Geodesicsokmope 1
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The document summarizes key concepts regarding geodesics and affine parametrization related to black holes. It discusses: 1) The action for a particle moving along a time-like curve in spacetime and how this relates to the metric and proper time. 2) How the action can be rewritten in terms of an independent function and the equations of motion that result from extremizing the action with respect to this function and spacetime coordinates. 3) This leads to a geodesic equation of motion involving the connection that is symmetric in its lower indices.
![Study notes
Blackholes
FJPRoa
I. GEODESICS/AFFINE PARAMETRIZATION
The action for a particle of mass m moving on a time-like curve C is given by
∫−=
B
A
dmcI
τ
τ
τ2
, (1)
where τ is the proper-time on C, c is the speed of light in the vacuum, which can be set c = 1 for
calculational convenience, and the metric signature is ( - + + + ). The two boundary values, Aτ and Bτ are
held fixed in taking the variation of I. Along the curve C we take λ as an arbitrary parameter and that for
notation we have, λddxx uu
/=& .
The fundamental line element
2
ds is related to the metric of a given space-time by
uv
vu
gdxdxdsd −=−= 2
τ . (2)
Thus, the action I can be re-written as
∫ −−=
B
A
uv
vu
gxxdmxI
λ
λ
λ &&)( . (3)
The two boundary values of λ , ),( BA λλ , correspond to the end-values Aτ and Bτ , and are
also held fixed in taking the variation of )(xI .
The time-like curve C, which defines the world-line of the particle, constitutes a set of coordinate
paths )(λu
x such that the extremum condition in the variation of )(xI is satisfied. That is,
0
)(
=
λδ
δ
u
x
I
, (4)
where each function )(λu
x is then a geodesic of space-time.
An equivalent action to (3) can also be defined by
[ ]∫ −= −
B
A
emgxxedexI vu
vu
λ
λ
λλλ )()(
2
1
),( 21
&& , (5)
where )(λee = is a new independent function of λ . Later on, we will show that by taking independent
variations (extremizations) of ),( exI in terms of eδ , and in terms of space-time coordinates
u
xδ ,
action (5) is equivalent to the original action (3).
As an exercise let us first tackle on a mathematical problem of extremizing (5) in terms of eδ .
That is, we take the variation eexI δδ /),( , and as for extremum condition, set this to zero and upon
noting that the end-values of eδ are zero: 0)()( == BA ee λδλδ .
For calculational convenience we define a function f(x, e) by
emgxxeexf vu
vu 21
),( −= −
&& (6)
so that the variation ],[ exIδ is to be given by
[ ]∫=
B
A
exfdexI
λ
λ
δλδ ),(
2
1
),( , (7)
where](https://image.slidesharecdn.com/geodesicsokmope1-160322164406/85/Geodesicsokmope-1-1-320.jpg)
![[ ] e
e
f
e
e
f
exf &
&
δ
δ
δ
δ
δ
δ
δ +=),( , (8)
and note that λδδ dede /)(=& .
The result of integration by parts of the integral, ∫
B
A
e
e
f
d
λ
λ
δ
δ
δ
λ &
&2
1
, noting that
0)()( == BA ee λδλδ , yields the variation
0
2
1),(
=
−= ∫
B
A
e
f
d
d
e
f
d
e
exI
λ
λ δ
δ
λδ
δ
λ
δ
δ
&
, (9)
which is set to zero for extremum condition. Thus, from this result we obtain the Euler-Lagrange equation
for ),( exf :
0=
−
e
f
d
d
e
f
&δ
δ
λδ
δ
, (10)
and note that f(x, e) does not depend on e& so that 0/ =ef &δδ . The variation ef δδ / is given by
022
=−−= −
mgxxe
e
f
vu
vu
&&
δ
δ
, (11)
which in turn is set to zero upon the extremum condition stated in (9), and also with the result
0/ =ef &δδ . Thus, in order for f(x ,e) and consequently for I(x, e) to be extremum in their variations in
terms of eδ is such that )(λe must satisfy
vu
vu
gxx
m
e &&−=
1
. (12)
As for the extremization of (5) in terms of space-time coordinates we apply the variational integral
0
2
1],[
=
−= ∫
B
A
uuu
u
x
f
d
d
x
f
d
x
exI
λ
λ δ
δ
λδ
δ
λ
δ
δ
&
, (13)
which is set to zero for extremum condition and the function f is that given in (6). We have set that the
end-values of
u
xδ vanish at the space-time boundaries: 0)()( == B
u
A
u
xx λδλδ . Thus, arriving at
the form of the integrand in (13) from which we have the corresponding Euler-Lagrange eq’n.
0=
− uu
x
f
d
d
x
f
&δ
δ
λδ
δ
, (14)
which ],[ exf u
needs to satisfy as it is varied in terms of the variations
u
xδ of space-time coordinates
u
x .
Let us take on the variation
u
xf &δδ / first, which yields
vu
v
u
gxe
x
f
&
&
1
2 −
=
δ
δ
(15)
with the factor 2 as we note that uvg is symmetric in the interchange of lower indices along the summation
over the dummy index v . From (15) we take the derivative of that with respect to λ so that we have](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Geodesicsokmope 1
- 1. Study notes Blackholes FJPRoa I. GEODESICS/AFFINE PARAMETRIZATION The action for a particle of mass m moving on a time-like curve C is given by ∫−= B A dmcI τ τ τ2 , (1) where τ is the proper-time on C, c is the speed of light in the vacuum, which can be set c = 1 for calculational convenience, and the metric signature is ( - + + + ). The two boundary values, Aτ and Bτ are held fixed in taking the variation of I. Along the curve C we take λ as an arbitrary parameter and that for notation we have, λddxx uu /=& . The fundamental line element 2 ds is related to the metric of a given space-time by uv vu gdxdxdsd −=−= 2 τ . (2) Thus, the action I can be re-written as ∫ −−= B A uv vu gxxdmxI λ λ λ &&)( . (3) The two boundary values of λ , ),( BA λλ , correspond to the end-values Aτ and Bτ , and are also held fixed in taking the variation of )(xI . The time-like curve C, which defines the world-line of the particle, constitutes a set of coordinate paths )(λu x such that the extremum condition in the variation of )(xI is satisfied. That is, 0 )( = λδ δ u x I , (4) where each function )(λu x is then a geodesic of space-time. An equivalent action to (3) can also be defined by [ ]∫ −= − B A emgxxedexI vu vu λ λ λλλ )()( 2 1 ),( 21 && , (5) where )(λee = is a new independent function of λ . Later on, we will show that by taking independent variations (extremizations) of ),( exI in terms of eδ , and in terms of space-time coordinates u xδ , action (5) is equivalent to the original action (3). As an exercise let us first tackle on a mathematical problem of extremizing (5) in terms of eδ . That is, we take the variation eexI δδ /),( , and as for extremum condition, set this to zero and upon noting that the end-values of eδ are zero: 0)()( == BA ee λδλδ . For calculational convenience we define a function f(x, e) by emgxxeexf vu vu 21 ),( −= − && (6) so that the variation ],[ exIδ is to be given by [ ]∫= B A exfdexI λ λ δλδ ),( 2 1 ),( , (7) where
- 2. [ ] e e f e e f exf & & δ δ δ δ δ δ δ +=),( , (8) and note that λδδ dede /)(=& . The result of integration by parts of the integral, ∫ B A e e f d λ λ δ δ δ λ & &2 1 , noting that 0)()( == BA ee λδλδ , yields the variation 0 2 1),( = −= ∫ B A e f d d e f d e exI λ λ δ δ λδ δ λ δ δ & , (9) which is set to zero for extremum condition. Thus, from this result we obtain the Euler-Lagrange equation for ),( exf : 0= − e f d d e f &δ δ λδ δ , (10) and note that f(x, e) does not depend on e& so that 0/ =ef &δδ . The variation ef δδ / is given by 022 =−−= − mgxxe e f vu vu && δ δ , (11) which in turn is set to zero upon the extremum condition stated in (9), and also with the result 0/ =ef &δδ . Thus, in order for f(x ,e) and consequently for I(x, e) to be extremum in their variations in terms of eδ is such that )(λe must satisfy vu vu gxx m e &&−= 1 . (12) As for the extremization of (5) in terms of space-time coordinates we apply the variational integral 0 2 1],[ = −= ∫ B A uuu u x f d d x f d x exI λ λ δ δ λδ δ λ δ δ & , (13) which is set to zero for extremum condition and the function f is that given in (6). We have set that the end-values of u xδ vanish at the space-time boundaries: 0)()( == B u A u xx λδλδ . Thus, arriving at the form of the integrand in (13) from which we have the corresponding Euler-Lagrange eq’n. 0= − uu x f d d x f &δ δ λδ δ , (14) which ],[ exf u needs to satisfy as it is varied in terms of the variations u xδ of space-time coordinates u x . Let us take on the variation u xf &δδ / first, which yields vu v u gxe x f & & 1 2 − = δ δ (15) with the factor 2 as we note that uvg is symmetric in the interchange of lower indices along the summation over the dummy index v . From (15) we take the derivative of that with respect to λ so that we have
- 3. vu v vu v vuw wv u gxeegxegxxe x f d d &&&&&& & 211 22)(2 −−− −+∂= δ δ λ , (16) while for u xf δδ / we have )(1 wvu wv u gxxe x f ∂= − && δ δ . (17) Plugging results (16) and (17) into (14) yields vu v vwu wv vuw wv vu v gx e e gxxgxxgx & & &&&&&& =∂−∂+ )( 2 1 )( , (18) and we can put ( )vuwuvw wv uvw wv ggxxgxx ∂+∂=∂ &&&& 2 1 )( (19) because of the symmetry property of uvg in the interchange of its lower indices, vuuv gg = . Then we can make some re-labeling of the indices in one of the summed up terms in (19), )()( uwv wv uvw wv gxxgxx ∂→∂ &&&& (20) since v and w are just dummy indices. From these we re-write (18) into ( ) vu v wvuwuvuvw wv uv v gx e e gggxxgx & & &&&& =∂−∂+∂+ 2 1 . (21) Upon summing up both sides of (21) with the conjugate metric tensor su g and noting that uv sus v gg=δ , which is the Kronecker delta tensor with the property that its component is zero when vs ≠ , and unity when vs = , we finally arrive at an equation of motion given by ss vw wvs x e e xxx & & &&&& =Γ+ , (22) where ( )vwuwuvvuw uss vw gggg ∂−∂+∂=Γ 2 1 , (23) which is a torsion-free connection that is symmetric in its lower indices. Recalling back what we have resulted in (12) and multiplying both sides of that by 2 m and substitute the result in the integrand of (5) we get vu vu vu vu gxxmemgxxe &&&& −−=−− 221 , (24) where vu vu vu vu gxxmgxxe &&&& −−=−1 , (25) so that action (3) is recovered from action (5), and by carrying out the variation (4) leads to the same equation of motion (22). Thus, we find that the original action (3) is equivalent to action (5) not only because action (3) is recoverable from (5) but these two actions lead to the same equation of motion as carried out in terms of the variation of space-time coordinates. KILLING VECTORS Ref’s [1]Townsend, P. K., Blackholes – Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012 [2]Carroll, S. M., Lecture notes On General Relativity, http://www.arxiv.org/abs/gr-qc/9712019 [3] ] J. Foster, J. D. Nightingale, A SHORT COURSE IN GENERAL RELATIVITY, 2nd edition copyright 1995, Springer-Verlag, New York, Inc.,
- 4. [4]Gravitation And Relativity, Bowler, M. G., Pergamon Press Inc., Maxwell House, Fairview Park, ElmsFord, New York 1053, U. S. A., copyright 1976, chap. 7, sec. 7.2- 7.3 [5] Ohanian, H. C., GRAVITATION AND SPACETIME, New York: W. W. Norton and Company Inc., copyright 1976 [6] Rainville, E. D., Bedient, P. E., Elementary Differential Equations, Macmillan Publishing Co., Inc., New York, USA, 1981