The Semi-Symmetric Metric Connection – Part IV

EinsteinEquation17

General Relativity

With the underlying geometry of the space defined by the semi-symmetric metric connection (SSMC) having been explored, we’re now in a position to examine how Einstein derived his field equations for GR. We will be seeking to follow similar physically-motivated reasoning, such as he used for GR, in our search for candidate field equations which might add electromagnetism to GR based on the geometrical properties of the SSMC.

The field equations of General Relativity

Einstein was attempting to generalise the Newtonian equation of gravitation, the so-named Poisson equation:

$\nabla^2 \Phi = 4\pi G \rho, \qquad (1)$

where $\nabla^2$ is the Laplacian differential operator $\delta_{ij}\partial^i \partial^j$ , $\Phi$ is the gravitational potential, $\rho$ is the mass density, and $G$ is the Newtonian gravitational constant. The equation is set in ordinary 3-dimensional space, so Einstein was seeking to generalise it both to 4-dimensional spacetime but also to be a (generally covariant) tensor equation. The motivations behind this can be found in many texts on GR (see, for example, Carroll 2004), so I won’t repeat them here.

Any generalisation of Newtonian gravitation will need to reduce to the form (1) in the appropriate limit – which is to say, quasi-static (i.e., ‘slow moving’) weak fields. The RHS of (1) is a description of the mass distribution, while the LHS is a second-order differential equation for the gravitational potential $\Phi$ engendered by that mass distribution.

In classical field theory, matter-energy fields can be represented by way of the so-named stress-energy, matter or energy-momentum tensor, conventionally denoted $T_{\mu\nu}$ , which is symmetric. It is a property of such fields that they are always conserved (albeit transformed); that is, a fundamental observation in physics is that energy can never be created or destroyed, only transformed from one form to another. This is expressed mathematically via the operation called ‘divergence’, which should yield a vanishing result for matter-energy fields; i.e., the divergence of the energy-momentum tensor is always zero. In Riemannian geometry, with a Christoffel connection, this can be written (using notation defined previously) as

$\overset{*}{\nabla}{}^\nu T_{\mu\nu} = 0. \qquad (2)$

If the mass distribution on the RHS of (1) is to be generalised to the 10-component symmetric energy-momentum tensor $T_{\mu\nu}$ , then this means that the LHS of (1) also needs to be generalised to 10 differential equations up to second-order for some as-yet unknown quantity which should generalise the gravitational field. Since $T_{\mu\nu}$ is a symmetric tensor, the form that the field equations should take is

$U_{\mu\nu} = k T_{\mu\nu}, \qquad (3)$

where the ‘unknown’ $U_{\mu\nu}$ will be also be a symmetric tensor object containing up to second-order partial derivatives of the quantity representing the gravitational field, and k is a constant of proportionality, to be determined a posteriori by comparison with the quasi-static, weak-field approximation of the field equations with Newtonian gravity. In addition, for the equation (3) to be a generally valid tensor equation, the object $U_{\mu\nu}$ would also have to have the mathematical property that its divergence also vanishes

$\overset{*}{\nabla}{}^\nu U_{\mu\nu} = 0, \qquad (4)$

because of (2).

In GR, the core assumption Einstein made is that the metric tensor $g_{\mu\nu}$ represents the potentials of the gravitational field. They are found in the equation that defines the invariant spacetime interval in Riemannian geometry

$ds^2 = g_{\mu\nu} dx^\mu dx^\nu, \qquad (5)$

which is a generalisation (both to arbitrary coordinates and curved spacetime) of the conventional spacetime interval from the (flat) Minkowski spacetime of Special Relativity

$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu, \qquad (6)$

where $\eta_{\mu\nu}$ can be represented by a matrix of constants, with 0s off the main diagonal and 1s along the diagonal, and the time-time component having the opposite sign to the space-space components: the two main forms used are

$\eta_{\mu\nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

$\eta_{\mu\nu} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix},$

usually rendered as $\eta_{\mu\nu} = \mathrm{diag}(-1, 1, 1, 1)$ and $\eta_{\mu\nu} = \mathrm{diag}(1,- 1,-1,- 1)$ respectively. Note that $\eta_{\mu\nu}$ differs from the familiar unity tensor/matrix (‘Kronecker delta’) $\delta^\mu_\nu$ where the 1s all have the same sign.

The 4×4 matrix representing the $g_{\mu\nu}$ can always be reduced to $\eta_{\mu\nu}$ at a point, by a suitable change of coordinate system (physically, this represents a change of reference frame to one that is ‘freely-falling’), an effect that encodes mathematically the so-named Equivalence Principle between a gravitational field and an accelerating reference frame (or, what is effectively the same thing, the equivalence of gravitational and inertial mass). As noted, local (i.e., at a point) conservation of energy and momentum is also encoded in (2). This principle is said to have come to Einstein while sitting in a chair in his job at the patent office in Bern, Switzerland, in 1907 (what he later called “the happiest thought of my life”). He wrote (Pais 1982, p.179):

all of a sudden a thought occurred to me: ‘If a person falls freely he will not feel his own weight’ … This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.

Thus, this was another consideration guiding Einstein’s search for a generalisation from the uniform motion (‘inertial reference frames’) distinguished in Special Relativity to non-uniform motion (non-inertial reference frames).

Einstein used the physics-based observation in (2) to look for a suitable object $U_{\mu\nu}$ that had the required mathematical property in (4), such that $U_{\mu\nu}$ is a function of the $g_{\mu\nu}$ found in (5). Therefore, he needed to know more about and utilise the mathematical properties of the geometrical objects found in Riemannian geometry (which, as noted previously in this series of posts, are all based upon the metric tensor $g_{\mu\nu}$ ). This was not a trivial undertaking, and he soon called upon his friend Marcel Grossman to help him (Pais 1982, p.212): “Grossman, Du musst mir helfen, sonst werd’ ich verrückt!” (G, you must help me or else I’ll go crazy!). I can say that generations of physics students since that time have shared that sentiment, even as we had the distinct advantage of knowing what the end-point would be, whereas Einstein was still very much searching for it.

After several years, and after considering a number of possibilities, Einstein was eventually led to what were the final field equations proposed at the end of November, 1915. With the infinite wisdom of hindsight, the derivation is relatively easy.

Recall that the Bianchi identity – the fifth identity discussed in Part III – is, for the Riemannian spacetime of GR with Christoffel connection, expressed as

$\overset{*}{\nabla}{}_{[\tau} K_{\nu\mu]\lambda}{}^\alpha = 0 \, . \qquad \textrm{(Vb)}$

Since the Riemann tensor is always anti-symmetric in its first two indices for any connection, from Identity (I), expanding out the anti-symmetrising brackets yields

$\overset{*}{\nabla}{}_{\tau} K_{\nu\mu\lambda}{}^\alpha + \overset{*}{\nabla}{}_{\nu} K_{\mu\tau\lambda}{}^\alpha + \overset{*}{\nabla}{}_{\mu} K_{\tau\nu\lambda}{}^\alpha = 0 \, . \qquad (7)$

Contracting on $\alpha$ and $\nu$ , and using the fact that $K_{\tau\alpha\lambda}{}^\alpha = - K_{\alpha\tau\lambda}{}^\alpha$ from Identity (I), yields

$\overset{*}{\nabla}{}_{\tau} K_{\mu\lambda} + \overset{*}{\nabla}{}_{\alpha} K_{\mu\tau\lambda}{}^{\alpha} - \overset{*}{\nabla}{}_{\mu} K_{\tau\lambda} = 0 \, . \qquad (8)$

Since $\overset{*}{\nabla}_{\alpha}g_{\mu\nu}=0$ (the metricity of the Christoffel connection), raising and lowering indices with the metric commutes with taking covariant derivatives, which simplifies tensor manipulation considerably. The middle term of (8) can thus be written as $\overset{*}{\nabla}{}^{\alpha} K_{\mu\tau\lambda\alpha}$ , which allows us to make use of Identity (II), namely that $K_{\mu\tau\lambda\sigma} = - K_{\mu\tau\sigma\lambda}$ . Equation (8) therefore takes the form

$\overset{*}{\nabla}{}_{\tau} K_{\mu\lambda} - \overset{*}{\nabla}{}^{\alpha} K_{\mu\tau\alpha\lambda} - \overset{*}{\nabla}{}_{\mu} K_{\tau\lambda} = 0 \, . \qquad (9)$

Contracting now on $\mu$ and $\lambda$ by taking $g^{\mu\lambda}$ of (9), yields

$\overset{*}{\nabla}{}_{\tau} K - \overset{*}{\nabla}{}^{\alpha} K_{\tau\alpha} - \overset{*}{\nabla}{}^{\mu} K_{\tau\mu} = 0 \, , \qquad (10)$

which, on lowering the derivative indices, re-writing dummy indices, and collecting terms, gives

$\overset{*}{\nabla}{}_{\tau} K - 2 \overset{*}{\nabla}{}_{\sigma} K_{\tau}{}^{\sigma} = 0 \, , \qquad (11)$

which shows that,

$\overset{*}{\nabla}{}_{\sigma} K_{\tau}{}^{\sigma} = \frac{1}{2} \overset{*}{\nabla}{}_{\tau} K \, . \qquad (12)$

Equation (11) can also be written as

$\overset{*}{\nabla}{}_{\sigma} ( K_{\tau}{}^{\sigma} - \frac{1}{2} \delta^{\sigma}_{\tau} K ) = 0 \, . \qquad (13)$

If a tensor $G_{\mu\nu}$ is defined, such that

$G_{\mu\nu} \overset{\mathrm{def}}{=} K_{\mu\nu} - \frac{1}{2} g_{\mu\nu} K \, , \qquad (14)$

then equation (13), which is derived from the Bianchi identity (Vb), shows that the tensor $G_{\mu\nu}$ always has vanishing divergence as an identity,

$\overset{*}{\nabla}{}_\nu \, G_{\mu}{}^{\nu} = 0, \qquad (15)$

which fulfills the condition (4) needed above. This strongly suggests that $G_{\mu\nu}$ could be our sought-after object $U_{\mu\nu}$ .

The symmetric tensor $G_{\mu\nu}$ is what is now called the Einstein tensor, since he realised the importance of the property (15) for the search for field equations. Making use of the property (15) in the equations (4) and (3) above yields the following proposed field equations:

$G_{\mu\nu} = k T_{\mu\nu} \, , \qquad (16)$

for some constant $k$ . Comparison with the Newtonian limit (a discussion of which can be found in any good textbook; again, see Carroll 2004) shows that $k$ is found to be

$k = \frac{8\pi G}{c^4} \, , \qquad (17)$

where $G$ is the Newtonian constant above from (1), and $c$ is the speed of light. Many texts and researchers use units where $G=c=1$ , so that (16) is often written in the simpler form

$G_{\mu\nu} = 8\pi T_{\mu\nu} \, . \qquad (18)$

This is a very elegant way to derive the field equations of General Relativity, and it stems from the mathematical properties of Riemannian geometry (left hand side) married to empirical properties of matter-energy fields (right hand side), based upon the fundamental assumption that spacetime is ‘curved’ (i.e., that the gravitational field is described by the spacetime metric which is engendered by the distribution of matter-energy). It is in this sense that I completely agree with Sean Carroll (2004, p.vii) that “General relativity is the most beautiful physical theory ever invented.” John Wheeler famously put it as: “spacetime tells matter how to move; matter tells spacetime how to curve”.

Of course, beauty and elegance are one thing; physical correctness is quite another. But, as is well-known at the time of writing this, so far GR has passed every experimental test it has ever been subjected to over the last century, so that it certainly appears to be a valid approximation to reality to the degree to which we have so far been able to test it (although it is still “only” a classical theory operating in what manifestly appears to be a quantum universe, which is clearly a problem).

At around the same time as Einstein wrote down his equation, Hilbert derived the field equations from a variational procedure using the Ricci scalar $K$ as the basis of the Lagrangian function. Any good textbook on GR will show how this is done, so I will not discuss it here; but also because I will not be seeking to use a Lagrangian for SSMC later. Rather, the idea will be to try to use a similar process of mathematics guided by physics to try to find appropriate field equations for a field theory combining GR and electromagnetism based upon the SSMC.

Later, Einstein added the so-called cosmological constant $\Lambda$ to the LHS, via a term $\Lambda g_{\mu\nu}$ , to comply with then-current statements by astronomers, something he later regretted (“the greatest blunder of my life”), but which has had a renewed interest in it since the discovery of ‘dark energy’ (essentially an anti-gravity effect) in the late 1990s. It may yet turn out not to have been a blunder after all…

Because the covariant derivative of the metric vanishes, and the covariant derivative of a constant also vanishes, the term $\Lambda g_{\mu\nu}$ could also be included in or added to the object $U_{\mu\nu}$ in (3) and still fulfill the required condition (4). With the cosmological term added to the LHS of (18), these days the full field equations of GR are usually written as (in appropriate units)

$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu} \, . \qquad (19)$

This expression can be rewritten in a different way that will be convenient for what is to come. By using Equation (16) (i.e., taking $\Lambda=0$ and returning to the use of $k$ for the constant) and writing out the expression for $G_{\mu\nu}$ in full, we get the form

$K_{\mu\nu} - \frac{1}{2} K g_{\mu\nu} = k T_{\mu\nu} \, . \qquad (20)$

Contracting on $\mu$ and $\nu$ yields (since $\delta^\mu_\mu = 4$ )

$K = - k T \, , \qquad (21)$

where $T \equiv T^\mu_\mu$ is the trace of $T_{\mu\nu}$ , which can be substituted back into (20) which is then rewritten as

$K_{\mu\nu} = k ( T_{\mu\nu} - \frac{1}{2} T g_{\mu\nu} ) \, . \qquad (22)$

This expresses the energy momentum tensor $T_{\mu\nu}$ and its trace $T_{}$ in terms of the Ricci tensor $K_{\mu\nu}$ alone, which is sometimes easier to work with, and will be especially so for the case below when $T_{\mu\nu}$ is the energy-momentum tensor for electromagnetism, which has vanishing trace, $T=0$ .

The equation (2a) shown in the main image at the top of this post uses $G_{im}$ to represent the Ricci tensor, and one can clearly see the resemblance of (22) to the (2a) in the image (up to a – sign caused by the index placement convention on the Riemann tensor). The paper from which the image is taken is from the proceedings of the Prussian Academy of Sciences, the first time the field equations of the theory were written down in their finalised form. It was submitted on Nov 25, 1915 (published on Dec 2) which is therefore, in effect, the ‘birthday’ of GR.

Gravity + electromagnetism – the Einstein-Maxwell theory

If we use the energy-momentum tensor describing the electromagnetic field in the Einstein field equations above, as well as the Maxwell equations to describe electromagnetism in curved spacetime, we get the standard approach to GR + electromagnetism known as Einstein-Maxwell Theory (EMT).

Given that we are seeking an extension to GR which will attempt in some way to include electromagnetism using a geometrically-motivated argument based upon the SSMC, it seems apposite for us to examine the situation in EMT, since such an extension will need to be able to account for EMT – and if not precisely then most assuredly not without good reason. For this, we will not need the cosmological term, and writing the proportionality constant as just $k$ will simplify the discussion.

[It turns out that EMT encompasses the totality of both classical dynamics and electrodynamics, because equations of motion for particles can be derived from these field equations without the need to postulate them separately (something necessary for both classical mechanics and Maxwellian electrodynamics). This surprising result flows from later work by Einstein and his collaborators Infeld and Hoffmann (see Einstein and Infeld 1949, and most especially the PhD thesis of Wallace 1940, supervised by Infeld), and is one of the most breathtakingly beautiful and elegant consequences of the non-linearity of the field equations of both GR and EMT.]

The field equations for the combined gravitational and electromagnetic field (in free space and with $\Lambda = 0$ ) can be found from equation (22), with the electromagnetic energy-momentum tensor for free space defined by

$T_{\mu\nu} = F_{\mu\sigma}F_\nu{}^\sigma - \frac{1}{4} g_{\mu\nu} F_{\sigma\tau}F^{\sigma\tau} \qquad(23)$

with the (obviously antisymmetric) Faraday tensor defined by

$F_{\mu\nu} = 2\overset{*}{\nabla}_{[\mu} A_{\nu]} \equiv 2\partial_{[\mu} A_{\nu]}, \qquad (24)$

where $A_\mu$ is the electromagnetic potential.

Note, however, that the trace of $T_{\mu\nu}$ vanishes, $T=0$ , so that equation (22) thereby reduces to

$K_{\mu\nu} = k T_{\mu\nu}. \qquad (25)$

The electromagnetic potentials $A_\mu$ obey the free-space Maxwell equations for curved spacetime:

$\overset{*}{\nabla}{}^\nu F_{\mu\nu} = 0, \qquad (26)$

so that (25) and (26) are the Einstein-Maxwell field equations for the combined gravitational and electromagnetic fields.

In Part V, we examine the experimental tests of GR over the last century, before moving on in Part VI to the issue of considering possible field equations for GR with electromagnetism based upon the SSMC.

Next time: Part V: Testing General Relativity.

References

Carroll, Sean M. 2004. Spacetime and geometry: An introduction to general relativity. San Francisco: Addison-Wesley.

Einstein, Albert, and Leopold Infeld. 1949. ‘On the motion of particles in General Relativity Theory’. Canadian Journal of Mathematics 1 (3): 209–41. doi:10.4153/CJM-1949-020-8.

Pais, Abraham. 1982. Subtle is the Lord: The science and life of Albert Einstein. Oxford University Press.

Wallace, Philip R. 1940. ‘On the relativistic equations of motion in electromagnetic theory’. PhD thesis, Dept of Mathematics, University of Toronto.

Image credit: from ‘The Field Equations of Gravitation’, The Collected Papers of Albert Einstein, vol. 6: The Berlin Years: Writings, 1914-1917, p.246. Available at: https://einsteinpapers.press.princeton.edu/vol6-doc/274.

General Relativity

The field equations of General Relativity

Gravity + electromagnetism – the Einstein-Maxwell theory

References

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