How generalized Minkowski four-force

leads to scalar-tensor gravity

György SzondyLogic, Relativity and Beyond

2nd International ConferenceAugust 9-13 2015, Budapest

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Theories of Gravitation

http://en.wikipedia.org/wiki/History_of_gravitational_theory

~50 alternatives and

extensions

A lot of results in addition to GR2015.08.09.

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• Special Relativity and Minkowski four-force

• Scalar-Tensor Gravity

• Quantum effects (causing this Scalar field)

Topics

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Minkowski (four)-force

P2

𝑭⊥𝒑↔𝐹 𝑖𝑝𝑖=0

𝑚0=√𝐸2−𝑝2

Special case

proper time m0

𝑐=1

REST MASS is INVARIANT

t

x

𝑭=𝑚𝒂=𝑚𝑑𝒖𝑑𝜏

=𝑑𝒑𝑑𝜏

FF

F

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orthogonal

𝑎𝑖𝑢𝑖=0↔𝒂⊥𝒖𝒂=

𝑑𝒖𝑑𝜏

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Generalized minkowski-force

m1

m2

F

Károly Novobátzky in the ´50s

General 4-force might change rest mass

t

x

𝐹 𝑖𝑝𝑖≠0

𝑭=𝑑𝒑𝑑𝜏

=𝑑 (𝑚𝒖)𝑑𝜏

=𝑚 𝑑𝒖𝑑𝜏

+𝒖 𝑑𝑚𝑑𝜏

𝑭=𝐹⊥+𝐹∥

𝑚 (𝑥 )=𝑚0+𝜙 (𝒙 )

𝑭 ∥

𝑭⊥

Similar to Higgs-field

𝜙𝑝 (𝒙 )≠𝜙𝑒 (𝒙 )

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6

𝐸=𝑚0𝑐

2

√1− 𝑣2𝑐2

Static gravitational field

m1m2

FE = const

(Brans & Dicke 1961)

𝐸=𝑚0𝑐

2√𝑔00

√1− 𝑣2𝑐2

𝑚 (𝑥 )=𝑚0√𝑔00

Scalar-tensor gravity

t

x

0=𝛿∫𝑚 √(𝑔𝑖𝑗𝑢𝑖𝑢 𝑗 )𝑑𝑠𝑑 (𝑚𝑢𝑖)𝑑𝑠

−12𝑚𝑔 𝑗𝑘 ,𝑖𝑢

𝑗𝑢𝑘−𝑚,𝑖=0

LRB15 - Szondy: Scalar-Tensor Gravity

𝐸=𝑚 (𝑥 )𝑐2

√1− 𝑣2𝑐2

𝑚(𝑥)=𝑚0 𝑓 (𝑥)

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7

𝐸=𝑚0𝑐

2

√1− 𝑣2𝑐2

Static gravitational field - remarks

(Brans & Dicke 1961)

𝐸=𝑚0𝑐

2√𝑔00

√1− 𝑣2𝑐2

𝑚 (𝑥 )=𝑚0√𝑔00

Exact transformation between STG & GR

0=𝛿∫𝑚 √(𝑔𝑖𝑗𝑢𝑖𝑢 𝑗 )𝑑𝑠𝑑 (𝑚𝑢𝑖)𝑑𝑠

−12𝑚𝑔 𝑗𝑘 ,𝑖𝑢

𝑗𝑢𝑘−𝑚,𝑖=0

LRB15 - Szondy: Scalar-Tensor Gravity

𝐸=𝑚 (𝑥 )𝑐2

√1− 𝑣2𝑐2

2015.08.09.

1. Light has no rest- mass same as GR

2. represents the Newtonian gravitational force.

3. Mercury perihelion advance can be calculated from the curvature effects

4. B&D in they original paper insisted on the constancy of rest mass, and used conform transformation to transform their result back to GR and tried to vary the gravitational constant

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Conformal transformation applied by Brans & Dicke

Scalar-Tensor theory:

Conformal   transformation : 𝑔𝑖𝑗= 𝑓2g𝑖𝑗

𝑑𝑠2= 𝑓 2d 𝑠2 ,        𝑢𝑖= 𝑓 − 1𝑢𝑖

Getting back General Relativistic equation of motion:

𝑚 (𝑥 )= 𝑓 (𝑥 )𝑚0

𝑚=𝑐𝑜𝑛𝑠𝑡

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𝑑 (𝑚𝑢𝑖)𝑑𝑠

−12𝑚𝑔 𝑗𝑘 ,𝑖𝑢

𝑗𝑢𝑘−𝑚,𝑖=0

We use the results of GR in STG

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How to transform Schwarzscild solution to Scalar-Tensor gravity

Schwarschild solution uses standard coordinates

Isotropic coordinates

)sin()()( 2222222 ddrdrrBdtrAds

)sin)((')(' 22222222 dddBdtAds

2)4

1(

srr

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𝑔𝑖𝑗= 𝑓2g𝑖𝑗

Coordinatetransformation

)sin(4

1

41

41

222222

4

2

2

2

ddd

rdt

r

r

ds g

g

g

Schwarzschild metric:

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Understanding Schwarzscild metric

0.5 1 1.5 2rg1

2

3

4

5

rrg𝑟 /𝑟 𝑠

𝜌 /𝑟 𝑠Event horizon

stan

dard

isotropic

2)4

1(

srr

… in isotropic coordinates

-5-2.5

0

2.5

5

-5

-2.5

0

2.55

-4

-2

0

2

4

-5-2.5

0

2.5

5

-5

-2.5

0

2.55

Einstein-Rosen bridge

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Scalar field - Coordinate independent?g00 in isotropic coordinates

(gravitational potential)

0.5 1 1.5 2rg0.2

0.4

0.6

0.8

1

g00

0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

1.2

R

Rest mass depends on Ricci scalar

𝒈𝟎𝟎=𝟏−𝟒√ 𝟐𝟑 (−𝑹)

𝒎 (𝒙 )=𝒎𝟎√𝒈𝟎𝟎

𝒎 (𝒙 )=𝒎𝟎√𝟏−𝟒√ 𝟐𝟑 (−𝑹 (𝒙))

√−𝑅𝑔00

𝑟 /𝑟 𝑠𝑟 /𝑟 𝑠

Ricci scalar in isotropic coords (after conformal transformation)

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How can curvature effect Rest-Mass?Particle model from caracteristic lengths

Different possible particles

0.5 1 1.5 2m

2

4

6

r

2

2

c

Gmrs

mc

hkkr ComptonQ

22

Schwarzschild-radius

Quantum-radius

Rest-mass and size depends on ‚k’ quantum number

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How can background-curvature effect Rest-Mass?

Particle model from caracteristic lengths

is different for p+ and e– 𝑚=𝑚0𝑠 (𝑟𝑝 ,𝑅𝑈)

𝑚(𝑞) /𝑚0

𝑞=𝑟𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 /𝑅𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒

𝑞=𝑟𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 /𝑅𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒

Background curvature

Modified Schwarzschild-radius

𝑟 𝑠 (𝑞 )=1−√1−16𝐺𝑚𝑞𝑐2

4𝑞

G might be different for different particles?2015.08.09.

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Conclusion

• General force field can change rest mass• In scalar-tensor gravity Newtonian gravity and

curvature-effects are seperated• Background curvature changes rest mass• Rest mass change are due to quantum effects• Gravitational constant is different for different

elementary particles

There is a lot to do2015.08.09.

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References1. Gy. Szondy, A Pure Geometric Approach to Derive Quantum Gravity from

General Relativity; viXra:1312.0222 (2013)2. Gy. David, Lecture 2011.04.27, https://www.youtube.com (Novobatzky)3. C. Brans and R. H. Dicke, Mach's Principle and a Relativistic Theory of

Gravitation, Phys. Rev. D 124 925-935 1961 4. Gy. Szondy, Linear Relativity as a Result of Unit Transformation,

arXiv:physics/0109038 (2001) 5. Gy. Szondy, Mathematical Equivalency of the Ether Based Gavitation

Theory of Janossy and General Relativity, arXiv:gr-qc/0310108

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Comments & Questions

2015.08.09.

Thank you for your attention!