .

As in the definite case we introduce as moments of a given direction the covariant quantities so that the quadratic identity (25) takes the form

If the quantities are the parameters of an orthogonal -uplet consisting of proper directions only, we have on account of the orthogonality of the -uplet, and also by (27).

The total number of negative (and consequently also of the remaining positive) quantities for a given is always equal to its index of inertia, and hence is always the same no matter what (proper) -uplet is considered.

The two groups of relationships between parameters and moments of an -uplet which we have just written down may be summarized in the single formula where the symbols have their usual meaning; or, since ,

From this we conclude that the elements reciprocal to the parameters are not exactly equal to the moments , but to . Thus the quantities are the elements reciprocal to the moments . If we imagine the equations (26) written down for every -uplet, we have (denoting the index of summation by ). By multiplying by and summing with respect to we obtain which replaces formula (2) for the definite case, and so on.

From this point it will suffice if I confine myself to quite brief hints, and I shall of course write down only those formulæ which do not remain unaltered throughout. These will be marked with an asterisk and given the same number as the corresponding formula referring to a definite metric.

In the first place, -uplet components of any given tensor and coefficients of rotation must in any case be introduced by the equations of definition (1) and (4); the solved expressions for the quantities , on the other hand, are in general

The covariant equations (6), and also the equations of definition of the 4-index symbols (7) are true without restriction; but the -uplet tensor expressions for the quantities suffer a small modification. In fact we must in general put

Of course these quantities are still connected by the relationships (9), in virtue of equations (7).

It is essential to note, however, that the local transference from one -uplet to another does not correspond to any orthogonal transformation, but to a pseudo-orthogonal transformation, i.e. to a transformation which leaves the quadratic form invariant. Thus the coefficients of a pseudo-orthogonal transformation of this kind must satisfy the conditions

The most general expression which can be attributed to the coefficients in the case of infinitesimal pseudo-orthogonal transformations follows immediately from the condition that the form is to be invariant. We have merely to put and to regard the quantities as indefinitely small. If in we carry out the substitution and require that should retain the form what we obtain (as in the case of pure orthogonal substitutions) is the condition of skew-symmetry, namely

The components of an -uplet tensor are systems of numbers which behave like tensors with respect to pseudo-orthogonal transformations; for local -uplet tensors this behaviour is maintained only with respect to pseudo-orthogonal transformations with constant coefficients. The operators behave like -uplet vectors.

If () and () denote any group of -uplet indices and , two local -uplet tensors, then contraction with respect to , is defined by the formula

We accordingly obtain instead of (10) and (11).

Further, the formulæ (12), (14), and (15) must be replaced by and while the expressions (13) for covariant and contravariant components in terms of the -uplet components are to be deduced from (1), the universally valid definition of the -uplet components of a tensor. Hence they become

As contraction of pseudo-orthogonal -uplet tensors is brought about by inserting the factor with the appropriate index, it is at once clear that (19), (20), and (24) take the forms

Of course the equations and (22), i.e. which express invariant properties, always remain valid.

4. Gravitational equations.

As usual, let the covariant components of the energy tensor be denoted by . If influences of any origin are admitted, these quantities are to be imagined broken up into two parts, one of which, , is purely electromagnetic, and the other, , represents the remainder, if any. We therefore put where is the well-known Maxwell tensor; further, for empty space is of course equal to zero.

As is well known, the Einstein equations (without the cosmological term) are where the constant of proportionality may be expressed in terms of , the gravitational constant, and , the velocity of light .

If we introduce the corresponding -uplet tensors in accordance with the formulæ we have, on the one hand, from (32), and (what is most important) the gravitational equations in the -uplet tensor form [8] where, in accordance with (10*) and (11*),

As the space-time manifold on which the general theory of relativity is to be based possesses an indefinite metric with an index of inertia 3, we have to put

The quantities are introduced by the equations (8*) as lattice differential elements of the second order. Their combinations behave like tensors with respect to all pseudo-orthogonal (i.e. in the present case Lorentz) transformations (even if the coefficients are permitted to vary in any way with position).

Accordingly, as indeed is clear from the outset, the ten equations (I) do not, as far as their original form is concerned, favour any special quadruplet. They are valid in one and the same form for all orthogonal quadruplets of the relativistic , and, as is well-known, serve to define their metric.

As in every case they give ten relationships between the sixteen parameters , we need only find six other apparently reasonable conditions connecting the latter, in order to mark out a special lattice (the world lattice) from among all the possible quadruplets and lattices corresponding to the space-time-manifold .

We shall shortly (§6) carry out this final step, which is in fact the only essential one. Meanwhile we may appropriately lead up to it by putting Maxwell's equations into a suitable form.

5. Electromagnetic equations.

Let be the (covariant, contravariant, and -uplet) components of the skew-symmetrical tensor which defines the electromagnetic field in the space-time world; let (a vector) be the current-vector[9] and , , its four components, where all the quantities are understood to be measured in so-called rational units.

Maxwell's equations (as adopted in the general theory of relativity after Einstein) then take the forms

Each group contains four equations, so that at first glance one would take the total number of equations to be eight. But we necessarily have , so that by (31) there must exist two identical relationships, namely those which express the fact that the divergences in question vanish. Thus two equations of the system (34) may (with appropriate subsidiary conditions) be regarded as resulting from the other six; and in fact we know that if is regarded as given or as associated in some other way with the tensor , then the equations (34) merely serve to determine the six components of for uniquely from their values for a given (and any ).

We have still to write down the symmetrical stress-energy tensor explicitly. As is well known, its covariant components are defined as follows:

By composition with (by replacing on the right-hand side by and by ) we obtain the required -uplet tensor formula:

6. Interpretation of the electromagnetic tensor in the world lattice. Purely geometrical formulation of the field equations.

A priori we may quite arbitrarily connect the six -uplet components of the electromagnetic field with any geometrical properties of a quadruplet (thereby defined) of the . A very simple way of doing this is to make the quantities proportional to the corresponding elements of a (differential) skew-symmetrical local -uplet tensor, e.g. to the differential expressions, of the second or first order respectively, which are defined by the equations of §3.

As we shall see, the best way is to select the first expression, and we accordingly put where denotes a constant.

As the Ricci coefficients of rotation are merely ratios of an angle and a length, the quantities are of dimensions . The quantities , on the other hand, behave like the square root of an energy-density. Consequently we have

Hence the factor of proportionality has dimensions which are those of an electric charge , e.g. the electronic charge, so that we may write where the factor of proportionality is now a pure number. Moreover, we may also replace in (36) by any other quantity of the same dimensions; e.g. we may put where is Planck's constant, the velocity of light in empty space, and a pure number.

Hence the final forms of the geometrical equations which arise from the Maxwellian system (34) and our proposed addition (P), are where means the local -uplet tensor (12*). In conclusion, then, the geometrical definition of the quadruplet (world lattice) associated with the field is to be taken from the two systems (I) and (II), which together give sixteen (apparently eighteen, but in reality only sixteen) differential equations (of the second and third order respectively) involving the sixteen -uplet parameters .

7. The case of empty space: absence of an electromagnetic field.

In empty space , (I) reduces in virtue of (32) to the form where the term on the right-hand side is given by by (35) and (P); while the system (II) becomes

If the electromagnetic field vanishes in addition to the external energy tensor , the quantities , and hence, by , the quantities also, are equal to zero. If this happens everywhere in the space-time world, we know [10] that the equations , which simply become , necessarily imply that the metric of the space is Euclidean or, more correctly, pseudo-Euclidean.

What, then, is the geometrical meaning of the absence of electromagnetic phenomena in this limiting case, i.e. what is the geometrical meaning of the equations

They simply state the fact that the world lattice is Cartesian or, more correctly, pseudo-Cartesian.

In order to give as concise a proof of this as possible, I shall only consider quadruplets in which the deviations from a pseudo-Cartesian lattice are infinitely small.

If, in particular, we take the co-ordinates to be Cartesian co-ordinates with respect to that lattice, we have for the parameters of the corresponding quadruplet.

Let be the parameters of any neighbouring quadruplet. Since the passage from the quantities to the quantities corresponds to an infinitesimal pseudo-orthogonal transformation, the quantities must, by (29), be expressible as follows: where the quantities form a skew-symmetrical -uplet tensor. From this we can immediately calculate the reciprocal elements. To a first approximation we obtain whence, multiplying by ,

On the other hand, if we altogether neglect infinitely small quantities, the operators reduce to the simple form and the covariant derivatives reduce to their usual forms.

Thus (4), the definition of the rotational invariants , gives (except for infinitely small quantities of the second order) and from (12*) we further obtain

The differential operator is none other than the Dalembertian or Lorentz operator . Thus the equations (37) take the form and together with suitable initial and boundary conditions they give i.e. the Cartesian (or, more correctly, pseudo-Cartesian) character of the world lattice. I think that this conclusion justifies our assumption (P). If we had put, say, where the quantities are given by the expressions (15*), we should not have obtained any satisfactory result.

A more general assumption, such as would, on the other hand, be more complicated, though just as admissible as (A) from the logical point of view. To a first approximation, in fact, we should obtain the same result, as the 's are of higher order than the 's.

FOOTNOTES:

[1] Berliner Berichte, I, 1929, pp. 1-8.

[2] Ger. Vierbein.

[3] See in particular my Absolute Differential Calculus (English translation by Miss Long), Chap. III. Blackie & Son, Ltd., 1927.

[4] Ger. n-Bein.

[5] Ger. Beintensor.

[6] Die Relativitätstheorie, Bd. II (2nd edition, Vieweg. Brunswick, 1923), § 14.

[7] Riemannian Geometry, Princeton University Press, 1926, Chap. III.

[8] Given in 1918 by Cisotti (Rend. Acc. Lincei, Vol. XXVII, pp. 366-371), but confined to the (imaginary) notation of (8), (10), (11).

[9] Ger. Viererstrom.

[10] Cf. Serini, Rend. Acc. Lincei, Vol. XX VII, 1918, pp. 235-238.

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Table of Contents

- A SIMPLIFIED PRESENTATION OF EINSTEIN'S UNIFIED FIELD EQUATIONS

- A SIMPLIFIED PRESENTATION OF EINSTEIN'S UNIFIED FIELD EQUATIONS

- FOOTNOTES:

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