Formula for the Riemann Tensor Components (RTC)

for orthogonal metrics of any dimensionality. [1] Air1@nyu.edu

Notation

For an orthogonal 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])

We also define h_D and w^D via:

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

ds² h_a (w^a) ² h_a [g_a]² [dx^a]². (summed over a: summation will be assumed in the below unless NS is specifically indicated) [The h_a contain the signature information.] (1) ^[2]

………

2-d and Gaussian Curvature

For a 2-d surface, using an orthogonal coordinate system (x_A, x_B), the metric is:

ds² = g_AAdx^A ² + g_BBdx^B ² (no sum on A or B [NS: A,B]).

Gauss’s curvature formula for this surface is: [See Barrett O’Neill: “Semi Riemannian Geometry” p81]

K_AB = - [g_A g_B]^-1{h_A [g_A,B /g_B],_B + h_B[g_B,A/g_A],_A} [NS: A,B].

The Gaussian curvature arises from the interconnection of the two dimensions represented by the coordinates corresponding to the two indices of K (or of the metric coefficients corresponding to the coordinates). Quantitatively, the Gaussian curvature arises via – or is dependent - the derivatives of the metric coefficient corresponding to one coordinate, derived w/r/t the other coordinate. In this sense the Gaussian curvature is the curvature arising from a “mutual” interconnection. (Then a second derivative is formed, giving rise to second order terms.)

Intrinsic Curvature: The above formula also provides the intrinsic curvature of a 2-d space described by the above metric.

Intrinsic Curvature for 3-d and higher

The formula for the Riemann Tensor Components (RTCs) for orthogonal metrics of any dimensionality is the Gauss term above plus an additional term which we call the ‘intermediary term’ I:

I^AB = h_Dh_B (h_A) g_D^-2 [g_A,D/g_A][g_B,D/g_B] _{D A,B} (summed on D A,B)

= [g_A g_B]^-1 h_Dh_B (h_A) g_D^-2 [g_A,D][g_B,D] _{D A,B}

This term vanishes unless both of the metric coefficients corresponding to the indices of the RTC have non-vanishing derivatives w/r/t the same coordinate(s). We will term “intermediary coordinates” those coordinates w/r/t which both of the metric coefficients 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 two dimensions represented by A and B via the dimensions represented by the “intermediary coordinates”.

We can therefore write: (Note: No Sum on A,B. However, D summed on DA,B)

R^AB_AB = K_AB + I^AB = - [g_A g_B]^-1{ [g_A,B /g_B],_B + h_A h_B[g_B,A/g_A],_A + h_Dh_B (h_A)g_D^-2 [g_A,D][g_B,D] _{D A,B} }

Where the first (Gaussian) part of the RTC is the curvature arising from the “mutual” interconnection, and the ‘additional’ (third) term is the curvature arising from the interconnection via the “intermediary” dimensions/coordinates.

· The terms are manifestly symmetric under A ßà B. [(AR to AR: check the signature part of the last term)] Note that this is not true of the usual formulae for the RTC.

· This division of the RTC into “mutual” and “intermediary” terms grants this form of the RTC formula a certain intuitive basis.

· Use of this formula facilitates intuiting 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 [R^t_t is a Laplacian]).

……………..

Derivation of the above RTC Formula

Introduction

We wish to find the Riemann tensor components (RTCs) R^A_BAB from the formula for the curvature two-form:

Term #1 Term #2

R^A_B = dw^A_B - w^A_D^ w^D_B (see eg MTW p351) (2a)

= - R^A_B|EF| w^E ^ w^F (E<F) (see eg MTW p352)

The (A,B,E,F) RTCs are obtained as the coefficients of w^E ^ w^F .

With two raised indices:

R^AB = - R^AB_|EF| w^E ^ w^F (see eg MTW p358) (2b)

= - R^AB_AB w^A ^ w^B _(NS:A,B) - R^AB_AD w^A ^ w^D _(D>A) - R^AB_DB w^D^ w^B _(D<B) - R^AB_CE w^C^ w^E _(C,EA,B)

For orthogonal metrics, we will be interested in calculating the RTCs with repeating indices, ie the R^A_BAB (NS: A,B) [3] and so we seek the coefficient of the term w^A ^ w^B . To do so we begin below with a calculation of w^A_B from which we will compute both terms in eq (2a) {ie: dw^A_B (term #1) and w^A_D^ w^D_B (term #2)}.

A) Computation of w^A_D: ^footnote [4]

For the specific coordinate x_A: [A:NS]

dw^A = d[g_Adx^A] [A:NS]

= d[g_A]dx^A + g_A ddx^A = g_A,D dx^D^dx^A _DA

= g_A,D w^D/g_D ^ w^A/g_A = [g_A,D/(g_Dg_A)] w^D ^ w^A _{D A}. (3)

However we also know that:

dw^A = - w^A_D^ w^D _{D A} = + w^D ^ w^A_{D D A} [5] (4)

By equating (3) and (4) we obtain:

w^D ^ w^A_{D D A} = w^D ^ w^A [g_A,D/(g_Dg_A)] _{[D not=A]} (5)

The first impression is that

w^A_D = w^A [g_A,D/(g_Dg_A)] _{[D not=A]}

however since by definition

w^D ^ w^D = 0

we can add a term proportional to w^D without affecting matters, ie:

[w^A + f w^D] ^ w^D = w^A ^ w^D

(where f is a function of the coordinates, we’ll signify it’s unknown nature by writing it below as [?] )

ie the more general solution of equation (5) above is:

w^A_D = [g_A,D/₍g_Dg_A)] w^A + [?] w^D (5b)

where [?] signifies an unknown term to be determined below.

Note that whereas A is a specific coordinate, D is summed over all coordinates other than A, and so for each coordinate value of D there will be a companion term: [?] w^D .

We now switch A ßà D to obtain:

w^D_A= [g_D,A/(g_Ag_D)] w^D + [?] w^A (5c)

We also know that:

w^A_D = - w^D_A h_A h_{D . D A}

We will assume from this point on that we sum over all values of D except for A.

Therefore:

w^A_D = [g_A,D/(g_Dg_A)] w^A + [?] w^D

= - w^D_A h_A h_D

= - h_A h_D { [g_D,A/(g_Ag_D)] w^D + [?] w^A }

So we can set:

[?] w^D = - h_A h_D [g_D,A/(g_Ag_D)] w^D (5d)

- h_A h_D [?] w^A = [g_A,D/₍g_Dg_A)] w^{A Footnote} [6]

Therefore inserting [?] w^D from (5d) into (5b) we can write the connection two-form as:

w^A_D = [g_A,D/(g_Dg_A)] w^A - h_A h_D [g_D,A/g_Ag_D)] w^{D Footnote [7]}

= [g_Ag_D] ^-1 {g_A,D w^A _- h_A h_D g_D,A w^D} . ^(6A1)

Written in terms of dx:

w^A_D = [g_Ag_D] ^-1 {g_A,D g_Adx^A _- h_A h_D g_D,A g_D dx^D } ^(6A2)

w^A_D = [g_A,D/g_D] [dx^A] _- h_A h_D [g_D,A/g_A][dx^D] . (6c)

· Note the symmetry of the two terms in A and D (other than for the signature symbols).

· Note from (6A1) that if both g_A,D and g_D,A are zero then w^A_D = 0. This makes geometric sense: if both of these derivatives vanish, there is clearly no direct ‘connection’ between the coordinates A and D. Often we will find that the two metric coefficients g_A and g_D are functions of only one of the coordinates (A,D), not both, and so one of the two terms in (6) vanishes.

Significance of the above equation:

_{· The above formulation allows use of a closed form of} w^A_{D for all further computations!}

_{· The development up to this point displays in a transparent manner the origin of each term in} w^A_D.

……………………….

· Note the relationship to the connections of the terms like g_A,D/g_D in the connection two-form:

G^D_AA = g_AA,D/[-2g_DD] = [h_Ag_A²]_,D/[-2h_Dg_D²] = 2h_Ag_Ag_A,D/[-2h_Dg_D²]

= -h_Dh_A [g_A/g_D][g_A,D/g_D]

If we define:

G^D_AA [g_A,D/g_D]

then we can write this as:

G^D_AA = -h_Dh_A [g_A/g_D] G^D_{AA .} [8]

…………………………………………….

B) Computation of w^A_D ^ w^D_B

From equation (6A1) w^A_D = [g_Ag_D] ^-1 {g_A,D w^A _- h_A h_D g_D,A w^D} _{(NS: A)}

we can write:

w^A_D ^ w^D_{B [D A,B]}

= [g_Ag_D] ^-1_{g_A,D w^A _- h_A h_D g_D,A w^D_} ^ [g_Bg_D] ^-1_{g_D,B w^D _- h_D h_B g_B,D w^B_{} (NS: A,B)} (7a)

[As a check: w^A_D and w^D_B have the same form, but with (A,D) à (D,B) ie [A à D, Dà B] .

We combine the two products [g_Ag_D] ^-1 and [g_Bg_D] ^–1 :

[g_Ag_D] ^-1 [g_Bg_D] ^–1 = [g_Ag_Dg_Bg_D] ^–1 = [g_Ag_Bg_D²] ^-1

so that we can write:

w^A_D ^ w^D_{B [D A,B]} = [g_Ag_Bg_D²] ^-1_{g_A,D w^A _- h_A h_D g_D,A w^D_} ^ _{g_D,B w^D _- h_D h_B g_B,D w^B_}

_{Term: #1 #2 #3 #4}

Multiplying the terms in eq. (7a) will give us these products:

(1) x (3) + (1) x (4) + (2) x (3) + (2) x (4)

Note that:

_· (2) x (3) = 0 because both have w^D _:

_· in (2) x (4) one has – x – = + , and h_D h_{D = 1 , leaving only} h_Ah_B;

Therefore:

w^A_D ^ w^D_{B D A,B} (NS: A,B) = (1) x (3) + (1) x (4) + (2) x (4)

= [g_A g_B g_D²]^-1{g_A,D g_D,B w^A ^ w^D _- h_Dh_B g_A,D g_B,D w^A ^ w^B + h_Ah_B g_D,A g_B,D w^D^w^B}_{D A,B} (7b)

Note that the above is term #2 in eq (2a) copied below:

R^A_B = dw^A_B - w^A_D^ w^D_B (see eg MTW p351) (2a)

_{C) Computation of} d w^A_B

w^A_B = [g_A,B/g_B] [dx^A] _- h_A h_B [g_B,A/g_A][dx^B] . (eq 6c with: D à B)

From equation 6c above we find:

d w^A_B = [g_A,B /g_B],_c dx^C ^ dx^A _{C A} - h_A h_B [g_B,A/g_A],_c dx^C ^ dx^B _{C B} (8a) (NS: A,B)

Changing the order of the first wedge product, and therefore changing the sign:

d w^A_B = - [g_A,B /g_B],_c dx^A ^dx^C _CA - h_A h_B [g_B,A/g_A],_c dx^C ^ dx^B _{C B} (8a) (NS: A,B)

Note that in order to take the derivatives we employed eq 6C with dx^A rather than 6A with w^A ; however now that we have the derivative we can put this back into terms of w^A and w^B_:

d w^A_B = -[g_Ag_C]^-1[g_A,B /g_B],_cw^A ^w^C _{C A} - h_A h_B[g_C g_B]^-1[g_B,A/g_A],_c w^C ^ w^B _{C B} (8b)

_{Term #1 Term #2}

Note:

_{· In the first term’s sum over C, C cannot be A, but since B is NOT A, C CAN be B.}

_· when in the first term C = B, and in the second term C = A _{both terms contain} w^A _^ w^B.

_· In the above case, [g_B g_A]^-1 is a common factor.

_{For term #1: We separate out the case C = B and C B.}

_{For term #2: we separate below the cases C = A, and CA. (NS: A,B)}

d w^A_B = a) Term #1 with C = B à -[g_A g_B]^-1 [g_A,B /g_B],_B w^A ^ w^B

b) + Term #1 with C B à - [g_A g_C]^-1[g_A,B /g_B],_c w^A ^ w^C _{C A,B}

c) + Term #2 with C = A à - h_A h_B [g_A g_B]^-1 [g_B,A/g_A],_A w^A ^ w^B

d) + Term #2 with C A à - h_A h_B[g_C g_B]^-1[g_B,A/g_A],_c w^C ^ w^B _{C A,B}

Collecting terms with the same wedge products:

a) and c) have the same wedge: also, [g_A g_B]^-1 is a common factor: so:

a) + c) = -[g_A g_B]^-1 {[g_A,B /g_B],_B + h_A h_B [g_B,A/g_A],_A] w^A ^ w^B

In d), switching w^C ^ w^B to w^B ^ w^C so that we need to introduce a ‘-‘, and then adding to b):

b) + d) = - g_C^-1 {g_A^-1[g_A,B /g_B],_c w^A - h_A h_B [g_B]^-1[g_B,A/g_A],_c w^B }^ w^C _{C A,B}

and so we can write: note: we change the summation index from C to D:

d w^A_B = - [g_A g_B]^-1 { [g_A,B /g_B],_B + h_A h_B [g_B,A/g_A],_A } w^A ^ w^B +

- {[g_A g_D]^-1[g_A,B /g_B],_D w^A - h_A h_B[g_D g_B]^-1[g_B,A/g_A],_D w^B}^ w^D _{D A,B} (NS: A,B) (8c)

Note that the above is term #1 in eq (2a) [rewritten below].

R^A_B = dw^A_B - w^A_D^ w^D_B (see eg MTW p351) (2a)

D) Finding the Riemann tensor components (RTCs)

Inserting the results of equations (7b) and (8c) into eq. (2): (we use the summation index D):(NS: A,B)

R^AB = dw^A_B - w^A_D^ w^D_B = - R^AB_AB w^A ^ w^B - R^AB_AD w^A ^ w^D - R^AB_DB w^D^ w^B

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

{-[g_A g_D]^-1[g_A,B /g_B],_D w^A + h_A h_B[g_D g_B]^-1[g_B,A/g_A],_D w^B}^ w^D _{D A,B}

- [g_Ag_Bg_D²]^-1{g_A,D g_D,B w^A ^ w^D _- h_Dh_B g_A,Dg_B,D w^A ^ w^B + h_Ah_B g_D,A g_B,D w^D^w^B}_{D A,B}

We now collect terms with the same wedge products:

Note the three terms in w^A ^ w^B : (the first two, and the second-to-last): adding them gives:

-[g_A g_B]^-1{[g_A,B /g_B],_B + h_A h_B[g_B,A/g_A],_A}w^A^w^B - {[g_Ag_Bg_D²]^-1 (-) h_Dh_B g_A,Dg_B,D w^A ^ w^B }_{D A,B}

Distributing the common w^A^w^B and - [g_A g_B]^-1 gives us:

- [g_A g_B]^-1 { [g_A,B /g_B],_B + h_A h_B[g_B,A/g_A],_A - g_D^-2 h_Dh_B g_A,Dg_B,D }w^A ^ w^B _{D A,B}

Note that there are only two terms in w^B^ w^D:

1) - h_A h_B[g_D g_B]^-1[g_B,A/g_A],_D w^B^ w^D _{D A,B}

2) - [g_Ag_Bg_D²]^-1 h_Ah_B g_D,A g_B,D w^D^w^B _{D A,B}

= h_Ah_B [g_Dg_Bg_Ag_D]^-1 g_D,A g_B,D w^B ^ w^D _{D A,B}

Terms 1) and 2) have h_Ah_B [g_Dg_B]^-1 w^B ^ w^D in common, and so adding 2) + 1) gives:

h_Ah_B [g_Dg_B]^-1 {[g_Ag_D]^-1 g_D,A g_B,D - [g_B,A/g_A],_D} w^B ^ w^D

so we find:

R^AB = - [g_A g_B]^-1 { [g_A,B /g_B],_B + h_A h_B[g_B,A/g_A],_A - g_D^-2 h_Dh_B g_A,Dg_B,D }w^A ^ w^B _{D A,B}

+ h_Ah_B [g_Dg_B]^-1 {[g_Ag_D]^-1 g_D,A g_B,D + [g_B,A/g_A],_D} w^B ^ w^D

- g_A ^-1{g_D^-1 [g_A,B /g_B],_D + [g_Bg_D²] ^-1 g_A,D g_D,B} w^A ^ w^D _{D A,B}

= - R^AB_AB w^A ^ w^B - R^AB_AD w^A ^ w^D - R^AB_DB w^D^ w^B

Therefore:

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

= + [g_A g_B]^-1{[g_A,B /g_B],_B + h_A h_B[g_B,A/g_A],_A} - h_Dh_B g_D^-2 [g_A,D/g_A][g_B,D/g_B] _{D A,B}

Note that the third term seems non-symmetric since only h_Dh_B appear without h_A. However we can see that this must be the case since the h’s appear in pairs, their presence makes a difference only if one of A or B is t, so that the product is – (whatever the choice of signature). If one or the other is t, then h_A h_B is necessarily –1 and the sum of the first two terms become the negative of its sum under interchange A à B. If the third term had h_A h_B in it then this third term would not change sign under A à B, so therefore it must have only one of them, and has one sign if A is t and another sign if B is t. ie when there are two terms involved as in the first part of R^AB_AB, the switch in sign can be overall by both terms conspiring to switch together, but for one term {the [g_A,D/g_A][g_B,D/g_B] term} the sign switch can happen only if there’s an h for only one of the two coordinates.

………..

Note:

• The first term is exactly Gauss’s curvature formula for a 2-d surface.
• For an orthogonal metric, the RTC’s indices involve only two coordinates, and the RTC itself involves only the metric coefficients corresponding to these two indices, with the other coordinates appearing only in derivatives wrt them.

…………

Note:

Similarly (check signs):

R^AB_BD = - h_Ah_B [g_Dg_B]^-1 {[g_Ag_D]^-1 g_D,A g_B,D + [g_B,A/g_A],_D}

R^AB_AD = - g_A ^-1{g_D^-1 [g_A,B /g_B],_D + [g_Bg_D²] ^-1 g_A,D g_D,B} w^A ^ w^D _{D A,B}

……………………………….

Appendix

Post-Hoc Heuristic ‘Derivation” of the repeated-index RTC

A) Finding w^A_D:

From the way that the switching of upper to lower indices of w^A_D changes the sign as follows:

w^A_D = - w^D_A h_A h_{D . D A}

we can justify the following structure for w^A_D:

w^A_D = [g_A,D/(g_Dg_A)] w^A - h_A h_D [g_D,A/g_Ag_D)] w^D

= [g_Ag_D] ^-1 {g_A,D w^A _- h_A h_D g_D,A w^D} . ^(6A1)

Written in terms of dx: ie: w^A = g_Adx^A ; w^D = g_D dx^D :

w^A_D = [g_Ag_D] ^-1 {g_A,D g_Adx^A _- h_A h_D g_D,A g_D dx^D } ^(6A2)

w^A_D = [g_A,D/g_D] [dx^A] _- h_A h_D [g_D,A/g_A][dx^D] . (6c)

B) Computation of the ‘wedge product’ w^A_D ^ w^D_B

To form w^D_B we simply transform w^A_D above by switching (A,D à D,B) ie (AàD), (DàB), so that the first term is in w^D and the second has w^B . Creating the ‘wedge product’ of the two will lead to only one term with w^A ^ w^B _, basically from the product of the first term of w^A_D (which has w^A) with the second term of w^D_B (which has w^B); ie

[g_A,D w^A ] ^ [_- h_D h_D g_B,D w^B] = _- h_D h_B g_A,D g_B,D w^A ^ w^B

with a factor of [gg] ^–1 for both. Thus the relevant part w^A_D ^ w^D_B of is:

_- h_D h_B g_A,D g_B,D [g_Ag_D] ^-1[g_Dg_B] ^-1 w^A ^ w^B _{D A,B}

= - h_Dh_B g_D^-2 [g_A,D/g_A][g_B,D/g_B] _{D A,B} (7e)

_C) d w^A_B :

d w^A_B = - [g_A,B /g_B],_c dx^A ^dx^C _CA - h_A h_B [g_B,A/g_A],_c dx^B ^ dx^C _{C B} (8a) (NS: A,B)

= - [g_A,B /g_B],_c [w^A/g_A] ^ [w^C/g_C] _CA - h_A h_B [g_B,A/g_A],_c [w^B/g_B] ^ [w^C/g_C] _{C B}

The terms of interest are for C = B in the first term, and for C = A in the second, which makes w^A ^ w^B a common factor, and turns the g_i’s under the wⁱ’s , ie both g_A g_C and g_Bg_C into g_Ag_B and thereby makes [g_Ag_B]^-1 , a common factor as well. Distributing these common factors we have:

d w^A_B = - [g_A g_B]^-1 {[g_A,B /g_B],_B - h_A h_B [g_B,A/g_A],_A}w^A ^ w^B (8c) (NS: A,B)

Note that the above is the (ABAB) part of term #1 in eq (2a).

R^A_B = dw^A_B - w^A_D^ w^D_B (see eg MTW p351) (2a)

D) The (ABAB) Riemann tensor component (RTCs)

Therefore, adding (7e) and (8c) we obtain::

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

---

[1] From file “My RTC Formula derivation 2005”.

[2] Note: 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 in field theory, and tetrad formulation of GR in field theory.]

[3] AR to AR: show that only these exist, or are needed to set up or solve the Einstein equation.

[4] For the calculation below: note that by definition, operating twice with d, ie dd, gives zero.

[5] _{If D = A:} w^A_{A = -} w_A^A _{= -} h_A h_A w^A_{A = -} h_A ² w^A_{A = -} w^A_{A , So we do not allow D = A.}

[6] Since - h_A h_D is simply a sign, it can be transferred from the rhs to the lhs without change. As a result one can easily see that these two equations are the same, they are consistent, since interchanging D and A transforms one into the other.

[7] Remember that D will be summed over, and each w^D will have the appropriate sign due to the accompanying h_D .

[8] Note also that (see MTW p354): w_AD g_ANw^N_D = G_ADNw^N .

Back to Home Page: Physics, Philosophy, Politics, Biblical Studies