Five Dimensional Metrics

Part I = the files: "Chatterjee" and "Five Dimensional Metrics":

Part II = the files: “5 D for EG first two pages” + "older R calculation" + "Last Piece for EG"

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

Higher-Dimensional Theories [1]

We've all heard of 'the fifth dimension' in some context or other - maybe in the show 'the Twilight Zone'.

Some physicists in the past showed that in some sense our existence can be considered five-dimensional, with four dimensions of spacetime, and one corresponding to electrical charge. More recently string theory points to the existence of ten dimensions, or more.

Is there really a fifth dimension? Can we visit it?

Well, actually there is no ordering to the dimensions of spacetime so there is no 1st 2nd 3rd or 4th dimension, space is not the first three or last three of the four, time is not the 1st or 4th of the four - rather there are four dimensions of spacetime, three space and one time. However if we assume that there are five dimensions, then the extra or fifth would be one not included in ordinary theories of spacetime, and in that sense it is a 'fifth' dimension.

However, one cannot 'visit' the fifth dimension any more than one can visit the first or other dimension, for the simple reason that if we are four-dimensional beings, we need to be in all four at all times - we cannot lose a dimension. However, some aspects of our exstence can involve change in one particular dimension more than another, for example involving change in time more than change in space or v.v. Similarly, if it turns out that we are actually five-dimensional beings, the four of spacetime plus an additional, then it may well be that some situation might stress this other dimension more than the other four. It is not the 'fifth dimension', we are not 'in the fifth dimension', but rather that phenomenon makes manifest our five-dimensional rather than four-dimensional natures.

Allowing the combining and mixing of the 3 dimensions of space and the one dimension of time give us phenomena which do not exist if they cannot mix, and a mathematical theory with a four dimensional mix can describe these phenomena in a way that a theory without this mix cannot. If we assume the existence of five dimensions which mix in the simplest way, the mathematics gives rise to phenomena which match the mathematics of electromagnetic theory as described by general relativity. If one allows a more sophisticated mix, the phenomena which arise from the math correspond in some ways to quantum effects of electromagnetic theory.

Superstring theory involves a mix of ten dimensions, and the math seems to lead to phenomena which may even describe all of physics.

These are the motivations for studying theories of spacetime dimensions higher than four. The simplest multidimensional cases are those with only five dimensions, and those are the ones we will look at first, with string theories of ten dimensions being dealt with later.

Five Dimensional Theories

The simplest case of 5-d is that of flat 5-d space with metric:

ds2 = - dt2 + dx2 + dy2 +dz2 + dX52

or what is the same metric but expressed in spherical coordinates:

ds2 = - dt2 + dr2 + r2 dW2 + dX52

but that is trivial........

Any metric with a general 4 metric form but a flat 5th dimensional part such as:

ds2 = - Tt2 Tr2 dt2 + Rt2 Rr2 (dr2 + r2 dW2) + dX52

would give the same mathematics as would the 4-space by itself and the 5th space as an additional one-dimensional space added on. That is, the physics would be nothing new, except that the additional feature would have a geometrical description.

We can say that this form of the theory is unappealing to physicists for a few reasons:

1) Just as gravity gets a geometrical descrition as the curvature of 4-d spacetime, one can interpret electromagnetism as the curvature of a one-dimensional space, and then tack this one-d space onto 4-d spacetime to get a 5-d spacetime. However, although the geometrical picture of curved spacetime is intuitively appealing, the 5th dimension picture of electromagnetic phenomena is mathematically nice but has no real intuitive appeal; we experience 4 dimensions of spacetime without regard to gravitational phenomena, and we do see gravity operating in this 4-d space to make things move in space and time, whereas we do not experience any additional dimensions other than the four of ordinary spacetime and so atributing electromagnetic phenomena to an additional dimension which is felt only by the fact that these phenomena exist is unsatisfying.

2) Although it is nice that one can express electromagnetic phenomena in this geometrical way as the curvature of a one-dimensional space, there is nothing new that is gained by this - no new eletromagnetic phenomena or other phenomena come out of this picture.

3) To get all of gravity theory out of the model of a curves spacetime one needs a mix of the all the 4-d among themselves, that is, a mix of 3-d space and 1-d time; simply keeping the 3-d of space and the 1-d of time separate do not result in the mathematical description of the effects of gravity. Thus general relativity theory is truly a 4-d theory, not two theories, one of a 3-d space and the other of a 1-d space which is time. On the other hand, all of gravity and all of electromagnetics can be described by 2 separate spaces, one of 4-d and the other of 1-d. Although the gravitational effect of the electric field comes out of combination of the two spaces into a 5-d space, there is no true interaaction between the 4-d part and the 1-d part.

To get something really new we would have to have a real mix of the 4-space of ordinary spacetime and the additional 1-d space. The question is whether this would give results consistent with nature. If so, then it would be a remarkable achievement: to combine gravity and electromagnetism in a geometric picture for aesthetic reasons, and then to find that a generalization of this theory actually correctly predicts previously unknown phenomena. or includes a description of known but supposedly unrelated phenomena. This would be a strong endorsement of the method of emplying aesthetic mathematical principles to discover more about nature.

Non-Trivial Five-Dimensional Space-Time Theory

The following is an example of a non-trivial type of metric - that is, a metric which has a mix of the 4-d and additional 5th dimension - and it has a familiar type form:

ds2 = - dt2 + Rt2 Rr2 (dr2 + r2 dW2) + [Rt2 Rr2]-1 dX52

This metric is non-trivial: even though there are no functions of the fifth coordinate in the metric coefficients, there are functions of r and t in the fifth-d part, and this causes a real mix. One could also have :

ds2 = - f(X5)dt2 + h(X5)Rt2 Rr2 (dr2 + r2 dW2) + b(X5)j(r,t) dX52

where there is a mix caused by the functions of the 5-d part appearing in metric coefficients for dr and dt as well as the functions of r,t appearing in the 5 part.

However, if there is a 5 part in the 4-part of the metric, then maintaining constant 4-d coordinates but changing the fifth would give different values for the metric. This would mean that two 4-d spacetime points would not be a set distance apart: they would have a different distance between them as the 5 part changed. However we do not observe that such things occur. As far as we can tell, the distance between two points remains constant over time if we do not change the matterenergy content of spacetime, and thus we do not accept that a change of some quantity, something related to the extra dimension, can have an effect on the 4-d part of the metric, and thus there should be no functions of the fifth coordinate in the 4-d part of the metric. Similarly, the time interval between the beginning and end of a process remains the same, whereas if there was a function of the fifth coordinate in the 4-d part it would change as a function of the change of the fifth coordinate. Thus we do not wish to include functions of the fifth coordinate in the coefficients of the four ordinary spacetime parts of the metric.

In that case, the only place we will find functions of X5 will be in the fifth component, in which case it can be absorbed into the definition of X5 so that f(X5)dX5 becomes dX'5 which then is relabeled to become simply dX5. Thus we will not have any functions of X5 at all. Instead, the mixing will come from the presence of functions of r and t in the 5 part (no functions of the angular coordinates in order to keep spherical symmetry). Thus the most general metric we would look at would be:

ds2 = - f(r,t)dt2 + g(r,t)dr2 + m(r,t) r2 dW2 + j(r,t) dX52

We will wish to find five-d spaces with 4-d subspaces which are familiar to us. For example a five-dimensional space with a four-dimensional subspace which is homogeneous and isotropic would have the metric:

ds2 = - dt2 + Rt2/(1+kr2)2 (dr2 + r2 dW2) + 5t2 5r2 dX52

The Einstein Tensor for Five-Dimensional Spaces

The Einstein tensor components for spherically symmetric 4 and 5 d spaces have a simple relationship to each other (see appendix for derivation). We will write down the relationship of the r part of the 5-d r-r tensor to the r part of the 4-d r-r tensor:

5rGrr = 4rGrr + 2(1/Rt2)(1/Rr2 )(5'r/5r)(qr'/qr) ,

which for this metric with Rr = (1+kr2) and qr = r/(1+kr2) is:

5rGrr = 4rGrr + (2/Rt2)(1+kr2)2(5'r/5r)(1-kr2)/[r(1+kr2)]

= 4rGrr + (2/Rt2)(5'r/5r)(1-k2r4)/r

For the 5-d tensor component to be homogeneous, it must be independent of r, and since the 4-part is homogeneous and r-independent it must be that

(5'r/5r)(1-k2r4)/r

is r-independent. That is,

(5'r/5r)(1-k2r4)/r = c , i.e 5'r/5r = cr/(1-k2r4).

integral(5'r/5r ) = ln5r = c integral[r/(1-k2r4)] + K

The integral can be easily performed using the substitution R=r2 , dR=2rdr.

The solution is:

5r = K [(1-kr2)/(1+kr2)]-c/4k

so that the 5-d metric for a homogeneous isotropic space is (squaring 5r makes the exponent -c/2k)

ds2 = -dt2 +Rt2/(1+kr2)2 (dr2+r2d2) + 5t2 K [(1-kr2)/(1+kr2)]-c/2k dX52

It is amazing that this solution can be obtained so simply - for an appreciation of how simple our method is, see Chatterjee where this result is derived using long and tedious calculations.

The other results derived there are also easily found in this way. Give presentation of all his results, and generalization.

5t = ? Grr = ? Gtt = ?

Try also: ds2 = -a(r,t)dt + (dr2+r2dW2) + a-1(r,t) dX52 and inverse with 5 & t parts.

[AR: The following was in the end of file "Five Dimensional Metrics"]

The Static Electric and Magnetic Field as A Five-DImensional Object

We saw previously that for the static magnetic field as source, the four-dimensional metric is:

ds2 = a2 [- ch2z dt2 + dz2 + dW2]. dW2 = dq2 + cos2q df2

We now look for the metric of the five-dimensional space containing the magnetic field.

.....

The metric can be written as....

ds2 = a2 [(dx5 + cosqdf)2 + sin2qdf2 + dq2 + dz2 - ch2z dt2]

which is the same as:

ds2 = a2 [(dx5)2 + 2 cosq dx5 df + df2 + dq2 + dz2 - ch2z dt2]

……………..

This section = “5 D for EG first two pages” + "older R calculation" + "Last Piece for EG"

5-D Inverse Metric Calculations (Folder: “GR Book”)

x = coordinate in the 5 dimension:

ds² = - A ²(r)dt² + A^-2(r)dr^{2 +} r²[dx² + sin²x(sin²qdq² + df²)]

ds² = g_AAdx^Adx^A = h^A_A (w^A) ² .

w^t = Adt, w^r = dr/A, w^q = r sinx dq, w^f = r sinx sinq df, w⁵ = rdx,

..........

dw^t =dA ^dt = A’ dr ^ dt = A’A w^r ^ w^t/A = A’ w^r ^ w^t.

However we also know that:

dw^t = - w^t_C^ w^C = - w^t_r^ w^t - w^t₅^ w⁵ - w^t_q^ w^q - w^t_f^ w^f

Therefore: dw^t = - w^t_r^ w^t - w^t₅^ w⁵ - w^t_q^ w^q - w^t_f^ w^f = A’ w^r ^ w^t.

Since the metric is static, there is no function of t in any metric coefficient, and only the r coordinate appears in functions in the t metric coefficient, so all the above two forms with t as one index [ie all w^t_m]will be zero except w^t_r.

Therefore: dw^t = - w^t_r^ w^r - w^t₅^ w⁵ - w^t_q^ w^q - w^t_f^ w^f = A’ w^r ^ w^t,

i.e.

- w^t_r^ w^r = A’ w^r ^ w^t = -A’ w^t ^ w^r --> w^t_r = A’ w^t = A’A dt.

R^a_b = w^a_m ^w^m_b + dw^a_b.

R^t_r = w^t_m ^w^m_r + dw^t_r. All w^t_m are zero except w^t_r , and so:

R^t_r = w^t_r ^w^r_r + dw^t_r, ie

R^t_r = dw^t_r = d A’A dt = (A’’+A’²)dr^dt .

Since dr ^dt = A w^r ^w^t/A = w^r ^w^t ,

R^t_r = (A’’+A’²) w^r ^w^t .

.......... .....................[2]........................................

STEP2:

d w^q = dr sinx dq + r d(sinx) dq = sinx dr ^ dq + r cosx dx ^ dq

= sinx A w^r ^ [w^q/r sinx] + rcosx[w⁵/r] ^ [w^q/r sinx]

= [A/r] w^r ^ w^q + [cotx/r]w⁵ ^ w^q

In addition:

d w^q = - w^q_t^ w^t - w^q₅^ w⁵ - w^q_r^ w^r - w^q_f^ w^f . Therefore:

d w^q = w^r ^w ^q_r + w⁵^ w^q₅ - w^q_f^ w^f = [A/r] w^r ^ w^q + [cotx/r]w⁵ ^ w^q

The simplest possibility is:

w ^q_r = [A/r] w^q

w^q₅ = [cotx/r] w⁵

w^q_f = a w^f.

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

STEP 3:

dw^f = dr sinx sinq df + r dsinx sinq df + r sinx dsinq df

= dr sinx sinq df + r cosxdx sinq df + r sinx cosqdq df

=Aw^rsinxsinq[w^f/rsinxsinq]+rcosx[w⁵/r]sinq[w^f/rsinxsinq]+rsinx cosq[w^q/rsinx][w^f/rsinx sinq]

=Aw^r [w^f/r]+cosx[w⁵][w^f/rsinx]+ cosq[w^q][w^f/rsinx sinq]

=[A/r]w^r ^ w^f+ [cotx/r] w⁵^ w^f + [cotq/rsinx] w^q ^ w^f

In addition:

d w^f = - w^f_q^ w^q - w^f₅^ w⁵ - w^f_r^ w^r - w^f_t^ w ^t . Therefore:

d w^f = - w^f_q^ w^q - w^f₅^ w⁵ - w^f_r^ w^r =[A/r]w^r ^ w^f+ [cotx/r] w⁵^ w^f + [cotq/rsinx] w^q ^ w^f

The simplest match is made when:

w^f_q=[cotq/rsinx] w^q

w^f₅=[cotx/r] w^f

w^f_r^ w^r =[A/r] w^f

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

STEP 4:

w⁵ = rdx, d w⁵ = drdx = A w^r^[w⁵ /r] = [A/r] w^r^w⁵

Using the results we found above:

d w⁵ = - w⁵_t^ w^t - w⁵_r^ w^r - w⁵_q^ w^q - w⁵_f^ w^f = - w⁵_r^ w^r -[] w^q ^ w^q -[]w^f ^ w^f =[A/r] w^r^w⁵.

Therefore: w⁵_r =[A/r] w⁵ .

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

Now that we know all the two forms, we can compute their derivatives, and then we shall be able to find the Riemann tensor components. First we find the derivatives:

..........

w⁵_r =[A/r] w⁵ = [A/r]rdx = Adx

Therefore: dw⁵_r = dAdx = A’drdx = A’A [w^r^ w⁵/r] = [A’A/r] w^r^ w⁵.

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

STEP 5:

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

w^q = r sinx dq, w^f = r sinx sinq df, w⁵ = rdx,

w^t = Adt, w^r = dr/A, w^q = r sinx dq, w^f = r sinx sinq df, w⁵ = rdx,

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

R^q_fqf = g_ff^-2[qqf”/qqf - (qqf’/qqf) fff’/fff]

-g_qq^-2 d(qqf’) h_qh_f[ffq”/ffq-(ffq’/ffq)qqq’/qqq]

-g_DD^-2 d(DDf’) h_qh_f(qqD’/qqD)ffD’/ffD SUMMED OVER D NOT=q,f

We want to compute the following:

R^r_5r5 , R^t_{ftf ,} R^t_{rtr ,}R^t_5t5 R^q_{rqr ,} R^q₅^q_fqf

G^r_r = R^t_ftf + R^t_5t5 +2 R^q_5q5 +2R^q_tqt + R^q_fqf

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

R^t_r = (A’’+A’²) w^r ^w^t .

R^t₅ = [-A’A/r] w^t ^ w⁵

R^t_q = [-A’A/r]w^t ^w^q

R^t_f = [-A’A/r]w^t ^w^f

R^r₅ = - [A’A/r] w^r^ w⁵

R^r_q = - [A’A/r] w^r^ w^q

R^r_f = - [A’A/r] w^r^ w^f

R⁵_q = R⁵_f = [(1-A²)/r²] w⁵^w^q

R^q_f = [(1-A²)/r²] w^q ^ w^f

.........

A²’ = 2A’A --> A’A/r = A²’/2r ,

A²’’ = [A²’]’ = 2[A’A]’ = 2[A’’A+A²] --> A’’A+A² = A²’’/2 .

..........

To make a distinction between the two forms (eg R^t_r) and the tensor components (eg R^t_r), we change the fonts of the above R’s to R:

R^tq = R^tq_tq w^t ^w^q

ie R^tq_tq = [-A’A/r]

since the metric coefficient are +1 except for h_tt which is -1, we can contract and raise or lower indices without changing the sign, unless we raise or lower t, in which case there is a sign change.

R^t_q = R^t_f = [-A’A/r]

Basically, since there is only one term in the two forms R, and the metric coefficients are 1, we can read the tensor components R^a_b directly off the list of the two form R‘s.

R^t_r = (A’’+A’²) w^r ^w^t --> R^t_r = (A’’+A’²) = A^2’’/2

R^t₅ = [-A’A/r] w^t ^ w^{5 -->} R^t₅ = [-A’A/r] = -A²’/2r

R^r₅ = - [A’A/r] w^r^ w^{5 -->} R^r₅ = [-A’A/r] = -A²’/2r

R^r_q = - [A’A/r] w^r^ w^{q -->} R^r_q = [-A’A/r] = -A²’/2r

R^r_f = - [A’A/r] w^r^ w^{f -->} R^r_f = [-A’A/r] = -A²’/2r

R⁵_q = R⁵_f = [(1-A²)/r²] w⁵^w^{q -->} R⁵_q = R⁵_f =[(1-A²)/r²]

R^q_f =[(1-A²)/r²] w^q ^ w^{f -->} R^q_f =[(1-A²)/r²]

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

R^t_q = R^t_f = R^t₅ = R^r₅ = R^r_q = R^r_f = -A²’/2r

R^t_r = (A’’+A’²) = A^2’’/2

R⁵_q = R⁵_f = R^q_f = [(1-A²)/r²]

..........

Using the list of these R’s and noting that to drop or raise the index t changes the sign of R (if we keep t as the first index, and do not lower it, we need not change any sign), we can easily find the Einstein curvature components.

From MTW: G^a_a = - d^abg_abg R^|bg|_|bg|

G⁵₅ = - d^5tr_5trR^tr_tr -2 d^5tq_{5t q} R^tq_tq -2 d^5rq_{5r q} R^rq_rq - d^5qf_5qf R^qf_qf

all the delta’s = 1, because the upper and lower three indices are identical, in same order, and this is because there is only one two-form term in each R ......

d^5tq_{5t q}

Reading off the terms, we find:

G⁵₅ = - R^tr_tr -2 R^tq_tq -2 R^rq_rq - R^qf_qf = [(A²-1)/r² + A²’’/2]

Writing the above in shorthand without the deltas and R’s:

G⁵₅ = -tr -2 tq -2 rq - qf = [(A²-1)/r² + A²’’/2]

In the same way we can find: (paying attention to the raising and lowering of t)

G^t_t = -5r - 2(5q) - 2(r q) - qf = 3 [(A²-1)/r² + A²’/2r]

Note that in 4-D the first two terms vanish and the last two remain, giving the correct result.

G^r_r = -[t5 + 2(5q) + 2(t q) + qf] = 3 [(A²-1)/r² + A²’/2r]

G^q_q = G^f_f = - [f5 + tf + fr +5r + 5t + tr] =[(A²-1)/r² +2A²’/r + A^2’’/2]

In 4-D the three terms in 5 vanish, and the G’s reduce to their correct 4-D expressions.

Note that the terms in G are the same ones because the terms in the R’s are identical; in fact all the G’s are simple multiples of the 4-D case except for the mix of fifth-dimensional coordinate and angular coordinate, and that is due to their mix in the metric.

R = {+-] G⁵₅ + G^t_t + G^r_r + G^q_q + G^f_f = 3[3(A²-1)/r² + A²’/r + A²’’/2]

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

File: 5-D Metric Calculations (Folder: “GR Book”)

Deriving a general formula to compute the Riemann Tensor Components (RTC) and the Einstein tensor components for orthogonal metric of any dimensionality.

INSERT DEFINITION SIGN. INSERT “NOT EQUAL TO” signs

For orthogonal metric of any dimensionality:

ds² = g_AAdx^Adx^A = h^A_A (w^A) ² = h^A_{A [}g_Adx^A]². (1) INSERT DEFINITION SIGN.

In this notation the h^A_A contain the signature information and g_A is the square-root of |g_AA|.

w^A = g_Adx^A (No summation = NS) (2) INSERT DEFINITION SIGN.

We wish to compute the RTC R^A_BAB according to the formula:

R^A_B = d w^A_B - w^A_C^ w^C_B = - R^A_BAB w^A ^ w^B

Therefore we will calculate w^A_B, then d w^A_B and - w^A_C^ w^C_B .

A) Computation of w^A_D: INSERT “NOT EQUAL TO” signs.

Add NS where appropriate

dw^A = d[g_A]dx^A = g_A,B dx^B^dx^A _{[B not=A]}[A:NS] = g_AA,B w^B/g_BB ^ w^A/g_AA = [g_AA,B/(g_BBg_AA)] w^B ^ w^A. (3)

However we also know that:

dw^A = - w^A_C^ w^C _{[C not=A]} (4)

Equation 3) = equation (4; this implies: (note: the summation dummy variable B,C à D)

[g_AA,D/(g_DDg_AA)] w^D ^ w^A _{[D not=A]} = + w^D ^ w^A_{D D not=A}

We use MTW[heeler]’s method to find w^A_D .

There are two situations:

1) [g_AA,D/(g_AAg_DD)] does not vanish, ie g_AA,D does not vanish: à w^A_D ~ w^A

2) g_AA,D = 0 à w^A_D ~ w^D .

We can therefore write:

w^A_{D =} [g_AA,D/(g_DDg_AA)] w^A_- d(g_AA,D) h^A_A h^D_D [g_DD,A/(g_AAg_DD)] w^{D Show how the second part of this term was arrived at [3] (6A)}

_{The above formulation allows us to use the same} w^A_{D for both cases.}

_{B) Computation of} w^A_D ^ w^D_B

_{From equation 6A:}

w^A_D ^ w^D_{B D not=A,B = {}g_AA,D [w^A/(g_AAg_DD)] _- d(g_AA,D) h^A_A h^D_D g_DD,A [w^D/(g_DDg_AA)]_}

^ _{g_DD,B [w^D/(g_DDg_BB)] _- d(g_DD,B) h^D_D h^B_B g_BB,D [w^B/(g_BBg_DD)]_{} (7a)}

_{[As a check:} w^A_{D and} w^D_{B should have the same form, just with (A,D) --> (D,B), ie whatever is A in the upper line --> D in the lower line, and the same for D --> B.]}

₌ g_AA,D w^A/(g_AAg_DD) ^ g_DD,B w^D/(g_DDg_BB)

_- d(g_AA,D) h^A_A h^D_D g_DD,A w^D/(g_DDg_AA) ^ g_DD,B w^D/(g_DDg_BB) _{<-- this line is ~} w^D^ w^D _{= 0.}

₊ g_AA,D w^A/(g_DDg_AA) ^ _- d(g_DD,B) h^B_B h^D_D g_BB,D [w^B/(g_BBg_DD)]

_- d(g_AA,D) h^A_A h^D_D g_DD,A w^D/(g_DDg_AA) ^ _- d(g_DD,B) h^B_B h^D_D g_BB,D w^B/(g_BBg_DD) _{D not=A,B}

w^A_D ^ w^D_{B D not=A,B = {}g_AA,D g_DD,B /(g_AAg_DDg_DDg_BB)] w^A ^ w^D

_- d(g_DD,B) h^B_B h^D_D g_AA,D g_BB,D /(g_BBg_DDg_DDg_AA) w^A ^ w^B

+ d(g_AA,D) d(g_DD,B) h^A_A h^B_B h^D_D² _{g_DD,A g_BB,D /(g_DDg_AAg_BBg_DD)_} w^D ^w^B _{. D not=A,B}

_{C) Computation of} d w^A_D

_{In order to compute the d} w^A_{B we need to first recast} w^A_{B from (6A) in terms of dx}^A_{, dx}^B_:

w^A_D = [g_AA,D/g_DD] [w^A/g_AA] _- d(g_AA,D) h^A_A h^D_D [g_DD,A/g_AA][ w^D/g_DD] ^(6B)

= [g_AA,D/g_DD] [dx^A] _- d(g_AA,D) h^A_A h^D_D [g_DD,A/g_AA][dx^D] . ^(6C)

_{From equation 6C above we find:}

d w^A_B= [g_AA,B /g_BB],_cdx^C ^dx^A _{[c not =A]} - d(g_AA,B)h^A_A h^B_B[g_BB,A/g_AA],_c dx^C ^dx^B _{[c not = B]} (8a)

_{Now that we have the derivative, we can put this back into terms in} d w^A _and d w^B_:

d w^A_B=[g_AAg_CC]^-1[g_AA,B /g_BB],_cw^C ^w^A _{C not =A} - d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_c w^C ^ w^B _{C not = B} (8b)

Note: [g_BB g_AA]^-1 is a common factor when in the first term C = B, and in the second term C = A.

_{For the term in the above left: in the sum over C, C cannot be A, but since B is NOT A, C CAN be B. We separate out the case C=B and C not = B (of course C is NOT A)}

_{For the term in the above right: we separate below the cases C = A, and C is NOT A.}

d w^A_B = -[g_BB g_AA]^-1 { [g_AA,B /g_BB],_B + d(g_AA,B) h^A_A h^B_B [g_BB,A/g_AA],_A } w^A ^ w^B

- { [g_CC g_AA]^-1[g_AA,B /g_BB],_c + d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_c w^B ^ w^{C }}_{C not = A,B} (8c)

D) Finding the RTC

We can find the Riemann tensor components via:

R^A_B = d w^A_B - w^A_C^ w^C_B = - R^A_BAB w^A ^ w^{B [(?) check this]} (9)

From a comparison of equations (7a) and (8c) we find:

R^A_B = - {[g_BB g_AA]^-1 { [g_AA,B /g_BB],_B + d(g_AA,B) h^A_A h^B_B [g_BB,A/g_AA],_A } w^A ^ w^B

- { [g_CC g_AA]^-1[g_AA,B /g_BB],_c + d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_c w^B ^ w^{C }}_{C not = A,B}

+{ [g_CC g_AA]^-1[g_AA,B /g_BB],_{C +-} (g_BBg_CC) ^-1[g_AA,C /(g_AA)] _[g_CC,B /g_CC]} w^A ^ w^C

- {d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_C + d(g_AA,B) h^A_A h^C_C (g_AAg_DC) ^-1 _{g_CC,A /g_CC} d(g_CC,B) h^B_B h^C_{C [}g_BB,C /g_BB]_} w^C ^w^B _{C not=A,B}

= - {[g_BB g_AA]^-1 { [g_AA,B /g_BB],_B + d(g_AA,B) h^A_A h^B_B [g_BB,A/g_AA],_A } w^A ^ w^B

+{ [g_CC g_AA]^-1[g_AA,B /g_BB],_{C +-} (g_BBg_CC) ^-1[g_AA,C /(g_AA)] _[g_CC,B /g_CC]} w^A ^ w^C

+ [{ [g_CC g_AA]^-1[g_AA,B /g_BB],_c + d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_c ^}_{C not = A,B} - {d(g_AA,B)h^A_A h^B_B[g_CC g_BB]^-1[g_BB,A/g_AA],_C + d(g_AA,B) h^A_A h^C_C (g_AAg_DC) ^-1 _{g_CC,A /g_CC} d(g_CC,B) h^B_B h^C_{C [}g_BB,C /g_BB]_}] w^C ^w^B _{C not=A,B}

………………………………………………………………………………….

= - {[g_BB g_AA]^-1 [g_AA,B /g_BB],_B

The term [g_BB,A/g_AA],_A is the same as part of the one above just with A<-->B

[g_CC g_AA]^-1[g_AA,B /g_BB],

[g_CC g_BB]^-1[g_BB,A/g_AA],_c

[g_CC g_AA]^-1[g_AA,B /g_BB],_C

(g_BBg_CC) ^-1[g_AA,C /g_AA] _[g_CC,B /g_CC

[g_CC g_BB]^-1[g_BB,A/g_AA],_C

(g_AAg_DC) ^-1 _{g_CC,A /g_CC[g_BB,C /g_BB

{[g_BB g_AA]^-1 { [g_AA,B /g_BB],_B

{[g_BB g_AA]^-1 [g_BB,A/g_AA],_A }

[g_CC g_AA]^-1[g_AA,B /g_BB],_C

(g_BBg_CC) ^-1[g_AA,C /(g_AA)] _[g_CC,B /g_CC

[g_CC g_AA]^-1[g_AA,B /g_BB],_c

[g_CC g_BB]^-1[g_BB,A/g_AA],_c ^}

[g_CC g_BB]^-1[g_BB,A/g_AA],_C

(g_AAg_DC) ^-1 _{g_CC,A /g_{CC [}g_BB,C /g_BB]_}]

Computing the various terms in the RTC:

Regarding the terms of the type [g_AA,B /g_BB],_c:

The metric type we will deal with has no cross terms (it is orthogonal), and in addition the coefficients are assumed to be separable functions, i.e. each metric coefficient is a product of one or more functions, each function being dependent on only one of the coordinates. We use the following notations:

for n dimensions, the coordinates are X₁ ...., X_n:

g_AA = F_AA(X₁)G_AA(X₂...H_AA(X_n) = AAX₁ AAX₂ ....AAX_n = p_XnAAX_n --> p_XAAX --> pAAX

Note that both A and X are the coordinates.

We can immediately see that:

g_AA,B /g_AA = [AAB’ p _Xnot=BAAX]/ [AAB p_X=notBAAX] = AAB’/AAB = A_B’/A_B.

Example: for spherical symmetry, g_ff = r²sin²q:

g_f = sqrt|g_ff| = rsinq.

f_q = the function of theta in the ff metric coefficient =sinq. Therefore:

g_f,q/g_f = f_q'/f_q = sinq'/sinq = cotq.

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

The term g_AA,B/g_BB is more complicated since there is not necessarily any cancellation involved.

We now compute the term:

g_AA^-1 g_BB^-1[g_AA,B/g_BB]_,B

We give a visual manipulation immediately below: further below it is done step by step.

Since both derivatives are with respect to the B coordinate, functions of all the other coordinates can be taken outside the brackets. When the complement to the function of B is separated off from g_{AA ,} (leaving behind A_B’), and is taken out of the brackets, and multiplied by g_AA^-1 , the only remaining function is 1/A_B. When the same is done for g_BB in the denominator, (leaving behind 1/B_B), and it is multiplied by g_BB^-1, the product can be written as B_B/g_BB² so that we obtain:

[1/A_B][B_B/g_BB²][A_B’/B_B]'

Since [A_B’/B_B]' = A_B’'/B_B - A_B’B_B’/B_B².

We obtain:

g_AA^-1 g_BB^-1[g_AA,B/g_BB]_,B = [1/A_B][B_B/g_BB²][A_B’'/B_B - A_B’B_B’/B_B²] = g_BB^-2 [A_B’'/A_B - (A_B’/A_B)(B_B’/B_B)]

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

Step by Step: We start by computing:

g_AA,B = [pAAX],_B = [AAC p_X

_BAAX],_B = [AAB,_B p_X= not _BAAX] = [AAB’ p_X=notBAAX],

With this we find:

[g_AA,B/g_BB]_,B ={[AAB’ p_X=notBAAX]/[BBB p_X=notBBBX]}_,B

= {p_X=notBAAX/ p_X=notBBBX} {[AAB’]/[BBB]}_,B

Where: {p_X=notBAAX/ p_X=notBBBX} = {[g_AA/AAB]/[g_BB/BBB]} = [g_AA/g_BB][BBB/AAB]

i.e.: [g_AA,B/g_BB]_,B =[g_AA/g_BB][BBB/AAB] {[AAB’]/[BBB]}_,B

Therefore:

g_AA^-1 g_BB^-1 [g_AA,B/g_BB]_,B = g_AA^-1 g_BB^-1 [g_AA/g_BB][BBB/AAB] {[AAB’]/[BBB]}_,B

= g_BB^-2 [BBB/AAB] {[AAB’]/[BBB]}_,B

= g_BB^-2 {[AAB’]/[BBB]}_,B/[AAB/BBB]

or:

[g_AA,B/g_BB]_,B = [g_AA,BB/g_BB]-[g_AA,B g_BB,B/g_BB²]= [g_AA,BB/g_BB]-[g_AA,B g_BB,B/g_BB²] =

= g_BB^-1 { [g_AA,BB]-[g_AA,B g_BB,B/g_BB] }

= (g_AA/g_BB) {[g_AA,BB/g_AA]-[g_AA,B/g_AA ][g_BB,B/g_BB] }

= (g_AA/g_BB) {[A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] }.,

and therefore:

g_AA^-1 g_BB^-1[g_AA,B/g_BB]_,B = g_AA^-1 g_BB^-1(g_AA/g_BB) {[A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] }

= g_BB^-2 {[A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] }.

= g_BB^-2 {[AAB’]/[BBB]}_,B/[AAB/BBB]

(we can use a notation of bar on top of a letter, e.g bar on top of A_B’ means A_B’/A_B).

An example:

For the metric:

ds² = - g_tt (r,t,q)dt² + g_rr (r,t,f)dr^{2 +} r² (sin²qdq² + df²)

= [TTT TTR TTq]² dt² + [RRT RRR RRf]² dr^{2 +} [qqR qqq] ²dq² + [FFR]²df^{2 =}

= T_t²T_r² T_q² dt² + R_t²R_r² Rf² dr^{2 +}[q_r q_q] ²dq² + [F_r]²df²

we find that:

g_rr^-1 g_tt^-1[g_rr,t/g_tt]_,t = g_tt^-2 {[R_t’’/R_t] - [R_t’/R_t] [T_t’/T_t] }.

_{...........................................}

We now compute, for

:

[g_AA,B/g_BB]_,C = [g_AA,B,C/g_BB]-[g_AA,B g_BB,C/g_BB²]

= g_BB^-1{[(p_(Xnot=B,C)AAX)AAB_,B AAC_,C]-[(p_(Xnot=B)AAX)AAB_,B (p_(Xnot=C)BBX)BBC_,C /g_BB]}=

= g_BB^-1 { (g_AA/AAB AAC)AAB_,B AAC_,C]-[g_AA/AAB) AAB_,B (g_BB/BBC)BBC_,C /g_BB]}=

= [g_AA/g_BB] [AAB_,B/AAB] {[AAC_,C/AAC]-[BBC_,C/BBC]}=

= [g_AA/g_BB] [A_B’/A_B] {[A_C’/A_C]-[B_C’/B_C]}.

_{...............}

We will find the need to compute [g_AA,B/g_BB]_,C for the case where C = B and where it is not. Specifically we shall need to sum [g_AA,B/g_BB]_,C Lw^c so that C takes all values, one of them being B. We will then write:

[g_AA,B/g_BB]_,C Lw^c =

= [g_AA/g_BB] {[ [A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] ] Lw^B

{[A_C’/A_C]-[B_C’/B_C]} Lw^c }.

+ [A_B’/A_B] _S

The above term will be a significant part of the Riemann tensor component R^A_BAB

[We can call this term PR^A_BAB.]

_........

for n dimensions, the coordinates are B₁ ...., B_n:

g_AA = F_AA(B₁)G_AA(B₂)...H_AA(B_n) = AAB₁ AAB₂ ....AAB_n = p_BnAAB_n --> p_BAAB --> pAAB

Note that A and B are coordinates.

g_AA,C = [pAAB],_C = [AAC p_{B C}AAB],_C = [AAC,_C p_B= not _CAAB] = [AAC’ p_B=notCAAB],

g_AA,C /g_AA = [AAC’ p

AAB]/ [AAC p_B=notCAAB] = AAC’/AAC = A_C’/A_C.

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

The Meaning of the R’s as seen from their form:

Writing R in this form allows us to see how it is dependent on the metric coefficient functions (see discussion below).

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

From .... we find:

R^A_BAB = g_BB^-2[AAB”/AAB - (AAB’/AAB) BBB’/BBB]

-g_AA^-2 d(AAB’) h_Ah_B[BBA”/BBA-(BBA’/BBA)AAA’/AAA]

-g_DD^-2 d(DDB’) h_Ah_B(AAD’/AAD)BBD’/BBD SUMMED OVER D NOT=A,B

The Riemann tensor has a second derivative, the two form connection involves the derivative of the metric coefficients, and the Riemann tensor components involve the derivative of this.

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

R^A_BAB will contain three basic terms: all three will be based on the two relevant metric coefficients: g_AA and ^-1 g_BB .

There will be two terms with second derivatives, another with a product of two first derivatives.

R^A_BAB has in it the term: g_BB^-2 {[AAB’]/[BBB]}_,B/[AAB/BBB]

The second-derivative terms: Take the B functional part of the A metric coefficient, take its derivative w/r/t B, divide by the B functional part of the B metric coefficient, and then take the derivative of this w/r/t B; then divide by the ratio of the two functional parts without derivatives, and the inverse square of the B metric coefficient.

Or: Take the derivative of the A metric coefficient w/r/t B, divide by the B metric coefficient as a form of normalization, and then repeat the process: take the derivative of this w/r/t B, and then as ‘normalization’ divide by the product of the A and B metric coefficients.

We can also put this as: where PPR means one part of the term R^A_BAB

PPR^A_BAB = g_AA^-1 g_BB^-1[g_AA,B/g_BB]_,B = g_AA^-1 g_BB^-1(g_AA/g_BB) {[A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] }

= g_BB^-2 {[A_B’’/A_B] - [A_B’/A_B] [B_B’/B_B] }.

Note that for the inverse metric case where for two coordinates C and D:

C_c=D_c^-1, and therefore [C_D’/C_D] = - [D_D’/D_D] :

PPR^C_DCD = g_DD^-2 {[C_D’’/C_D] - [C_D’/C_D] [D_D’/D_D] } = g_DD^-2 {[C_D’’/C_D] + [C_D’/C_D]²} .

...........

R^A_BAB = g_BB^-2[AAB”/AAB - (AAB’/AAB) BBB’/BBB]

-g_AA^-2 d(AAB’) h_Ah_B[BBA”/BBA-(BBA’/BBA)AAA’/AAA]

-g_DD^-2 d(DDB’) h_Ah_B(AAD’/AAD)BBD’/BBD SUMMED OVER D NOT=A,B

h_t = -1, all other h are +1, and so with the products of the h,including h_t h_t so the only combination/product which is not +1 will be those with one and only one h_t in them.

This R formula can be used for any amount of dimensions, in any metric form without cross terms. (try for cylindrical , spherical etc...)

It would not be difficult to write a computer program which would add these R^A_BAB’s to compute G, and to add the G’s to find the curvature R.

We use familiar symbols for the coordinates (eg when using spherical coordinates) but we are NOT assuming spherical symmetry: the coordinates symbolized by q and f can be any coordinates, not necessarily angles, or they can be angles but in a non-sph. sym. coordinate system; we will also be using the symbols t and r, without necessarily giving them physical interpretation, except that t will be the coordinate whose associated metric coefficient is time-like. For the general case we can substitute any four coordinates for t,r,q,f - we will make actual specific choices only when we eg limit ourselves to spherical coordinates and set various factors to zero as a result.

R^q_fqf = g_ff^-2[qqf”/qqf - (qqf’/qqf) fff’/fff]

-g_qq^-2 d(qqf’) h_qh_f[ffq”/ffq-(ffq’/ffq)qqq’/qqq]

-g_DD^-2 d(DDf’) h_qh_f(qqD’/qqD)ffD’/ffD SUMMED OVER D NOT=q,f

R^q_tqt = g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt’/ttt]

-g_qq^-2 d(qqt’) h_qh_t[ttq”/ttq-(ttq’/ttq)qqq’/qqq]

-g_DD^-2 d(DDt’) h_qh_t(qqD’/qqD)ttD’/ttD SUMMED OVER D NOT=q,t

R^q_5q5 = g₅₅^-2[qq5”/qq5 - (qq5’/qq5) 555’/555]

-g_qq^-2 d(qq5’) h_qh₅[55q”/55q-(55q’/55q)qqq’/qqq]

-g_DD^-2 d(DD5’) h_qh₅(qqD’/qqD)55D’/55D SUMMED OVER D NOT=q,5

R^q_rqr = g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr’/rrr]

-g_qq^-2 d(qqr’) h_qh_r[rrq”/rrq-(rrq’/rrq)qqq’/qqq]

-g_DD^-2 d(DDr’) h_qh_r(qqD’/qqD)rrD’/rrD SUMMED OVER D NOT=q,r

R^t_5t5 = g₅₅^-2[tt5”/tt5 - (tt5’/tt5) 555’/555]

-g_tt^-2 d(tt5’) h_th₅[55t”/55t-(55t’/55t)ttt’/ttt]

-g_DD^-2 d(DD5’) h_th₅(ttD’/ttD)55D’/55D SUMMED OVER D NOT=t,5

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

-g_tt^-2 d(ttr’) h_th_r[rrt”/rrt-(rrt’/rrt)ttt’/ttt]

-g_DD^-2 d(DDr’) h_th_r(ttD’/ttD)rrD’/rrD SUMMED OVER D NOT=t,r

R^t_ftf = g_ff^-2[ttf”/ttf - (ttf’/ttf) fff’/fff]

-g_tt^-2 d(ttf’) h_th_f[fft”/fft-(fft’/fft)ttt’/ttt]

-g_DD^-2 d(DDf’) h_th_f(ttD’/ttD)ffD’/ffD SUMMED OVER D NOT=t,f

R^r_5r5 = g₅₅^-2[rr5”/rr5 - (rr5’/rr5) 555’/555]

-g_rr^-2 d(rr5’) h_rh₅[55r”/55r-(55r’/55r)rrr’/rrr]

-g_DD^-2 d(DD5’) h_rh₅(rrD’/rrD)55D’/55D SUMMED OVER D NOT=r,5

For the general 5-D case we need to compute: R^r_5r5 R^t_ftf R^t_rtr R^t_5t5 R^f_5f5 R^f_rfr R^q_rqr R^q_fqf R^t_qtq R^q_5q5

For spherical symmetry:

G^r_r= R^t_ftf + R^t_5t5 +2 R^q_5q5 +2R^q_tqt + R^q_fqf

We want to find the above RTC.

Of the four combinations of f and q only ffq‘ is not = 0 = (sinq)‘ . Therefore:

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

R^q_fqf = -g_qq^-2 [ffq’’/ffq]

-g_DD^-2 d(DDf’) (qqD’/qqD)ffD’/ffD SUMMED OVER D =r,t,5.

= g_qq^-2 - g_DD^-2 [(qqr’/qqr)ffr’/ffr + (qqt’/qqt)fft’/fft+ (qq5’/qq5)ff5’/ff5].

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

R^q_tqt = g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt’/ttt]

--g_DD^-2 d(DDt’) (qqD’/qqD)ttD’/ttD SUMMED OVER D NOT=q,t

= g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt’/ttt] + g_rr^-2 d(rrt’) (qqr’/qqr)ttr’/ttr + g₅₅^-2 d(55t’) (qq5’/qq5)tt5’/tt5

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

R^q_5q5 = g₅₅^-2[qq5”/qq5 - (qq5’/qq5) 555’/555]

-g_qq^-2 d(qq5’) h_qh₅[55q”/55q-(55q’/55q)qqq’/qqq]

-g_DD^-2 d(DD5’) h_qh₅(qqD’/qqD)55D’/55D SUMMED OVER D NOT=q,5

= g₅₅^-2[qq5”/qq5 - (qq5’/qq5) 555’/555]

-g_rr^-2 d(rr5’) h_qh₅(qqr’/qqr)55r’/55r - g_tt^-2 d(tt5’) h_qh₅(qqt’/qqt)55t’/55t

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

R^q_rqr = g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr’/rrr]

-g_qq^-2 d(qqr’) h_qh_r[rrq”/rrq-(rrq’/rrq)qqq’/qqq]

-g_DD^-2 d(DDr’) h_qh_r(qqD’/qqD)rrD’/rrD SUMMED OVER D NOT=q,r

= g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr”/rrr] -g₅₅^-2 d(55r’)(qq5’/qq5)rr5’/rr5

-g_tt^-2 d(ttr’) (qqt’/qqt)rrt’/rrt

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

R^t_5t5 = g₅₅^-2[tt5”/tt5 - (tt5’/tt5) 555’/555]

-g_tt^-2 d(tt5’) h_th₅[55t”/55t-(55t’/55t)ttt’/ttt]

-g_DD^-2 d(DD5’) h_th₅(ttD’/ttD)55D’/55D SUMMED OVER D NOT=t,5 ,and not q,f because tt q‘=0, so the sum is only over r:

= g₅₅^-2[tt5”/tt5 - (tt5’/tt5) 555’/555]

- - g_tt^-2 d(tt5’) [55t”/55t-(55t’/55t)ttt’/ttt]

- -g_rr^-2 d(rr5’) (ttr’/ttr)55r’/55r

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

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

-g_tt^-2 d(ttr’) h_th_r[rrt”/rrt-(rrt’/rrt)ttt’/ttt]

-g_DD^-2 d(DDr’) h_th_r(ttD’/ttD)rrD’/rrD SUMMED OVER D NOT=t,r ,and not q,f because tt q‘=0, so the sum is only over 5:

= g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

- - g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

- - g_DD^-2 d(55r’) (tt5’/tt5)rr5’/rr5

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

R^t_ftf = g_ff^-2[ttf”/ttf - (ttf’/ttf) fff’/fff]

-g_tt^-2 d(ttf’) h_th_f[fft”/fft-(fft’/fft)ttt’/ttt]

-g_DD^-2 d(DDf’) h_th_f(ttD’/ttD)ffD’/ffD SUMMED OVER D NOT=t,f

= - - g_tt^-2 [fft”/fft-(fft’/fft)ttt’/ttt]

-g_DD^-2 d(DDf’) (ttD’/ttD)ffD’/ffD D =r,5[not q because ttq‘=0]

= + g_tt^-2 [fft”/fft-(fft’/fft)ttt’/ttt]

-g_rr^-2 d(rrf’) (ttr’/ttr)ffr’/ffr -g₅₅^-2 d(55f’) (tt5’/tt5)ff5’/ff5

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

R^r_5r5 = g₅₅^-2[rr5”/rr5 - (rr5’/rr5) 555’/555]

-g_rr^-2 d(rr5’) h_rh₅[55r”/55r-(55r’/55r)rrr’/rrr] -g_DD^-2 d(DD5’) h_rh₅(rrD’/rrD)55D’/55D SUMMED OVER D NOT=r,5 [not q,f because rrq‘=0] , i.e. D = t.

= g₅₅^-2[rr5”/rr5 - (rr5’/rr5) 555’/555]

-g_rr^-2 d(rr5’) [55r”/55r-(55r’/55r)rrr’/rrr] -g_tt^-2 d(tt5’) (rrt’/rrt)55t’/55t.

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

We now compute the Riemann tensor components for the metric:

(written in three equivalent ways, to demonstrate the meaning of the notation we employ)

ds² = -g_tt(t,r)dt² + g_rr(t,r)dr² + f(t) r²dW₃²: dW₃²= d5² + sin²5 dW₂²

ds² = - ttt² ttr²dt² + rrt² rrr²dr² + r² d5² + f(t) r² sin²5 dq² + f(t) r² sin²5 sin²q df²

ds² = - ttt² ttr²dt² + rrt² rrr²dr² +55t² r²d5² + qqT² qqR² qq5² dq² + fft ffr ff5 ffq df²

no f(5) in r, t, and 5 metric coefficients:

i.e, have f(5) only in angular coordinates metric coefficients: i.e.:

555= RR5 = TT5 = 1, i.e. 555’ = RR5’=TT5’=0, but qq5, ff5 not = 1.

and also with spherical symmetry:

[qqR = ffR=55R = r, qq5 = sin5, ffq‘ = (sinq)‘ .]

Note: remember that the g’s here are the squareroot of the actual g’s, and all are positive - the h carries the negative sign of g_tt:

g_rr^-2 = (rrt rrr)^-2 , g_tt^-2= (ttt ttr)^-2 , g₅₅^-2 = r^-2.

We want to compute the following:

G^r_r= R^t_ftf + R^t_5t5 +2 R^q_5q5 +2R^q_tqt + R^q_fqf

So we wish to find: R^r_5r5 R^t_ftf R^t_rtr R^t_5t5 R^q_rqr R^q_5q5

Of the four angular combinations only ffq‘ not = 0 = (sinq)‘ .

[ffq”/ffq] = -[sinq/sinq] = -1. Therefore:

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

R^q_fqf = -g_qq^-2 [ffq”/ffq]

-g_DD^-2 d(DDf’) (qqD’/qqD)ffD’/ffD SUMMED OVER D =r,t,5.

= - - g_qq^-2 - g_rr^-2 (qqr’/qqr)ffr’/ffr - g_tt^-2(qqt’/qqt)fft’/fft - g₅₅^-2 (qq5’/qq5)ff5’/ff5.

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

R^q_tqt = g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt”/ttt]

--g_DD^-2 d(DDt’) (qqD’/qqD)ttD’/ttD SUMMED OVER D NOT=q,t

= g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt”/ttt] + g_rr^-2 d(rrt’) (qqr’/qqr)ttr’/ttr

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

R^q_5q5 = g₅₅^-2[qq5”/qq5 - (qq5’/qq5) 555’/555]

-g_DD^-2 d(DD5’) (qqD’/qqD)55D’/55D SUMMED OVER D NOT=q,5

= g₅₅^-2[qq5”/qq5]

-g_rr^-2 d(rr5’) (qqr’/qqr)55r’/55r - g_tt^-2 d(tt5’) (qqt’/qqt)55t’/55t

R^q_rqr = g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr’/rrr]

-g_DD^-2 d(DDr’) (qqD’/qqD)rrD’/rrD SUMMED OVER D NOT=q,r

= g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr’/rrr] -g_tt^-2 d(ttr’) (qqt’/qqt)rrt’/rrt

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

R^t_5t5 = --g_tt^-2 d(tt5’) [55t”/55t-(55t’/55t)ttt’/ttt]

--g_DD^-2 d(DD5’) (ttD’/ttD)55D’/55D SUMMED OVER D NOT=t,5

= g_tt^-2 d(tt5’) [55t”/55t-(55t’/55t)ttt’/ttt] + g_rr^-2 d(rr5’) (ttr’/ttr)55r’/55r

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

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

-g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

-g₅₅^-2 d(55r’) (tt5’/tt5)rr5’/rr5 [it was SUMMED OVER D NOT=t,r, nor the angular coordinates, i.e.it was summed only over 5.]

= g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr] -g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

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

R^t_ftf = + g_tt^-2 [fft”/fft-(fft’/fft)ttt’/ttt] -g_rr^-2 (ttr’/ttr)ffr’/ffr

-g₅₅^-2 (tt5’/tt5)ff5’/ff5

= + g_tt^-2 [fft”/fft-(fft’/fft)ttt’/ttt] -g_rr^-2 (ttr’/ttr)ffr’/ffr

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

R^r_5r5 = -g_rr^-2 d(rr5’) [55r”/55r-(55r’/55r)rrr’/rrr] - g_tt^-2 d(tt5’)(rrt’/rrt)55t’/55t

= -g_rr^-2 [55r”/55r-(55r’/55r)rrr’/rrr] - g_tt^-2 (rrt’/rrt)55t’/55t

..........

5-D: STATIC: Sph'ly Sym: S³ formed from the angular and fifth coordinates:

There is no f(5) in r, t, and 5 metric coefficients:

The metric is written in three equivalent ways, to demonstrate the meaning of the notation we employ:

ds² = -g_tt(r)dt² + g_rr(r)dr² + r²dW₃²: dW₃²= d5² + sin²5 dW₂²

ds² = -ttr²dt² + rrr²dr² + r² d5² + r² sin²5 dq² + r² sin²5 sin²q df²

ds² = -ttr²dt² + rrr²dr² + 55r²d5² + qqR² qq5² dq² + ffr ff5 ffq df²

There is no f(5) in r, t, and 5 metric coefficients:

i.e: qqR = ffR=55R = r, qq5 = sin5, ffq‘ = (sinq)‘ .

555= RR5 = TT5 = 1,

i.e. 555’ = RR5’=TT5’=0,

Note: remember that the g’s here are the squareroot of the actual metric coefficient g’s, and all are positive - the h carries the negative sign of g_tt:

g_rr^-2 = (rrr)^-2 , g_tt^-2= (-ttr)^-2 , g₅₅^-2 = r^-2.

We want to compute the following: R^r_5r5 R^t_ftf R^t_rtr R^t_5t5 R^q_rqr R^q₅^q_fqf

G^r_r= R^t_ftf + R^t_5t5 +2 R^q_5q5 +2R^q_tqt + R^q_fqf

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

R^q_fqf = - - g_qq^-2 - g_rr^-2 (qqr’/qqr)ffr’/ffr - g_tt^-2(qqt’/qqt)fft’/fft - g₅₅^-2 (qq5’/qq5)ff5’/ff5

= g_qq^-2 - g_rr^-2 (1/r²) - g₅₅^-2(qq5’/qq5) ²

= g_qq^-2 - g_rr^-2 (1/r²) - g₅₅^-2(qq5’/qq5) ²

Since: qqR = ffR=55R = r, qq5 = sin5, 555= RR5 = TT5 = 1

g_qq = r sin5, g_rr^-2 = (rrr)^-2 , g_tt^-2= (-ttr)^-2 , g₅₅^-2 = r^-2.

R^q_fqf = r^-2(1/sin5)² - g_rr^-2/r² - r^-2(cos5/sin5) ²

= r^-2{- g_rr^-2 + (1/sin²5) - (cos²5/sin²5) } = r^-2{- g_rr^-2 + (1 - cos²5/sin²5) }

i.e.

R^q_fqf = (1 - g_rr^-2)/r² ,

which is linear in g_rr^-2, which is g_rr^-1 as usually defined.

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

R^q_tqt = g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt’/ttt] + g_rr^-2 d(rrt’) (1/r)ttr’/ttr

=g_rr^-2 (1/r) ttr’/ttr

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

R^q_5q5 = g₅₅^-2[qq5”/qq5] - g_rr^-2 d(rr5’) 1/r(55r’/55r) - g_tt^-2 d(tt5’) (qqt’/qqt)55t’/55t

= g₅₅^-2[qq5”/qq5] - g_rr^-2(1/r²).

Since for this metric: qq5 = sin5, g₅₅^-2 = r^-2.

R^q_5q5 = r^-2[sin5”/sin5] - g_rr^-2(1/r²) = {[-1] - g_rr^-2}(1/r²)

R^q_5q5 = - (1+g_rr^-2)/r².

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

R^q_rqr = g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr’/rrr] - g_tt^-2 d(ttr’) (qqt’/qqt)rrt’/rrt

= g_rr^-2[r”/r - (r’/r) rrr’/rrr]

= -g_rr^-2 (1/r) rrr’/rrr

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

R^t_5t5 = g_tt^-2 [55t”/55t-(55t’/55t)ttt’/ttt] + g_rr^-2 (ttr’/ttr)r’/r

= g_rr^-2 (1/r) (ttr’/ttr)

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

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

-g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

-g_DD^-2 d(DDr’) (ttD’/ttD)rrD’/rrD SUMMED OVER D NOT=t,r

= g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

-g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

-g₅₅^-2 d(55r’) (tt5’/tt5)rr5’/rr5

= g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

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

R^t_ftf = g_ff^-2[ttf”/ttf - (ttf’/ttf) fff’/fff]

+g_tt^-2 d(ttf’) [fft”/fft-(fft’/fft)ttt’/ttt]

+g_rr^-2 d(rrf’) (ttr’/ttr)r’/r

= g_rr^-2 (1/r)(ttr’/ttr)

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

R^r_5r5 = - g_rr^-2 d(rr5’) [r”/r-(r’/r)rrr’/rrr] -g_tt^-2 d(tt5’)(rrt’/rrt)55t’/55t

= g_rr^-2(1/r)[rrr’/rrr]

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

Inverse Metric

R^r_5r5 = - R^q_rqr = g_rr^-2(1/r)[rrr’/rrr]

Remember that rrr is the square-root of the r function part of g_rr. We can express this R in terms of the function itself, i.e. rrr². Sometimes the function is an inverse, and we can express R^r_5r5 and R^q_rqr in terms of rrr^-1 or rrr^-2

In terms of rrr²:

D^2'/D² = 2DD’/D² = 2D’/D. or D’/D = D^2'/2D² .

I.e. changing it to the square makes a change only in a factor of 2 in the denominator. Therefore we can write [rrr’/rrr] as [rrr²’/2rrr²].

In terms of rrr^-1:

D^-1'/D^-1 = - D^-2D’/D^-1 = - D’/D.

I.e. changing it to the inverse makes a change only in the sign. Therefore we can write [rrr’/rrr] as [- rrr^-1’/rrr^-1].

In terms of rrr^-2:

Combining these, we obtain: [rrr’/rrr] = [rrr²’/2rrr²] = -[rrr-²’/2rrr-²].

Where rrr = ttr^-1 : we can express [rrr’/rrr] in terms of ttr²:

Since in this case ttr² = rrr^-2:

[rrr’/rrr] = -[rrr^-2’/2rrr^-2] = ttr²’/2ttr².

the R was for static metric, no 5, sph sym so no angular functions, so is only r function, i.e. g_rr = rrr = ttr^-1:

g_rr^-2 [rrr’/rrr] = rrr^-2 [ttr²’/2ttr²] = ttr² [ttr²’/2ttr²] = ttr²’/2

= g_tt²’/2

= in the ordinary notation = g_tt’/2.

In this form we can see that R^r_5r5 and R^q_rqr are linear in g_tt as it is usually defined.

........

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

Conundrum:

R^t_rtr contains a second derivative as well as a product of first derivatives. Only this R has such a product term, and so in summing the R’s for computing G, it cannot cancel, and so one expects that this will inevitably be a term in G, and so G can therefore not be linear.

Solution:

The second derivative in ttr when expressed in terms of ttr² is actually composed of two terms, one of which is the square of the first derivative, and when rrr = ttr^-1 it actually exactly cancels the other term and so G can be linear!

Note that since rrr’/rrr = - rrr^-1’/rrr^-1:

when rrr = 1/ttr we can set: rrr’/rrr = - rrr^-1’/rrr^-1 = -ttr/ttr, and so:

i.e. for rrr = ttr^-1:

R^t_rtr = g_rr^-2[ttr”/ttr + (ttr’/ttr)²] .

Since g_rr^-2 = g_tt² = ttr² : and from algebra: (ttr’/ttr) = (ttr²’/2ttr²) :

Performing the multiplication by g etc we obtain:

R^t_rtr = ttr²[ttr”/ttr + (ttr²’/2ttr²) ²]

= ttr” ttr + ttr’² .

As we mentioned above, the second term will be exactly cancelled by a part of the first term, as shown below:

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

ttr” = [ttr²]^1/2” = {(1/2)[ttr²]^-1/2 [ttr²]’ }’

= (1/2){(-1/2)[ttr²]^-3/2 [ttr²]’² + [ttr²]^-1/2 [ttr²]’’ }

Therefore:

ttr” ttr = [ttr²]^1/2” [ttr²]^1/2

= - [ttr²]’²/4[ttr²] + ttr²’’/2 .

ttr^’2 = {[ttr²]^1/2}’² = {(1/2)[ttr²]^-1/2 [ttr²]’}² = (1/4)[ttr²]^-1 [ttr²]’²

= [ttr²]’²/4[ttr²] .

Therefore:

R^t_rtr = ttr” ttr + ttr’² = - [ttr²]’²/4[ttr²] + ttr²’’/2 + [ttr²]’²/4[ttr²]

= - [ttr²]’²/4[ttr²] + ttr²’’/2 + [ttr²]’²/4[ttr²]

= ttr²’’/2 .

A beautiful cancellation, getting rid of the term with the square of the first derivative.

...........

Expressing ttr”/ttr in terms of [ttr²].

ttr”/ttr = [ttr²]^1/2”/[ttr²]^1/2 = [ttr²]^1/2”/[ttr²]^1/2

And:

[ttr²]^1/2” = {(1/2)[ttr²]^-1/2 [ttr²]’ }’

= (1/2){(-1/2)[ttr²]^-3/2 [ttr²]’² + [ttr²]^-1/2 [ttr²]’’ }

Therefore:

ttr”/ttr = [ttr²]^1/2”/[ttr²]^1/2

= (1/2){(-1/2)[ttr²]^-2 [ttr²]’² + [ttr²]^-1 [ttr²]’’ }

= (-1/4)[ttr²]^-2 [ttr²]’² + (1/2)[ttr²]^-1 [ttr²]’’

= -[ttr²]’²/4[ttr²]² + (1/2) [ttr²]’’/[ttr²] .

Since g_rr^-2 = g_tt² = ttr² :

R^t_rtr = g_rr^-2[ttr”/ttr + (ttr’/ttr)²]

= ttr²{-[ttr²]’²/4[ttr²]² + (1/2) [ttr²]’’/[ttr²] + (ttr’/ttr)²}

= {-[ttr²]’²/4[ttr²] + (1/2) [ttr²]’’ + ttr’².

.........

g_rr^-2= rrr^-2 Prrc^-2 = ttr² Prrc^-2

Therefore: g_rr^-2[rrr’/rrr] = ttr² Prrc^-2[ttr²’/2ttr²] = (1/2) Prrc^-2[ttr²’]

but actually this is not needed because the R was for static metric, no 5, sph sym so no angular functions, so is only r function, i.e. g_rr = rrr.

Where all of the functions in g_rr and g_tt are inversely related......

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

rrr = ttr^-1.

G^r_r= R^t_ftf + R^t_5t5 +2 R^q_5q5 +-2R^q_tqt + R^q_fqf

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

Because the metric is static it has no t dependence, and it has little 5 dependence, and small angular dependence bec it is sph. sym, the only coordinate playing a major role in the metric coefficient functions is r, and therefore g_rr is prominent.

R^q_fqf = (1 - g_rr^-2)/r² ,

R^q_tqt =g_rr^-2 (1/r) ttr’/ttr

R^q_5q5 = - (1+g_rr^-2)/r².

R^q_rqr = -g_rr^-2 (1/r) rrr’/rrr

R^t_5t5 = g_rr^-2 (1/r) (ttr’/ttr)

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr’/rrr]

R^t_ftf = g_rr^-2 (1/r)(ttr’/ttr)

R^r_5r5 = g_rr^-2(1/r)[rrr’/rrr]

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

For Inverse Form of the Metric

g_rr^-2 [rrr’/rrr] = rrr^-2 [ttr²’/2ttr²] = ttr² [ttr²’/2ttr²] = ttr²’/2

= g_tt²’/2

= in the ordinary notation = g_tt’/2.

In this form we can see that R^r_5r5 and R^q_rqr are linear in g_tt as it is usually defined.

........

g_rr^-2 [rrr’/rrr] = g_tt²’/2 = in the ordinary notation = g_tt’/2.

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

R^q_fqf = (1 - g_rr^-2)/r² = (1 - g_tt^-2)/r² ,

R^q_tqt = g_rr^-2 (1/r) ttr’/ttr = g_tt² (1/r) ttr’/ttr = ttr² (1/r) ttr’/ttr = ttr ttr’/r

= ttr^2’/2r = in the ordinary notation = g_tt’/2r.

R^q_5q5 = - (1+g_rr^-2)/r².

R^q_rqr = -g_rr^-2 (1/r) rrr’/rrr = - g_tt²’/2r = in the ordinary notation = - g_tt’/2r.

R^t_5t5 = g_rr^-2 (1/r) (ttr’/ttr) = ttr^2’/2r = in the ordinary notation = g_tt’/2.

R^t_rtr = ttr²’’/2 .

R^t_ftf = g_rr^-2 (1/r)(ttr’/ttr) = ttr^2’/2r = in the ordinary notation = g_tt’/2r.

R^r_5r5 = g_rr^-2(1/r)[rrr’/rrr] = g_tt²’/2r = in the ordinary notation = g_tt’/2r.

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

rrr = ttr^-1.

G^r_r= R^t_ftf + R^t_5t5 +2 R^q_5q5 +2R^q_tqt + R^q_fqf

= ttr^2’/2r + ttr^2’/2r - 2 (1+g_rr^-2)/r² + 2 ttr^2’/2r + (1 - g_rr^-2)/r² ,

= 2 ttr^2’/r - 2 (1+g_rr^-2)/r² + (1 - g_rr^-2)/r² ,

R^q_rqr = -g_rr^-2 (1/r) rrr’/rrr = g_tt²’/2r = in the ordinary notation = g_tt’/2r.

R^t_rtr = ttr²’’/2 .

R^r_5r5 = g_rr^-2(1/r)[rrr’/rrr] = g_tt²’/2r = in the ordinary notation = g_tt’/2r.

Example: 4 dimensions

G^r_r = R^t_ftf +2R^q_tqt + R^q_fqf

and of the four combinations of f and q only ffq‘ is not = 0 = (sinq)‘ . We can see from the below that all the R’s with a 5 index do in fact vanish, as should be the case for 4 dimensions.

Note that the R’s with one angular coordinate will have summation terms involving the other angular coordinate, (e.g. ttq’) and these will vanish for Sph. Sym., and so the terms in the 5 coordinate will not play a factor. The only place the 5 comes in will be where 5 is one of the indices of R, and this R will vanish entirely in 4-d, and also in the R with both angular coordinates, i.e. R^q_fqf, since here only mixtures of r and t and 5 will occur. Therefore, the 4-d R’s are given by the 5-d R’s with the R’s having a 5 index set to zero, and the term in R^q_fqf with 5 in it also set to zero. All the other R’s are the same in 4 and 5 dimensions. When computing the actual values of the terms, e.g. g_qq, of course the value will be different for the case where g_qq has a functional dependance on 5 and when it does not (as in the 5-d case of interest to us with inverse metric form and f(5) only in the angular metric coefficients).

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

R^q_fqf = g_qq^-2 - g_rr^-2 (qqr’/qqr)ffr’/ffr - g_tt^-2(qqt’/qqt)fft’/fft -g₅₅^-2 (qq5’/qq5)ff5’/ff5.

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

R^q_tqt = g_tt^-2[qqt”/qqt - (qqt’/qqt) ttt”/ttt] + g_rr^-2 d(rrt’) (qqr’/qqr)ttr’/ttr

+ g₅₅^-2 d(55t’) (qq5’/qq5)tt5’/tt5

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

As it should be, all terms here are zero:

R^q_5q5 = g₅₅^-2[qq5”/qq5 - (qq5’/qq5) 555”/555]

-g_rr^-2 d(rr5’) (qqr’/qqr)55r’/55r - g_tt^-2 d(tt5’) (qqt’/qqt)55t’/55t

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

R^q_rqr = g_rr^-2[qqr”/qqr - (qqr’/qqr) rrr”/rrr] -g₅₅^-2 d(55r’)(qq5’/qq5)rr5’/rr5

-g_tt^-2 d(ttr’) (qqt’/qqt)rrt’/rrt

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

As should be the case, all terms here are zero:

R^t_5t5 = g₅₅^-2[tt5”/tt5 - (tt5’/tt5) 555”/555]

- - g_tt^-2 d(tt5’) [55t”/55t-(55t’/55t)ttt’/ttt]

- -g_rr^-2 d(rr5’) (ttr’/ttr)55r’/55r

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

R^t_rtr = g_rr^-2[ttr”/ttr - (ttr’/ttr) rrr”/rrr]

- - g_tt^-2 d(ttr’) [rrt”/rrt-(rrt’/rrt)ttt’/ttt]

- - g_DD^-2 d(55r’) (tt5’/tt5)rr5’/rr5

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

R^t_ftf= g_tt^-2 [fft”/fft-(fft’/fft)ttt’/ttt] -g_rr^-2 d(rrf’) (ttr’/ttr)ffr’/ffr

-g₅₅^-2 d(55f’) (tt5’/tt5)ff5’/ff5

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

R^r_5r5 = g₅₅^-2[rr5”/rr5 - (rr5’/rr5) 555”/555]

-g_rr^-2 d(rr5’) [55r”/55r-(55r’/55r)rrr’/rrr] -g_tt^-2 d(tt5’) (rrt’/rrt)55t’/55t.

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

[1] The following was file "Chatterjee", beginning with section "Higher dimensional theories"

[2] NOTE: We use MTW[heeler]’s method to find w^A_D .

NOTE: can be ignored: There are two categories here: where g_AA,D does not vanish, and where it does vanish, and these correspond to the cases where w^A_D will be proportional to w^A or to w^D respectively. We find that where g_AA,D = 0:

w^A_{D =} [g_AA,D/g_DDg_AA] w^A

If it is not zero:

w^A_{D =} - h^A_A h^D_D [g_DD,A/g_AAg_DD] w^D .

[3] Clearly whereg_AA,D = the first term above will vanish, whereas if it is not zero, the second will vanish.