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Electro-Magnetic Field Equation and Wave Function, Equation in
Rindler spacetime
Sangwha-Yi
Department of Math , Taejon University 300-716
ABSTRACT
In the general relativity theory, we find the electro-magnetic field transformation and the
electro-magnetic field equation (Maxwell equation) in Rindler spacetime. We find the
electro-magnetic wave equation and the electro-magnetic wave function in Rindler
space-time. Specially, this article say the uniqueness of the accelerated frame because
the accelerated frame can treat electro-magnetic field equation.
PACS Number:04,04.90.+e,03.30, 41.20
Key words:General relativity theory,
Rindler spacetime,
Electro-magnetic field transformation,
Electro-magnetic field equation
Wave function
Wave equation
e-mail address:[email protected]
Tel:051-624-3953
1. Introduction
In the general relativity theory, our article’s aim is that we find the electro-magnetic field equation in
Rindler space-time.
The Rindler coordinate is
)sinh()(0
01
0
2
c
a
a
cct
0
2001
0
2
)cosh()(a
c
c
a
a
cx
,
32, zy (1)
In this time, the tetrad ae is
][1 222
2
22 dzdydxc
dtd
dd
xx
c
ba
ab
2
1
ddgc
ddeec
baab 22
11 ,
aa xe (2)
)0,0),sinh()1(),cosh()1(()(0
01
2
00
01
2
0
0
00
c
a
c
a
c
a
c
a
c
xe
(3)
About y -axis’s and z -axis’s orientation
)0,1,0,0()(2
02
xe , )1,0,0,0()(
3
03
xe (4)
The other unit vector )( 01 e is
)0,0),cosh(),(sinh()(0
00
0
1
01
c
a
c
axe
(5)
Therefore,
10
01
2
000
0 )sinh()1()cosh(
dc
a
c
ad
c
accdt
10
000
0 ˆ)sinh(ˆ)cosh(
dc
ad
c
ac
10
01
2
000
0 )cosh()1()sinh(
dc
a
c
ad
c
acdx
10
000
0 ˆ)cosh(ˆ)sinh(
dc
ad
c
ac ,
22 ̂ dddy , 33 ̂ dddz (6)
The vector transformation is
V
x
xV
'
' ,
Ux
xU
''
(7)
Therefore, the transformation of the electro-magnetic 4-vector potential AA ),(
is
AeA
xA
x
xA
''
,
x
e
ded
xdx
x
xdx
''
,
x
e (8)
Hence, the transformation of the electro-magnetic 4-vector potential ),( A
in inertial frame and the
eletro-magnetic 4-vector potential ),( A
in uniformly accelerated frame is
4)1( 2
2
2
2
tc
jc
Atc
4)
1( 2
2
2
2
4-vector
d
dxjc 0),(
1)sinh()1)(cosh(0
01
2
00
0
A
c
a
c
a
c
a
1ˆ)sinh(ˆ)cosh(
00
00
A
c
a
c
a
1)cosh()1)(sinh(0
01
2
00
0
A
c
a
c
a
c
aAx
1ˆ)cosh(ˆ)sinh(
00
00
A
c
a
c
a
3322ˆ,ˆAAAAAA zy (9)
1000
0100
0010
000)1( 2
2
10
c
a
g
,
1000
0100
0010
0001
ba
ba ee
,
aa ee
gee abba gAAT
abba gee
1111 )()( AAAAgAA TTT
baba ege gAAAAA TTT 1111 )()( (10)
dz
dy
dx
cdt
1000
0100
00)cosh()1)(sinh(
00)sinh()1)(cosh(
00
2
10
00
00
2
10
00
c
a
c
a
c
ac
a
c
a
c
a
3
2
1
0
d
d
d
cd
3
2
1
0
d
d
d
cd
A
1000
0100
00)cosh()sinh(
00)sinh()cosh(
00
00
00
00
c
a
c
ac
a
c
a
3
2
1
0
ˆ
ˆ
ˆ
ˆ
d
d
d
cd
(11)
zyxtc
zyxtc
zyxtc
z
c
y
c
x
c
tc
c
Ax
e
3333
2222
1111
0000
1
1000
0100
00)cosh()sinh(
00
)1(
)sinh(
)1(
)cosh(
00
00
2
10
00
2
10
00
c
a
c
ac
ac
a
c
ac
a
(12)
z
y
x
tc
1
3
2
1
0
1
1
)(
c
A T
3
2
1
0
1
1
)(
c
AT
1000
0100
00)cosh(
)1(
)sinh(
00)sinh(
)1(
)cosh(
00
2
10
00
00
2
10
00
c
a
c
ac
a
c
a
c
ac
a
3
2
1
0
1
c
1000
0100
00)co s h ()s inh (
00)s inh ()co s h (
00
00
00
00
c
a
c
ac
a
c
a
3
2
1
0
ˆ
ˆ
ˆ
ˆ1
c
(13)
1
1
0
0 11
tcctc
c
tc
1
00
0
2
10
00
)sinh(
)1(
)cosh(
c
a
c
c
ac
a
1
00
0
00
ˆ)s inh (
ˆ)co s h (
c
a
cc
a
1
1
0
0 1
xcx
c
x
1
00
0
2
10
00
)cosh(
)1(
)sinh(
c
a
c
c
ac
a
1
00
0
00
ˆ)cosh(
ˆ)sinh(
c
a
cc
a
3322 ˆ
,ˆ
zy
2
ˆ2
02
22
021
2
02
2
2
2
2)
ˆ(1
)(
)1(
11
c
c
ac
tc
),,(zyx
, ),,(
321
, )
ˆ,
ˆ,
ˆ(
321ˆ
(14)
2. Electro-magnetic Field in the Rindler space-time
The electro-magnetic field ),( BE
is in the inertial frame,
tc
AE
, AB
(15)
tc
A
xE xx
])sinh()1)([cosh(])cosh(
)1(
)sinh([ 1
00
2
10
00
1
00
0
2
10
00
Ac
a
c
a
c
a
c
a
c
c
ac
a
])co s h ()1)([sinh(])sinh(
)1(
)cosh([ 1
00
2
10
00
1
00
0
2
10
00
Ac
a
c
a
c
a
c
a
c
c
ac
a
2
0
12
10
01
2
0
2)1(
)1(
1 1
c
a
c
a
c
A
c
a
0
2
10
21
2
0
1
2
10
1
)1(
1])1[(
)1(
1
c
A
c
ac
a
c
a
0
1
2
0
1
2
10
ˆ
ˆ]ˆ)1[(
ˆ)1(
1 1
c
A
c
a
c
a (16)
2
tc
A
yE yy ])sinh()1)([cosh( 1
001
2
00
0
A
c
a
c
a
c
a
2])sinh(
)1(
)cosh([
1
00
0
2
10
00
Ac
a
c
c
ac
a
0
00
2
10
2
00
2
10
2
)cosh(
)1(
1)cosh()1(
c
A
c
a
c
ac
a
c
a
])[sinh(21
0012
AA
c
a
]
)1(
1])1([
)1(
1)[cosh(
0
2
10
2
2
10
21
2
0
002
c
A
c
ac
a
c
ac
a
])[sinh(21
0012
AA
c
a
]ˆ
ˆ)]1(ˆ[
ˆ)1(
1)[cosh(
02
10
21
2
0
002
c
A
c
a
c
ac
a
]ˆ
ˆ
ˆ
ˆ)[sinh(
21
0012
AA
c
a (17)
3
tc
A
zE zz ])sinh()1)([cosh( 1
001
2
00
0
A
c
a
c
a
c
a
3])sinh(
)1(
)cosh([
1
00
0
2
10
00
Ac
a
c
c
ac
a
0
00
2
10
3
00
2
10
3
)cosh(
)1(
1)cosh()1(
c
A
c
a
c
ac
a
c
a
])[sinh(31
0013
AA
c
a
]
)1(
1])1([
)1(
1)[cosh(
0
2
10
2
2
10
31
2
0
003
c
A
c
ac
a
c
ac
a
])[sinh(31
0013
AA
c
a
]ˆ
ˆ)]1(ˆ[
ˆ)1(
1)[cosh(
02
10
31
2
0
003
c
A
c
a
c
ac
a
]ˆ
ˆ
ˆ
ˆ)[sinh(
31
0013
AA
c
a (18)
32
23
AA
z
A
y
AB yzx 32 ˆ
ˆ
ˆ
ˆ23
AA (19)
x
AA
x
A
z
AB xzxy
3
3
3
])co s h ()1)([sinh( 1
001
2
00
0
A
c
a
c
a
c
a
3])cosh(
)1(
)sinh([
1
00
0
2
10
00
Ac
a
c
c
ac
a
])[cosh(13
0031
AA
c
a
]
)1(
1])1([
)1(
1)[sinh(
0
2
10
2
2
10
31
2
0
003
c
A
c
ac
a
c
ac
a
]ˆ
ˆ
ˆ
ˆ)[cosh(
13
0031
AA
c
a
]ˆ
ˆ)]1(ˆ[
ˆ)1(
1)[sinh(
02
10
31
2
0
003
c
A
c
a
c
ac
a (20)
2
2
xxy
z
A
x
A
y
A
x
AB
3])cosh(
)1(
)sinh([
1
00
0
2
10
00
Ac
a
c
c
ac
a
2
])cosh()1)([sinh( 1
001
2
00
0
A
c
a
c
a
c
a
])[cosh(21
0012
AA
c
a
]
)1(
1])1([
)1(
1)[sinh(
0
2
10
2
2
10
21
2
0
002
c
A
c
ac
a
c
ac
a
]ˆ
ˆ
ˆ
ˆ)[cosh(
21
0012
AA
c
a
]ˆ
ˆ)]1(ˆ[
ˆ)1(
1)[sinh(
02
10
21
2
0
002
c
A
c
a
c
ac
a (21)
Hence, we can define the electro-magnetic field ),( BE
in Rindler spacetime.
0
2
10
2
2
10
2
10 )1(
1})1({
)1(
1
c
A
c
ac
a
c
aE
02
10
ˆ
2
10
ˆ
ˆ)}1(ˆ{
)1(
1
c
A
c
a
c
a
AABˆ
ˆ
In this time, ),,(321
, ),,( 321 AAAA
)ˆ
,ˆ
,ˆ
(321ˆ
, )ˆ,ˆ,ˆ(ˆ
321 AAAA
(22)
We obtain the transformation of the electro-magnetic field.
0
2
10
2
2
10
1
2
10
1
)1(
1})1({
)1(
1
c
A
c
ac
a
c
aE x
1
1
02
10
1
2
10
ˆ
ˆ)}1(ˆ{
ˆ)1(
1
E
c
A
c
a
c
a
,
)sinh()cosh(0
00
032
c
aB
c
aEE y
,
)sinh()cosh(0
00
023
c
aB
c
aEE z
1BBx ,
)sinh()cosh(0
00
032
c
aE
c
aBBy
)sinh()cosh(0
00
023
c
aE
c
aBBz
(23)
Hence,
1EE x , 1
BBx ,
3
3
2
2
B
E
B
E
H
B
E
B
E
z
z
y
y
)cosh(00)sinh(
0)cosh()sinh(0
0)sinh()cosh(0
)sinh(00)cosh(
00
00
00
00
00
00
00
00
c
a
c
ac
a
c
ac
a
c
ac
a
c
a
H
(24)
The inverse-transformation of the electro-magnetic field is
xEE 1 , xBB 1
z
z
y
y
B
E
B
E
H
B
E
B
E
1
3
3
2
2
)cosh(00)sinh(
0)cosh()sinh(0
0)sinh()cosh(0
)sinh(00)cosh(
00
00
00
00
00
00
00
00
1
c
a
c
ac
a
c
ac
a
c
ac
a
c
a
H
(25)
xEE 1 , xBB 1
)sinh()cosh(0
00
02
c
aB
c
aEE zy
,
)sinh()cosh(0
00
02
c
aE
c
aBB zy
)sinh()cosh(0
00
03
c
aB
c
aEE yz
)sinh()cosh(0
00
03
c
aE
c
aBB yz
(26)
3. Electro-magnetic Field Equation(Maxwell Equation) in the Rindler space-time
Maxwell equation is
4 E
(27-i)
jctc
EB
4
(27-ii)
0 B
(27-iii)
tc
BE
(27-iv)
1. 4 E
1EE x ,
)sinh()cosh(0
00
032
c
aB
c
aEE y
,
)sinh()cosh(0
00
023
c
aB
c
aEE z
z
E
y
E
x
E zyx
4
1])cosh(
)1(
)sinh([
1
00
0
2
10
00
Ec
a
c
c
ac
a
)]sinh()cosh([0
00
0
2 32
c
aB
c
aE
)]sinh()cosh([0
00
0
3 23
c
aB
c
aE
]
)1(
1)[sinh())(cosh(
0
2
10
32
0000123
c
E
c
a
BB
c
aE
c
a
]ˆˆˆ
)[sinh())(cosh(032
00ˆ
00123
c
EBB
c
aE
c
a (28)
2. jctc
EB
4
1BBx
)sinh()cosh(0
00
032
c
aE
c
aBBy
)sinh()cosh(0
00
023
c
aE
c
aBBz
X-component)z
B
y
B yz
2
)]sinh()cosh([
00
00
23
c
aE
c
aB
3
)]sinh()cosh([
00
00
32
c
aE
c
aB
])[sinh(])[cosh(32
00
32
003223
EE
c
aBB
c
a
xx jctc
E 4
xjcE
c
a
c
c
ac
a
4])sinh(
)1(
)cosh([ 11
00
0
2
10
00
Hence,
xjc
4
]
)1(
1)[cosh())(sinh(
0
2
10
32
00
00
123
c
E
c
a
BB
c
aE
c
a
]ˆˆˆ
)[cosh())(sinh(032
00
ˆ
00
123
c
EBB
c
aE
c
a (29)
Y-component)x
B
z
B zx
3
1
B
])cosh(
)1(
)sinh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
023
c
aE
c
aB
yy jctc
E 4
])sinh(
)1(
)cosh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
032
c
aB
c
aE
yjc
4
01
2
02
0
1
2
013
2
3
31
)1(
1
)1(
14
c
E
c
aB
c
a
c
a
BBj
c y
01
2
02
10
11
2
0
1
2
0
31
2
0
2
31
)1(
1)}1({
)1(
1)}1({
)1(
1
c
E
c
ac
aB
c
ac
aB
c
a
02
10
11
2
0
1
2
0
31
2
0ˆ
)}1({ˆ
)1(
1)}1({
ˆ)1(
1 2
31
c
E
c
aB
c
ac
aB
c
a
(30)
Z-component)y
B
x
Bxy
])cosh(
)1(
)sinh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
032
c
aE
c
aB
2
1
B
zz jctc
E 4
])sinh(
)1(
)cosh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
023
c
aB
c
aE
zjc
4
01
2
02
0
1
2
021
3
2
12
)1(
1
)1(
14
c
E
c
aB
c
a
c
a
BBj
c z
01
2
02
10
21
2
0
1
2
0
11
2
0
3
12
)1(
1)}1({
)1(
1)}1({
)1(
1
c
E
c
ac
aB
c
ac
aB
c
a
02
10
21
2
0
1
2
0
11
2
0ˆ
)}1({ˆ
)1(
1)}1({
ˆ)1(
1 3
12
c
E
c
aB
c
ac
aB
c
a
(31)
3. 0 B
z
B
y
B
x
BB zyx
1])cosh(
)1(
)sinh([
1
00
0
2
10
00
Bc
a
c
c
ac
a
)]sinh()cosh([0
00
0
2 32
c
aE
c
aB
)]sinh()cosh([0
00
0
3 23
c
aE
c
aB
0]
)1(
1)()[sinh())(cosh(
01
2
023
00
00
132
c
B
c
a
EE
c
aB
c
a
0]ˆ
)ˆˆ
()[sinh())(cosh(023
00
ˆ
00
132
c
BEE
c
aB
c
a
(32)
4.tc
BE
1EE x ,
)sinh()cosh(0
00
032
c
aB
c
aEE y
,
)sinh()cosh(0
00
023
c
aB
c
aEE z
X-component)z
E
y
E yz
2
)]sinh()cosh([
00
00
23
c
aB
c
aE
3
)]sinh()cosh([
00
00
32
c
aB
c
aE
])[sinh(])[cosh(32
00
32
003223
BB
c
aEE
c
a
tc
Bx
1])sinh(
)1(
)cosh([
1
00
0
2
10
00
Bc
a
c
c
ac
a
Hence, ))(sinh(0
0
B
c
a ]
)1(
1)) [ (c o s h (
0
2
10
32
00
123
c
B
c
a
EE
c
a
))(sinh( ˆ
00
B
c
a 0]
ˆ)
ˆˆ)[(cosh(
032
00
123
c
BEE
c
a (33)
Y-component)x
E
z
E zx
3
1
E
])cosh(
)1(
)sinh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
023
c
aB
c
aE
tc
B y
])sinh(
)1(
)cosh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
032
c
aE
c
aB
01
2
02
0
1
2
013
2
3
31
)1(
1
)1(
1
c
B
c
aE
c
a
c
a
EE
01
2
02
10
11
2
0
1
2
0
31
2
0
2
31
)1(
1)}1({
)1(
1)}1({
)1(
1
c
B
c
ac
aE
c
ac
aE
c
a
02
10
11
2
0
1
2
0
31
2
0ˆ
)}1({ˆ
)1(
1)}1({
ˆ)1(
1 2
31
c
B
c
aE
c
ac
aE
c
a0
(34)
Z-component)y
E
x
Exy
])cosh(
)1(
)sinh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
032
c
aB
c
aE
2
1
E
tc
Bz
])sinh(
)1(
)cosh([
1
00
0
2
10
00
c
a
c
c
ac
a
)]sinh()cosh([0
00
023
c
aE
c
aB
01
2
02
0
1
2
021
3
2
12
)1(
1
)1(
1
c
B
c
aE
c
a
c
a
EE
01
2
02
10
21
2
0
1
2
0
11
2
0
3
12
)1(
1)}1({
)1(
1)}1({
)1(
1
c
B
c
ac
aE
c
ac
aE
c
a
02
10
21
2
0
1
2
0
11
2
0ˆ
)}1({ˆ
)1(
1)}1({
ˆ)1(
1 3
12
c
B
c
aE
c
ac
aE
c
a0
(35)
Therefore, we obtain the electro-magnetic field equation by Eq (28)-Eq(35) in Rindler spacetime .
)1(42
10
ˆc
aEE
(36-i)
j
cc
E
c
ac
aB
c
a
4
)1(
1)}1({
)1(
10
2
10
2
10
2
10
j
cc
E
c
aB
c
a
4ˆ
)}1({
)1(
102
10
ˆ
2
10
(36-ii)
0ˆ BB
(36-iii)
0
2
10
2
10
2
10 )1(
1)}1({
)1(
1
c
B
c
ac
aE
c
a
02
10
ˆ
2
10
ˆ)}1({
)1(
1
c
B
c
aE
c
a
(36-iv)
),,( 321 EEEE
, ),,( 321 BBBB
,
),,(321
, )
ˆ,
ˆ,
ˆ(
321ˆ
Hence, the transformation of 4-vector
d
dxjc 0),(
is
)sinh()cosh()1(0
00
0
2
10
1
c
a
c
j
c
a
c
a
)sinh()1()cosh(0
01
2
00
01
c
a
c
ac
c
ajj x
, 32 , jjjj zy
In this time, 4-vector
d
djc 0),(
(37)
4. Electro-magnetic wave equation in Rindler space-time
The electro-magnetic wave function is
sin0xx EE , sin0yy EE , sin0zz EE
sin0xx BB , sin0yy BB , sin0zz BB
xEE 1 , xBB 1
'sin01 XEE
, 'sin01 XBB
)sinh()cosh(0
00
02
c
aB
c
aEE zy
,
)sinh()'sin()cosh()'sin(0
00
00
0 c
aB
c
aE zy
)sinh()cosh(0
00
02
c
aE
c
aBB zy
)sinh()'sin()cosh()'sin(0
00
00
0 c
aE
c
aB zy
)sinh()cosh(0
00
03
c
aB
c
aEE yz
)sinh()'sin()cosh()'sin(0
00
00
0 c
aB
c
aE yz
)sinh()cosh(0
00
03
c
aE
c
aBB yz
)sinh()'sin()cosh()'sin(0
00
00
0 c
aE
c
aB yz
(38)
)(c
zn
c
ym
c
xlt ,
)ˆ'
ˆ'
ˆ'ˆ(''
3210
cn
cm
cl
In this time,
)s inh ()(0
01
0
2
c
a
a
cct
,
0
2001
0
2
)cosh()(a
c
c
a
a
cx
32, zy
)(tanh
0
2
1
0
0
a
cx
ct
a
c
,
0
2222
0
21 )(
a
ctc
a
cx
tc
t
c
taca
c
ctaca
cxa
ctac
aaaa
20
002
01
002
0
01
0
0
0
)(1
1lim/)(tanhlim/)(tanhlimlim0000
02
2
02
002
22
02
2
02
0
1
0/)1)1((l im/)1)1((l iml im
000
axc
aca
c
tax
c
ac
aaa
xaxc
ac
a
02
02
0/)(lim
0
Hence,
tddc
ad
aaaaa
0
0
0
0
01
2
0
0
0
0
0
0 00000
limlim)1(limˆlimˆlim
xaa
1
0
1
0 00
l imˆl im , 3322 ˆ,ˆ zy
Therefore, electro-magnetic wave function is
)'''(')ˆ'
ˆ'
ˆ'ˆ('lim'lim
3210
00 00 c
zn
c
ym
c
xlt
cn
cm
cl
aa
)(c
zn
c
ym
c
xlt
nnmmll ',',','
1222 nml , 1''' 222 nml (39)
Hence,
1])(
)1(
11[
22
021
2
02
E
c
ac
0])ˆ
(1[ 1
2
ˆ2
02
Ec
1])(
)1(
11[
22
021
2
02
B
c
ac
0])ˆ
(1[ 1
2
ˆ2
02
Bc
yE
c
ac])(
)1(
11[
22
021
2
02
0])ˆ
(1[
2
ˆ2
02
yEc
yB
c
ac])(
)1(
11[
22
021
2
02
0])ˆ
(1[
2
ˆ2
02
yBc
zE
c
ac])(
)1(
11[
22
021
2
02
0])ˆ
(1[
2
ˆ2
02
zEc
zB
c
ac])(
)1(
11[
22
021
2
02
0])ˆ
(1[
2
ˆ2
02
zBc
(40)
The electro-magnetic wave equation is in vacuum
)}1({)1( 1
2
01
2
0 c
aE
c
a
)}1({)1( 1
2
01
2
0 c
aE
c
a
)}1({)1( 1
2
01
2
0 c
aE
c
a
Ec
a
c
a )1()1( 1
2
01
2
0
Ec
a
c
a )1()1( 1
2
01
2
0
Ec
a
c
a )1()1( 1
2
01
2
0
Ec
a 21
2
0 )1(
Ec
a
c
a )1()1( 1
2
01
2
0 E
c
a 21
2
0 )1(
Ec
a
c
a
c
aE
c
a )]1()1([)1(])1([ 1
2
01
2
01
2
01
2
0
])([)1(221
2
0 EE
c
a
)}]1({[1
2
10
0 c
aB
c
E
c
2
02)(
1
,
In this time, )0,0,()1(2
01
2
0
c
a
c
a
(41)
Hence,
Ecc
aE
c
a 2
02
1
2
01
2
0 )(1
)}1({)1(
Ec
a
c
a
c
aE
c
a )]1()1([)1(])1([ 1
2
01
2
01
2
01
2
0
Ec
EEc
a 2
02
221
2
0 )(1
])([)1(
),,()0,0,( 3211 4
2
0
4
2
0
EEE
c
aE
c
a
E
c
acc
a ])(
)1(
11[)1(
22
021
2
02
21
2
0
),,0( 324
2
0
EE
c
a
E
c
acc
a ])(
)1(
11[)1(
22
021
20
2
21
20
),,0( 324
2
0
EE
c
a
Ecc
a ])
ˆ(1[)1(
2
ˆ2
02
21
2
0
0
(42)
Hence, the magnetic wave equation is in vacuum
Bcc
aB
c
a 2
02
1
2
01
2
0 )(1
)}1({)1(
Bc
a
c
a
c
aB
c
a )]1()1([)1(])1([ 1
2
01
2
01
2
01
2
0
Bc
BBc
a 2
02
221
2
0 )(1
])([)1(
),,0( 324
2
0
BB
c
a
B
c
acc
a ])(
)1(
11[)1(
22
021
2
02
21
2
0
),,0( 324
2
0
BB
c
a
Bcc
a ])
ˆ(1[)1(
2
ˆ2
02
21
2
0
0
(43)
The electromagnetic wave function, Eq(38),Eq(39) satisfy the electromagnetic wave equation,
Eq(42),Eq(43).
5. Conclusion
We find the electro-magnetic field transformation and the electro-magnetic equation in uniformly
accelerated frame.
Generally, the coordinate transformation of accelerated frame is
(I) )sinh()(0
01
0
2
c
a
a
cct
0
2001
0
2
)cosh()(a
c
c
a
a
cx
,
32, zy (44)
(II) )sinh()exp(0
01
2
0
0
2
c
a
c
a
a
cct
0
2001
2
0
0
2
)cosh()exp(a
c
c
a
c
a
a
cx
,
32, zy (45)
Hence, this article say the accelerated frame is Rindler coordinate (I) that can treat electro-magnetic field
equation.
Reference
[1]S.Weinberg,Gravitation and Cosmology(John wiley & Sons,Inc,1972)
[2]W.Rindler, Am.J.Phys.34.1174(1966)
[3]P.Bergman,Introduction to the Theory of Relativity(Dover Pub. Co.,Inc., New York,1976),Chapter V
[4]C.Misner, K,Thorne and J. Wheeler, Gravitation(W.H.Freedman & Co.,1973)
[5]S.Hawking and G. Ellis,The Large Scale Structure of Space-Time(Cam-bridge University Press,1973)
[6]R.Adler,M.Bazin and M.Schiffer,Introduction to General Relativity(McGraw-Hill,Inc.,1965)
[7]A.Miller, Albert Einstein’s Special Theory of Relativity(Addison-Wesley Publishing Co., Inc., 1981)
[8]W.Rindler, Special Relativity(2nd ed., Oliver and Boyd, Edinburg,1966)
[9]Massimo Pauri, Michele Vallisner, "Marzke-Wheeler coordinates for accelerated observers in special
relativity":Arxiv:gr-qc/0006095(2000)
[10]A. Einstein, “ Zur Elektrodynamik bewegter K¨orper”, Annalen der Physik. 17:891(1905)