OK, with the mathematical preliminaries suitably dealt with, let’s go…
As noted previously in Part II, I’ll be using the tensor component notation and assuming a co-ordinate basis (together with the misuses of geometrically precise language that comes with that choice; my bad).
A general connection can be written in the form
where is the Christoffel connection of the metric and T is a 3rd-rank tensor.
The torsion tensor is defined by
For a semi-symmetric connection, the torsion tensor takes the form
where is a vector (field).
If the semi-symmetric connection is also metrical, i.e., , then T takes the specific form
Hence, from (4), the (i.e., unique) semi-symmetric metric connection (SSMC) is thus:
Also note that the covariant derivative of the metric with respect to the Christoffel connection , is also metrical:
which implies that raising and lowering indices commutes with both the SSMC covariant derivative as well as the Christoffel-covariant derivative . This makes tensor manipulations considerably easier, and means that we may choose, as convenient, between equivalent representations of tensors containing either form of covariant derivative.
From (5) it is clear that the SSMC adds only a single new object (over and above the metric and its related form ) – namely, the vector – to the Christoffel connection of GR, which is its major appeal, and principally why it is being investigated, as described in the first post of this series.
since , and also that
The Riemann (curvature) tensor can be found, from a general connection possessing no assumed symmetries, in the following way (although different authors use different index placements and sometimes a different sign convention; I am following Schouten (1954) for ease of comparison of results):
which is Schouten III.4.2 (p.138). Expanding the index square brackets, this takes the more detailed form
The Riemann tensor is always anti-symmetric in the first two indices, regardless of the symmetry properties of the connection from which it is formed, as should be clear from the square brackets in (8a), whence the relation
which is one of the four important identities that the Riemann tensor obeys (see below).
Schouten denotes the Riemann tensor formed from the Christoffel connection by , hence,
or, expanded out:
From (1) substituted into (8), we get
or more compactly, with the derivative (note the + sign on the third term below compared to the – sign above),
Two independent tensors can in principle be formed by simple contraction from the Riemann tensor – the Ricci tensor:
and the “second contraction” (sometimes known as the homothetic curvature tensor)
Because of the anti-symmetry (9) of the Riemann tensor in its first two indices, a contraction on its second and fourth indices would just produce another Ricci tensor with the opposite sign, which is of course not independent of (12).
Note that from (10) and the symmetry of the as direct substitution verifies (although this is also a very well-known result from GR).
From (11b), we find, for the SSMC (since ) that vanishes identically, owing to the derivative term vanishing from (7b) as well as the term also vanishing when expanded out and the dummy repeated indices and are rewritten:
[The identical vanishing of the tensor for the SSMC reveals an important property of the space of the SSMC: the connection is, in this case, called volume-preserving (Schouten, Chap.III, exercise 4,5, p.144), and this allows a covariant constant “scalar density-field” to be defined (p.155). Scalar densities are the mathematical objects used to derive field equations from an integral variational procedure, by way of a “Lagrangian” function (strictly, a Lagrangian density function). In this instance, such a Lagrangian density would depend only upon position, which is considered desirable for deriving equations to describe the field behaviour as it varies from place to place throughout the space.]
This means that the Ricci tensor is the sole independent tensor that can be derived by simple contraction from the Riemann tensor of the SSMC. The same is also true for the case of the Riemannian geometry of GR.
Substituting (11a) into (12) yields
while (11b) into (12) gives (again noting signs)
where is the Ricci tensor of :
and is symmetric:
From (15) we can see that when then, from (4), , and so the Riemann (11) and Ricci (15) tensors reduce to their familiar forms from GR, being comprised solely of the Christoffel connection , which is comprised solely of the metric tensor .
We would now like to investigate (11) more fully in order to see exactly how the Riemann tensor of the SSMC deviates from the case of pure gravitation in GR, . From (4) is it clear that this deviation will be linear in derivatives of the vector and contain quadratic products of the vector along with various factors of the metric. Already this hints that the electrodynamics implied by this geometry might be non-linear, which is quite suggestive of the Born-Infeld electrodynamics investigated in the 1930s, and opens the possibility of eventual experimental results that might be able to distinguish between an SSMC-based electrodynamics as compared to the electrodynamics in conventional Einstein-Maxwell theory.
[An interesting coincidence is that, following his collaboration with Born, Infeld later ended up working with Einstein on the approximation procedure for GR (the so-named “EIH [Einstein-Infeld-Hoffmann] approximation” – see Einstein & Infeld, 1949) which demonstrated that the equations of motion of non-ideal (i.e., massive) particles in GR can be derived from the field equations alone, without needing to postulate them separately. As it turns out, this is due to the non-linearity of the field equations of GR together with the fact that the field object is a second-rank tensor, something later demonstrated by Bergmann (1949) in his work on generalised field theories of tensor fields of arbitrary rank. A vector object (i.e., a tensor of rank 1), even with non-linear field equations, is not sufficient to enable equations of motion to be derived from the field equations.]
Substituting equation (4) into (11a), the Riemann tensor is then:
where (to use a term much beloved by some mathematicians, ‘after some calculation’)
or, putting (4) into (11b) instead,
It is much clearer now from (18) and (19) that the “deviation from GR”, contained in the tensor , indeed shows itself to be linear in covariant derivatives of , and quadratic in products of , with various factors of the metric mixed in. The substitution of with (using, e.g., (12) from Part II) simply introduces more explicit quadratic products of as comparison of (19a) with (19b) shows. Clearly, by inspection of (19), if as in conventional GR, then the tensor vanishes identically, and so the Riemann tensor (18) thereby reduces to the more familiar case (10) of GR.
The Ricci tensor (15) is then
One very interesting thing to note from (20) and (21), is that the antisymmetric part of the Ricci tensor reduces to
That is, the anti-symmetrisation cancels out both forms of covariant derivative (as direct substitution reveals) leaving only the anti-symmetrised partial derivatives of the vector remaining.
In other words, the anti-symmetric part of the Ricci tensor of the SSMC has the same form as the familiar Faraday tensor from electromagnetism, if the vector is somehow identified with the usual electromagnetic 4-potential .
This implies, of course, that the Ricci tensor – whose symmetric part will, as with GR, contain the purely gravitational terms made up of only functions of (along with some others containing the vector field not found in GR) – will now also have an anti-symmetric part (which it does not possess in GR) that has the familiar form shown above of the Faraday tensor of Maxwell theory. Recalling the discussion near the end of Part II, we note that this implies that there exists an anti-symmetric “invariant sub-space” of the Ricci tensor, along with the trace and trace-free part of it, which is “rotated into itself” under a change of coordinates (as are the two other parts) implying a certain quasi-independence. Again, this is a suggestive observation, if we are to seek some correspondence between the mathematics of the SSMC and real-universe physics.
This “unification” of gravity and electromagnetism, if it turns out to be viable (see the Gell-Mann quote in the latter part of Part I), would therefore come about through a “unified connection” containing both the metric (including via the “pure-g” Christoffel objects) as well as the electromagnetic potential, from which the more familiar field objects, the Riemann and Ricci tensors (and other objects found from them) are derived. This is in contrast to Einstein’s non-symmetric unified field theory (EUFT), which posited that the unification of these two fields was to be found in the non-symmetric fundamental tensor – which is to say the non-symmetric . This observation will have implications later in the search for field equations.
Finally, the Ricci scalar (or ‘curvature scalar’) is found from contracting the indices of the Ricci tensor, thus
where K is the Ricci scalar of GR formed from , and M is found to be
The Ricci scalar is important in searching for field equations in GR in two ways. Firstly, it is the primary object which is used to form a Lagrangian scalar density in GR in order to derive field equations from a variational approach. Secondly, it is used in conjunction with the Ricci tensor to form a new tensor (now called the ‘Einstein tensor’) that has certain desirable properties which Einstein used in order to postulate the initial field equations of GR based on an appeal to physics, before a variational procedure was used to do the same thing (notably by Hilbert).
To close out the discussion of the geometry of the space of the SSMC, we now examine the main identities of the Riemann tensor.
Identities of the Riemann tensor
Here the main identities obeyed by the Riemann tensor are listed, for the cases of both the SSMC and the Christoffel connection . Detailed discussions and derivations can be found in Schouten III.Section 5 (p.144ff). (Schouten’s second and third identities are interchanged in the discussion here, since I believe they work better this way, from the point of view of simply reporting the main findings, whereas Schouten’s original order is actually better for carrying out the calculations that lead to their derivations.)
The first identity is
, for all
As noted above, this is always true for any connection, irrespective of the symmetry properties of that connection, and is clear from the definition of given above in (8a).
then the ‘all-downstairs-indices’ form of the Riemann tensor is also anti-symmetric in the last two indices, if the connection is metrical, which both the SSMC and the Christoffel connection are. Thus, the second identity:
, for metrical
For the case of the Riemann tensor of the Christoffel connection, this takes the form
and was implicit in the derivation of (14) above.
So far, both of these identities are familiar from (and indeed have the same form as in) the Riemannian space of GR. The third and fourth identities, however, are not quite as simple as their GR counterparts.
The third identity is
, for semi-symmetric
By using the index permutation procedure described in Part II, and by noting from (20) and (21a) that , this expression can also be rendered in the somewhat ‘friendlier’ form:
while, as noted in (22) above, since reduces to , this can also be written as:
When viewed in either this form, or the one in (IIIa1), it is very clear that when this identity reduces to the more familiar one from GR:
The fourth identity is considerably more complicated to obtain. Schouten derives this on p.145 using his equation III.5.14, which is an identity obeyed by any quantity that is anti-symmetric in . The object , and the related object , which is found from the first by swapping the two pairs of indices, are related through an equation of the form:
That is, there are four terms that are (completely) anti-symmetric in the first three indices and six terms which are symmetric in the last two indices (I’ve not listed what those indices actually are, since they are not required for this discussion and would only complicate this post with unnecessary detail; the full details are found in Schouten). However, if the object is chosen to be the Riemann tensor of either the SSMC or the Christoffel connection, then the six terms of the form will all vanish, from Identity (II) above, since they are both metrical connections. For the SSMC, each of the four terms can be substituted with a version of (IIIa) with the index suitably lowered, as in (25), which essentially ‘converts’ the Kronecker deltas into metrics . This substitution produces an initial tensor terms which, ‘after some calculation’, simplifies – for the form given by (IIIa2) – to:
Again, when , the SSMC reduces to the familiar Christoffel connection, vanishes, and so the well-known identity from GR,
is suitably recovered.
There is also a fifth identity, widely known as the Bianchi identity, which takes the form
noting the reversed position of the indices in the term on the RHS compared to the LHS. This is Schouten’s equation III.5.20 (p.147). When the connection is symmetric there is no torsion, so , the term therefore also vanishes, and the identity reduces to
If the symmetric connection is also metrical, then we have precisely the Christoffel connection , and this becomes the well-known identity from GR:
The form (Vb) and its contractions were especially important in Einstein’s considerations leading up to his proposal for the field equations in GR, so further examination of the Bianchi identity (V) will be delayed until the next part, when those field equations are discussed.
With the basic geometric properties of the space of the SSMC now described, it is to the search for candidate field equations that we now turn – in the first instance by examining how they were derived in General Relativity.
Next time: Part IV: Towards Field Equations
Bergmann, PG 1949, ‘Non-linear field theories’, Physical Review, vol. 75, no. 4, pp. 680–685. doi:10.1103/PhysRev.75.680.
Einstein, A & Infeld, L 1949, ‘On the motion of particles in General Relativity Theory’, Canadian Journal of Mathematics, vol. 1, no. 3, pp. 209–241. doi:10.4153/CJM-1949-020-8.
Schouten, JA 1954, Ricci-calculus: An introduction to tensor analysis and its geometrical applications, 2nd edn, Springer-Verlag, Berlin.
Main image: Generic calculations of components of the Riemann tensor in GR.