Download Non-localizability of Electric Coupling and Gravitational Binding of

Non-Localizability of Electric
Coupling and Gravitational
Binding of Charged Objects
Matthew Corne
Eastern Gravity Meeting 11
May 12-13, 2008
Purpose of Investigation
To determine whether or not static,
spherically symmetric extended objects
comprised of a charged perfect fluid possess
non-localizable contributions to their total
gravitational mass
Does electrostatic coupling energy contribute
to gravitational binding energy?
Historically, known that purely
electromagnetic classical models unstable
(hence Poincaré stress)
Conjectured (and assumed) instead,
electrostatic coupling energy contribution to
gravitational binding
E.g., Lorentz-Abraham model of electron,
certain models of spherical stars
• Know in general, total mass-energy not
localizable
• Gravitational energy-momentum not
localizable
• Exceptions: static, spherically symmetric
objects composed of neutral perfect fluid,
gravitational waves up to wavelength
Einstein’s Equations
G  8T
Einstein Tensor
Components
Energy-Momentum Tensor
Components
Energy-momentum tensor
for a charged perfect fluid:
Proper pressure
Metric tensor
components
1
1


T  (  p)u u  g p 
(F F  F F g )
4
4
Proper
density of the
fluid
Components
of fourvelocity
Electromagnetic field
tensor components
Due to symmetries, quantities radially
dependent:
• Matter density
• Charge density
• Pressure
• Electric field
Line element describing a static,
spherically symmetric spacetime:
ds  e
2
2(r)
g00  e
dt  e
2
dr  r (d  sin d )
2
2
2
2
2(r)
grr  e
2(r)
g  r
2
g  r sin 
2
2(r)
2
Diagonal
components of
metric tensor
2
Electric field determined by
Maxwell’s Equations:
  E  4
Other three are satisfied trivially
q(r)
F  E(r)  2
r
Radial electric field as
measured in the orthonormal
frame
0ˆ rˆ
q(r) 
r
 (r)4r e
0
2 (r )
dr
Charge inside sphere of
radius r
4 r 2e (r)dr
Element of proper
volume after
integration over
angles
Getting to the energy:
1 2(r) d
2(r)
G00  2 e
[r(1 e
)]
r
dr
T00  e
E
[(r)  ]
8
2(r)
• 00-component of Einstein’s tensor related to
energy density/00-component of energymomentum tensor (localizable contribution of
matter)
Boundary of object at r = R:
(r)  (r)  0
rR
Outside of object:
q(R)  Q 
R
 (r)4r e
0
2  (r )

dr
Charge Q
constant
Consider expressions
2
Q
m˜ (r)  
M
2r

e
2(r )
Constant M
interpreted as total
mass (Keplerian
motion)
1
1


2
˜ (r)
2m
2M Q
1
1
 2
r
r
r
Substitution into line element - get Reissner-Nordstrøm (RN)

Reissner-Nordstrøm (RN)
Line Element
2
2
2M
Q
2M
Q
ds2  (1
 2 )dt 2  (1
 2 )1 dr 2  r 2 (d 2  sin 2 d 2 )
r
r
r
r
• Describes spherically symmetric, static
gravitational field with radial electric field present
• Applies to exterior solution of extended object here
Integrating with respect to r :
m(r) 
r
r
 4r (r)dr  
2
0
0
q(r)  (r)4 r e
r
2  (r )
• Not total mass-energy inside radius r
• Method absent for defining total mass-energy with
surrounding material (Keplerian orbits indeterminable)
dr
Can interpret this as total gravitational mass
entering external RN solution:
m(R)  M 
R
(r)
2
(r)
e
4

r

(r)e
dr 

R
0
0
Contains contribution of
gravitational binding
energy

q(r) (r)4 r 2e (r )
dr
r
Contains contribution of
electrostatic coupling
energy
What is gravitational binding
energy?
• Negative of gravitational potential energy
• Holds together all components of object
m(r)  m0 (r)  U(r)  (r)
Total mass-energy
contributing to
gravitational field
Rest mass-energy
Internal energy,
in our case
electrostatic
Gravitational
potential
energy
Re-write equation in terms of binding energy:
m(r)  m0 (r)  E GravBinding  U(r)
 E GravBinding 
r
 (1 e
 (r )
)(r)4 r e
2 ( r )
0

r

0
m
r
4 r dr
2
• Gravity binds only localizable part of mass-energy (perfect fluid)
• Electrostatic coupling energy does not appear in gravitational
binding energy
dr
What we notice:
• Vanishing electrostatic coupling energy (pure
EM mass) at every point inside of object
• Charge: integral of density
• Contribution to total mass only outside of
object - not inside
Implications
• No purely electromagnetic mass
• Need gravitational binding energy
• Results independent of thermodynamic
considerations (i.e., pressure absent from
equations)
Important: Effects not coordinate dependent
r - given by the expression
r
A
4
A - proper area of an appropriate sphere

t - appropriate slicing by proper spaces of comoving observers
Schwarzschild coordinates - natural choice
i 1
ti1
Lapse
i
ti
i
t
Unit
Normal
i
Nn
N
Spacelike
hypersurfaces
i
Shift Vector
Further Results
• Removing charge - same result as a neutral
spherical star with external Schwarzschild solution
• Introducing rotation (angular momentum) such as
in Kerr-Newman - still have charge
• Electron (or any such particle) cannot be modeled
as a field that holds itself together
• Vacuum fluctuations inadequate
Conclusions
• Existence of non-localizable contributions to
total mass-energy
• Must always check configuration - no
assumptions
• Cannot model electrons this way!
• Cannot model stars this way!
Acknowledgments
NCSU Department of Mathematics
REG program (supported by NSF
MTCP grant)
References
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