Remarkable parallel between formulae for Riemann tensor component and Gauss's 2-d curvature

For diagonal metrics:

Riemann curvature = 2-d Gauss curvature + ‘intermediary-dimension’ term

Contents:

1. The formula.

2. Calculation of the Riemann tensor components (RTCs) for some simple diagonal metrics, using this formula - to display properties of the formula, to help others who wish to learn how to apply it, and to illustrate how much of a simplification results form the use of this formula compared to standard versions.

3. We'll discuss what this formula reveals regarding some of the properties of the diagonal RTCs for diagonal metrics.

4. To touch base with results published elsewhere, we'll provide Landau & Lipschitz's (LL) formula (for the diagonal RTCs for diagonal metrics), and show how some of the properties mentioned in 3 are evident from their formula as well.

5. We'll show how to get from LL's formula to ours.

Preface:

AR's form of the Riemann Tensor Component (RTC) formula for diagonal metrics of any dimensionality Send feedback to: air1@nyu.edu

The formula (see below for definition of the single-index 'square-root' g's) has a manifest symmetry of A & B, symmetry we expect in the RTC, but that is not directly evident in the usual form. Also, it can be seen as a direct extension of Gauss’s formula (see below). In addition, it cleanly separates out two types of effect (see discussion below).

R^AB_AB = -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} _- h_Dh_Ah_B g_D^-2 [g_A,D/g_A][g_B,D/g_B] _D≠_A,B

…………..

Example: ds² = dt² – t²(dr² + sinh²r d²W)

gt = 1, h_t= +1; gr = t , h_r= -1; g_q= t sinh r , g_f= t sinh r sinq.

This certainly seems curved, but all the RTCs vanish, so it is a flat spacetime! Using our formula the calculation of the RTCs is rather simple.

R^tr_tr = -[g_t g_r]^-1{h_t[g_t,r /g_r],_r + h_r[g_r,t/g_t],_t} _- h_Dh_th_r g_D^-2 [g_t,D/g_t][g_r,D/g_r] _D≠ _t,r

= -[t]^-1{ [0/g_r],_r -[1/1],_t} ₊ h_D g_D^-2 [0_/g_t][g_r,D/g_r] _D≠ _t,r = 0

R^rf_rf = -[g_r g_f]^-1{h_r[g_r,f /g_f],_f + h_f [g_f,r/g_r],_r} _- h_Dh_rh_f g_D^-2[g_r,D/g_r][g_f,D/g_f] _D=_t,q

In this case, both g_r and g_f ARE functions of one of the two intermediate coordinates (t,q), namely t, but g_r is not a function of the other intermediate coordinate, q, and so one of the intermediate terms vanish, ie:

= -[t g_f]^-1{h_r[t_,f /g_f],_f + h_f [g_f,r/g_r],_r} _- h_t g_t^-2[g_r,t/g_r][g_f,t/g_f] _- h_q g_q^-2[g_r,q/g_r][g_f,q/g_f]

= -[t(t sinh r sinq)]^-1{h_r[0 /g_f],_f+h_f[(t sinh r sinq),_r/t],_r}_-h_tg_t^-2[t_{,t /}t][t sinh r sinq]_,t/[t sinh r sinq]_-h_qg_q^-2[t_,q/g_r][g_f,q/g_f]

= -[t² sinh r sinq)]^-1h_f [(t sinh r sinq),_r/t],_{r -} h_t [1] [t_{,t /}t][t_,t/t] – h_qg_q^-2 [0] [g_f,q/g_f]

= - t^-2 [sinh r sinq]^-1h_f sinq (sinh r),_r,_{r -} h_t t^-2

= - t^-2{h_f (sinh r)^-1(sinh r),_r,_{r +} h_t}

= - t^-2{h_{f +} h_t} = 0.

Our intuition is not capable of deciphering a metric directly, looking at it and determining whether or not it represents curvature of flatness. The Riemann tensor provides a fully dependable way to make such a determination: if it vanishes there’s no curvature, if it doesn’t there IS curvature. It is non-zero if and only if there is non-zero curvature. Whatever the functions g_A and g_B are, however much they make the metric look as though it is that of a curved surface, if all components of the Riemann tensor vanishes the space is in fact flat; if even one component is non-zero, it is NOT flat.

…

For angular 2-space in flat 4-d:

R^rf_rf = -[1x g_f]^-1{h_r[1_,f /g_f],_f + h_f [g_f,r/1],_r} _- h_t g_t^-2[g_r,t/g_r][g_f,t/g_f] _- h_q g_q^-2[1_,q/1][g_f,q/g_f]

= h_f [1],_{r -} h_t g_t^-2[1_,t/g_r][g_f,t/g_f] = 0.

…

flat 4-d in polar coordinates: ds² = -dt² + dr² + r²dq² + r²sin²qdf²

g_r = 1, g_t = 1, g_q = r, g_f = r sinq

R^rq_rq = -[g_r g_q]^-1{h_r[g_r,q /g_q],_q + h_q [g_q,r/g_r],_r} _- h_Dh_rh_q g_D^-2[g_r,D/g_r][g_q,D/g_q] _D=_t,f

= -[1x g_q]^-1{h_r[1_,q /g_q],_q + h_q [g_q,r/1],_r} _- h_t g_t^-2[g_r,t/g_r][g_q,t/g_q] _- h_f g_f^-2[1_,f/1][g_q,f/g_q]

= 0 + h_q [1],_{r -} h_t g_t^-2[1_,t/g_r][g_f,t/g_f] = 0.

…….

R^rf_rf = -[g_r g_f]^-1{h_r[g_r,f /g_f],_f + h_f [g_f,r/g_r],_r} _- h_Dh_rh_f g_D^-2[g_r,D/g_r][g_f,D/g_f] _D=_t,q

Since g_r is not a function of f, obviously the first part of the { } will vanish, and since g_r is not a function of either of the two intermediate coordinates (t,q), the last two terms vanish, ie:

= -[1 x g_f]^-1{h_r[1_,f /g_f],_f + h_f [g_f,r/1],_r} _- h_t g_t^-2[g_r,t/g_r][g_f,t/g_f] _- h _q g _q ^-2[g_{r, q /}g_r][g_{f, q}/g_f]

= -g_f^-1h_f g_f,r,_{r -} h_t g_t^-2[0_/g_r][0/g_f] _- h _q g _q ^-2[0 _/g_r][g_{f, q}/g_f]

= -[rsinq]^-1h_f [rsinq]_,r,_{r ,}

however: [rsinq]_,r,_r = [sinq] r_,r,_r = 0, so R^rf_rf = 0.

After some experience with this type of calculation, just by inspection one can write:

R^rf_rf = -[rsinq]^-1h_f [rsinq]_,r,_r = 0

………….

R^qf_qf = -[g_q g_f]^-1{h_r[g_q,f /g_f],_f + h_q [g_f,q/g_q],_q} _- h_th_qh_f g_t^-2 [g_q,t/g_q][g_f,t/g_f]- h_rh_qh_f g_r^-2 [g_q,r/g_q][g_f,r/g_f]

= -[g_q g_f]^-1{h_r[0/g_f],_f + h_q [rcosq/r],_q} _- h_th_qh_f g_t^-2 [0_/g_q][g_f,t/g_f]- h_rh_qh_f 1^-2 [r_,r/g_q][(rsinq)_,r/g_f]

= -[g_q g_f]^-1h_q [-sinq]} - h_rh_qh_f [1_/g_q][sinq/g_f] = 0

Analysis: the cancellation is from: -[g_q g_f]^-1h_q [g_f,q/g_q],_q} ₌ h_rh_qh_f g_r^-2 [g_q,r/g_q][g_f,r/g_f]

- [g_f,q/g_q],_{q =} h_rh_f g_r^-2 g_q,r g_f,r

- [g_f,q/g_q],_q/[h_f g_q,r g_f,r] ₌ h_r g_r^-2 : for h_r g_r = 1, this is: [g_f,q/g_q],_q= -[h_f g_q,r g_f,r]

For g_q = r : [g_f,q/r],_q= -[h_f r_,r g_f,r] , [1/r]g_f,q,_q= -h_f g_f,r

For h_f = 1, g_f = rf(q) : f(q)_,q,_q= -[rf(q)]_,r = - f(q) and so f = sin (maybe cos also?)

…..

Terminology: Diagonal metrics are possible when one can choose orthogonal coordinates.

.....

Speculations:

• Can it be that there is some invariant characterization of those spaces (curvatures) which allow for diagonal metrics?
• What is the equivalent in a non-metric connected space which allows for orthogonal coordinates?
• Are the Riemann curvatures of generally-curved metric spaces expressible as the curvature for a diagonal metric plus some other (perhaps intuitively-meaningful) term?
• Can the off-diagonal components for a diagonal metric also be expressed in a simple form? For them, LL gives:

Discussion: Properties of the Riemann curvature evident from AR's formula (above): The signs of the three terms: For 4-d spacetime, whatever the choice of signature:

• If both (l,i) are spatial we can ignore el & ei. For this case there is a term in the summation m = t, and so if that term is non-zero then el ei em = – 1, and so the sign of the summation term will change.
• At most only one of (l,i) can be time, and if indeed one of them is, then there will be a relative negative sign between the first two subterms; but in that case m cannot be t and so there is no sign change in the last term.

Note that all second order derivatives originate with Fl,i,i and Fi,l,l , however these are 2nd derivatives of the exponential function, not of the metric coefficients.

We can see that the second order effect, the true Riemann curvature, arises from the metric coefficients of the two dimensions represented in the RTC indices (e,i), whereas the ‘intermediary’ dimensions (represented by m) gives rise only to products of first derivatives.

Also: (and this is true not only for diagonal metrics): since the second derivative is with respect to the ‘other’ coordinate, one only gets first derivatives of a metric function with respect to ‘its’ coordinate, for example the time-metric coefficient g00 will not have 2ndderivatives with respect to time, etc (see LL sec 95 re ‘peculiarities of the structure of the Einstein equaitons’). However, time-spatial Riemann components will have 2nd time derivatives with respect to the spatial coordinate. For example, for the FRW metric, Rtrtrhas R..(t).

....

Notation: Many textbook calculations write the metric coefficients (especially g_tt and g_rr) in exponential form, for example g_tt = -e^f(r) , and compute the Riemann tensor components in terms of f(r). We find however that using (g_tt and g_rr , and even moreso their vierbein/tetrad-like) square roots, the Riemann tensor components end up taking a simpler form. For a diagonal metric of any dimensionality we can write:

ds² = g_aadx^adx^a (summed on a)

We define: g_D ≡ √|g_DD| ; No summation over D: to indicate this in the following we will insert [NS].

Example of the notation: g_{t ≡} √| g_tt | ^[1][2]

We also define w^D via:

w^D = g_Ddx ^D (No summation over D [NS: D]),

and we define the h_a containing the signature information:

ds² = g_aadx^adx^a (summed on a)

≡ h_a (w^a) ² ≡ h_a [g_a]² [dx^a]².

Summation will be assumed in the below unless NS is specifically indicated. (1) ^[2][3]

AR's form of the RTC formula where -[g_A g_B]^-1 is distributed over the entire expression:

[4] This has a manifest symmetry of A & B, symmetry we expect in the RTC, but that is not directly evident in the usual form. Also, it can be seen as a direct extension of Gauss’s formula (see below). In addition, it cleanly separates out two types of effect (see discussion below).

By inspection we see that R_ABAB is the same except with the power +1 instead of -1 for the coefficient term -[g_A g_B].

We'll now arrive at the above formula via the one below given by L. D. Landau, E. M. Lifshitz (Abbrev: “LL”) (section 92): For _i≠ _{l :}

R_lili = e_l e^2F_l (F_i,iF_l,i – F²_l,i – F_l,i,i) + e_i e^2F_i(F_l,lF_i,l – F²_i,l – F_i,l,l) - e_l e^2F_lS_m≠ _i,l e_me^2(F_i ^{- F}_m⁾ F_i,m F_{l,m i}≠ _l (B)

Note that all second order derivatives originate with F_l,i,i and F_i,l,l . We can see that the second order effect, the true Riemann curvature, arises from the metric coefficients of the two dimensions represented in the RTC indices (l,i), whereas the ‘intermediate’ dimensions (represented by m) gives rise only to products of first derivatives.

We’ll rewrite LL’s formula, changing only the third (last) term by taking 2 terms out of the summation for clarity:

R_lili = e_l e^2F_l (F_i,iF_l,i – F²_l,i – F_l,i,i) + e_i e^2F_i(F_l,lF_i,l – F²_i,l – F_i,l,l) - e_i e^2F_i e_l e^2F_l S_m≠ _i,l e_me^-2F_m F_i,m F_{l,m i}≠ _l

R^li_li = g^ll gⁱⁱ R_lili = (1/g_llg_ii) R_lili = (e^-2F_i e^-2F_l) R_lili

= e_l e-^2F_i(F_i,iF_l,i – F²_l,i – F_l,i,i) + e_i e^2F_l(F_l,lF_i,l – F²_i,l – F_i,l,l) - e_i e_l S_m≠ _i,l e_me^-2F_m F_i,m F_{l,m i}≠ _l

With the metric coefficients expressed using LL’s signature and notation:

g_ii = e_i e^2Fi ; e₀ = 1, e_a = - 1. a = spatial coordinate indices ; (6)

using our “square-root g’s”:

g_ii = e_i g_i ² , g_i = e^Fi , g_i,m = F,_m e^Fi .

g_i,m/g_i = F,_m . (7)

RTC formulas (A) and (B) have three terms; from (7) we can see that the third (and last) terms of both are the same.

To transform this into our formula, we'll utilize three simple identities (identities valid for any function, not just the metric coefficients, but we'll use g's to simplify their substitution into the above RTC formula).

For any function f(x^j), j = 1,4, where we will use g_e instead of f, so we write f(x^j) as g_e(x^j): from simple differentiation with respect to a specific coordinate xⁱ (no summation):

(1)

Rearranging terms in (1) we have:

. (Identity 1)

We now have two different functions g_e(x^j) and g_i(x^j). The ‘i’ in g_i may refer to a specific coordinate xⁱ, but the function g_i(x^j) has no a priori relationship to that coordinate.

Given the expression: (2a)

we can use Identity 1 to rewrite the factor in { }, so that (2a) becomes:

, (2b)

which we rewrite as: . (2c)

To reduce the above, we’ll use another result. From simple differentiation:

, (3)

so we can rewrite (2c) as: (2d).

In summary, writing (2a) = (2d) we have three terms collapsing to one:

(Identity 2).

………

From simple differentiation: ,

, or in the other notation:.

………………………

Using identity 2 we can write the three sub-terms which appear in the first term of LL’s formula (B) as one term (note that our e is LL’s l):

F_i,iF_l,i – F²_l,i – F_l,i,i à (e^Fi/e^Fl) [F_l,ie^Fl/e^Fi]_,i

which expressed in our notation is:

-[g_A g_B]^-1[g_A,B /g_B],_B (8)

The first two terms of (B) are identical to each other just with i and l interchanged, and when cast in terms of our metric coefficient notation with single indices, they can be transformed into the negative of:

_{A,B .}

The RTC will contain a product term for each coordinate D which is neither A nor B: in 4-d, there are therefore at most two such coordinates, D = C and D = E, and therefore only two ‘additional terms’.

Note that Gauss curvature involves connections of the type √G^A_BB whereas the connections in this ‘additional’ term are of other type, where the coordinate wrt which we take the derivative is also the coordinate of the metric term in the denominator: eg:

√G^a_ba = g_a,b/g_a

Substituting A for a and D for b gives √G^A_AD, and then substituting B for a and D for b gives √G^B_BD . The full ‘additional’ term for the interconnection of the dimensions represented by the coordinates A and B is:

{h_Dh_B (h_A) g_D^-2 }√G^A_AD √G^B_{BD D} ≠_A,B = (h_Ah_Bh_D) g_D^-2 [g_A,D/g_A][g_B,D/g_B] _D ≠_A,B

where D is the ‘Dummmy’ summation index.

This additional term vanishes if one or both of the terms g_A,D/g_A and g_B,D/g_B vanish, ie if either one or both of g_A,D and g_B,D vanishes. Thus the additional term vanishes unless both of the metric coefficients corresponding to the indices of the RTC (ie g_AA and g_BB) have non-vanishing derivatives wrt the same coordinate(s) D.

· We will term “intermediary coordinates” those coordinates D wrt which both of the metric coefficients g_A and g_B corresponding to the indices of the RTC have non-vanishing derivatives. We can then say that this term relates to the curvature arising from the interconnection of the (A,B) 2-section with the “intermediary dimensions” represented by the “intermediary coordinates” D.

We can call the new term the “Intermediary Term”: Both the A and B metric coefficients must have derivatives w.r.t. the “intermediary” coordinate D, otherwise this whole term vanishes

Summary: Although the formulae for the curvature of 2-d surfaces in a 3-d flat space is the same as for the curvature of an intrinsically-curved 2-d space not embedded in any higher dimension, for a 2-d section in a 3 or higher-dimensional space such as the 4 dimensions of spacetime, we must take into account the indirect curvature deriving from the way that the various dimensions interact with each other, and this necessitates adding a term onto the curvature formula for curved 2-d surfaces.

Keep this structure of the Riemann curvature component in mind.

· The terms are manifestly symmetric under A ßà B. [(AR to AR: check the signature part of the last term)]

· The first (Gaussian) part of the RTC is dependent on (or quantify the curvature arising from) the interconnection of the two coordinates corresponding to the two indices of the RTC (or of the metric coefficients corresponding to the coordinates), specifically on the derivatives of the metric coefficient corresponding to one coordinate, derived w/r/t the other coordinate. In this sense it is the curvature arising from the “mutual” interconnection. The product term relates to the curvature arising from the interconnection of the (A,B) 2-section with the “intermediary dimensions” represented by the “intermediary coordinates” D.

· This division of the RTC into “mutual” and “intermediary” terms grants this form of the RTC formula a certain intuitive basis, and also helps intuit shortcuts in calculations of the RTC (some RTCs can be computed mostly ‘by inspection’ using this formula; the terms arising in the RTCs can be easily traced to the metric [some properties of special metrics are easily discernible eg for metrics with inverse relationship eg g_tt = -g_rr^-1 one obtains a Laplacian].

The Riemann Tensor Components (RTC) are composed of terms interrelating the metric coefficients of two dimensions at a time (eg one term is a measure of the curvature of the 2 dimensions represented by the coordinates A and B, another is a measure of the curvature of the 2 dimensions represented by the coordinates by A and D, another for B and D). In general one must compute the Riemann components for all pairs of two indices, that is, one computes the curvatures in each “2-section” (A,B) as follows:

R^AB_AB = -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} _- h_Dh_Ah_B g_D^-2 [g_A,D/g_A][g_B,D/g_B]} _D≠_A,B ^[6][4]

The Riemann tensor as a whole takes into account the curving of the whole spacetime by computing the curvature of each individual 2-surface of the whole spacetime. The field equation will have on the left hand side a combination of these individual curvatures, adding them together and making the sum equal to the tensor element on the right hand side, which is some aspect of the source (an element of the source tensor).

…..

Introduction: Define the basis one-form w^A and the connection two-form w^A_D.

w^A = g_A dA

w^A_D =

w^A_D clearly involves a first derivative of the one of the two metric coefficients involved, w.r.t. the other coordinate…..

dw^A_B involves a second derivative of each of the two metric coefficients involved, w.r.t. the other coordinate. Therefore the terms in the RTC coming from dw^A_B will involve a second derivative of each of the two metric coefficients involved, w.r.t. the other coordinate. This makes sense as curvature since it takes into account a type of twisting of the metric basis.

The RTC can be read from the coefficients in the “curvature two-form” R^A_B :

R^A_B = dw^A_B - w^A_D^ w^D_B : summed over D≠A,B (1)

The origin of the term in the RTC involving the product of the connection two-forms can be seen via construction of the RTC via parallel transport around a closed circuit, or via the two-form method, there will also be a In general the Riemann Tensor Components (RTC) for an orthogonal metric are composed of terms coming from one or both of dw^A_B and w^A_D ^ w^D_{B D} ≠ _A,B ; for some special symmetry situations only one of these terms survives, whereas for non-orthogonal coordinates there are more than these two terms.

.............................................

The RTC are given by: [I have to check the signs]:

R^AB_AB = -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} _- h_Dh_Ah_B g_D^-2 [g_A,D/g_A][g_B,D/g_B] _D≠_A,B (2)

Defining:

g_A,B /g_B ≡√^B_AA

the relationship to the connection coefficients (Christoffel symbols) is:

G^B_AA = ½ [g_AA,B /g_BB] = ½ [g_A²_,B /g_B²] = ½ 2 g_Ag_A,B /g_B² = [g_A/g_{B ]} g_A,B /g_B = [g_A/g_{B ]} √^B_{AA ;}

G^B_BD = ½ [g_BB,D /g_BB] = ½ [g_B²_,D /g_B²] = ½ [2g_Bg_B,D /g_B²] = [g_B,D /g_B] ≡ √^B_BD

See file: “Comparing my RTC formula to the usual one”

Note:

v No matter how many dimensions the space each term in the RTC above is computed only in a 2-d subspace, either (A,B), or (A,D), or (B,D) ; this makes each such calculation relatively simple.

[ADVANCED: This is so since each connection two-form w^A_D depends only on the two dimensional subspace (A,D), and the “curvature two-form” R^A_B is composed of these connection two-forms]

v The first two terms of the RTC above are simply the Gaussian curvature for the 2-d surface (A,B):

K_AB = -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} (3) ^[7][5]

[ADVANCED The whole term [g_A,D/g_A][g_B,D/g_B] vanishes unless both the A and B metric coefficients have derivatives w.r.t. the “intermediary” coordinate D. From now on, this is the sense in which we will use the term “intermediary” coordinate.

v [ADVANCED w^A_D ^ w^D_{B D} ≠ _A,B contributes a term to R^A_BAB proportional to: [g_A,D/g_A][g_B,D/g_B] (summed over D) ^[8][6]

Curvature of a 2-d Surface: Subspace of a Higher-d Space

For a 2-d space, ie a surface, there are only two orthogonal coordinates A and B, and therefore there is no D≠ A,B coordinate to be summed over, and so there can be no “intermediary” term [ADVANCED: w^A_D ^ w^D_B]. Therefore the RTC for a 2-d space (A,B) [ADVANCED: depend only on the terms arising from dw^A_B , namely]

R^AB_AB = -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A}

which is in fact Gauss’s formula for 2-d surfaces.

In a higher-d space containing this 2-d subspace (A,B), the RTC will then simply be the Gaussian curvature for the 2-d surface with the addition of the intermediary term:

R^AB_AB = K_AB + [g_A,D/g_A][g_B,D/g_B] _{(summed over D)} (3)

v We will term “self-contained” those subspaces in which the intermediary term vanishes (ie one or both of the A and B metric coefficients have no derivatives w.r.t. the “intermediary” coordinate D). Thus, “self-contained” 2-d subspaces have the same curvature as they would if there were no other dimensions: ie their RTC is equal to their Gaussian curvature.

v The term in the RTC coming from the dw^A_B is:

h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A (4)

When in the metric of the 2-d surface (A,B) the metric coefficients are functions of each other’s coordinate, then both terms survive. If neither is a function of the other, both terms vanish. [And of course if only one is a function of the other’s coordinate then only one term survives.]

[In general the RTC are composed of terms coming from both dw^A_B and w^A_D ^ w^D_{B D} ≠ _A,B .]

v But note that the first derivative is ‘normalized’ before the second derivative is taken: eg [g_A,B /g_B],_B , and the resulting second derivative is normalized by the product of the two metric coefficients: -[g_A g_B]^-1{[g_A,B /g_B],_B .

……………….

Appropriateness of the Word ‘Curvature’ as Applied to Spacetime Warping

The Gaussian curvature measure applies to an ordinary 2-dimensional surface in ordinary three-dimensional space, and we are very easily intuitively aware of the meaning of the word ‘curvature’ in that context. Although the meaning of the word ‘curvature’ is in a sense allegorical when applied to higher dimensions since we can never observe it directly as we do for 2-surfaces, and is certainly allegorical when applied to a combination of space and time as for spacetime, nevertheless since the Riemann ‘curvature’ is so closely modeled on the 2-d Gaussian curvature, we have good reason to maintain usage of the word ‘curvature’ even when applied to higher-dimensional spacetime warping.

≠⅞∂∆√∫

[1] See for example Synge: Ch VIII section 5 (see also section 1): He defines V as positive sqrt of g_tt, and writes eq (166) p339: R⁴₄ = V^-1del²V which is my R⁴₄ = g_t^-1del²g_t . R⁴₄ = g_t^-1del²g_t . See also Landau Lifshitz (LL) re this.

Synge also write similar equations (all are included as eq 166) for the other Ricci tensor components, eg the spatial Ricci tensor components = a spatial Ricci scalar plus the above time one.

[Note that the Newtonian limit of del2 has square root…]

[2] Since I define a one-index quantity g_m = sqrt|g_mm| , it is of interest to see that Landau Lifschitz (LL): p277: uses a similar notation for something else: g_a = g_0a , where _a are the spatial indices only.

[Also: tetrad notation - and tetrad formulation of GR - in field theory.]

[3] Using different font-layout: R^AB_AB = -[g_A g_B]^-1{h_A [g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} - h_Dh_Ah_B g_D^-2 [g_A,D/g_A][g_B,D/g_B] _D≠_A,B (A)

Or: -[g_A g_B]^-1{h_A[g_A,B /g_B],_B + h_B[g_B,A/g_A],_A + h_Dh_Ah_B g_D^-2 g_A,D g_{B,D D}≠_A,B}

[4] See my formula for the Ricci tensor components for a diagonal metric: LL gives the formula:

R_ii = S_l≠ _i (F_i,iF_l,i – F²_l,i – F_l,i,i + e_i e_l e^2(F_i ^{- F}_l⁾ ) [F_l,iF_i,l – F²_i,l – F_i,l,l - F_i,lS_m≠ _i,l F_m,l]

[5][1] See eg O’Neill p81

[6][4] The relationship to the common expression in terms of Christoffel symbols is made when we see that:

G^B_AA = ½ [g_AA,B /g_BB] = ½ [g_A²_,B /g_B²] = ½ 2 g_Ag_A,B /g_B² = [g_A/g_{B ]} g_A,B /g_B ;

G^B_BD = ½ [g_BB,D /g_BB] = ½ [g_B²_,D /g_B²] = ½ [2g_Bg_B,D /g_B²] = [g_B,D /g_B]

[7][5] w^A_D ^ w^D_{B D not=A,B} = [g_Ag_Bg_D²] ^-1_{g_A,D g_D,B w^A ^ w^D _- h_D h_B g_A,D g_B,D w^A ^ w^B + h_A h_B g_D,A g_B,D w^D ^ w^B_{}D A,B (7b)}

[8][6] w^A_D ^ w^D_{B D not=A,B} = [g_Ag_Bg_D²] ^-1 _{g_A,D g_D,B w^A ^ w^D _- h_D h_B g_A,D g_B,D w^A ^ w^B + h_A h_B g_D,A g_B,D w^D ^ w^B_{}D A,B (7b)}