Deutsche Physikalische Gesellschaft

*Frühjahrstagung **Gravitation
und Relativitätstheorie*

Jena, 25. Februar - 1. März 2013

** **

**GRT - well proven and also incomplete? **

** **

J. Brandes^{}

26.2.2013; last update: 7.8.2013

**Content:**

Contradictory results of total energy

Rewording of equivalence principle

Modification of measuring instruments
within gravitational fields

More about modification of measuring
instruments within gravitational fields

Many thanks for having a talk together with so many professionals.

Incompleteness means some imperfections even
contradictions of a theory but which can be overcome by some extension.

The elementary question: What is the total
energy *E _{G}* of a particle resting in the

(1) _{ }

Derived from the formulas of free, radial fall
one gets [1], [3], [2] 312:

(2) _{}

This is at least qualitatively correct since
removing the particle from the gravitational field needs energy. Doing this the
total energy_{} of the particle becomes _{}and therefore within the gravitational field _{} has to be lower.

On the other side, there is the equivalence
principle. A particle resting in its local inertial system (i. e. the free
falling particle) has a total energy equal to its rest mass:

(3)
_{}

Formula (2) and (3) contradict each other.
Certainly, they belong to different reference systems with one of them being
accelerated, in fact. But: At time point _{} the free falling
particle is also a resting one within the *r,t*-reference system since its
velocity _{}. Only its acceleration _{}. Special theory of relativity is applicable and therefore
the free falling particle at _{} and an always resting
particle at the same place possess identical total energies (3). **Formula (2)
and (3) contradict each other.**

On account
of the qualitative argument above, formula (2) is the correct one. One can see
it again by series expansion of (2):

(4) _{}

The second
term describes the negative gravitational energy. Approximately formula (2) becomes
the rest mass minus Newtonian gravitational energy. Therefore formula (2) meets
the Newtonian limit but formula (3) does not.

“For **measurements** within
gravitational fields the **measuring** results within local inertial systems
are predicted by special relativity.”

Concerning our application this
means: The measurement of *E _{G}* with measuring instruments resting in the
gravitational field yields

Let us
consider possible modifications of measuring instruments during measurement of *E _{G
}*. Let us choose some intellectually simple measuring procedure.
Transfer an antiparticle to the resting particle and perform the measurement of
annihilation frequency of the two resulting photons. One gets:

(5) _{}

_{} : Annihilation
frequency, measured by a clock resting in the gravitational field

_{}: proper time of a clock resting in the gravitational field (_{}-clock, standard clock)

On the
other side, it is derivable from SM:

(6) _{}

With
words. Time passing of _{} of a *t*-clock
outside the gravitational field means time passing of a _{}-clock less by a factor

(7) _{}

Therefore,
the measured frequencies are:

(8) _{}

_{}: ‘real’ frequency, slowing down of clocks by gravitational
fields eliminated.

_{}: ‘real’ energy

(8)
inserted into (5) yields

_{}, identical with formula (2).

With
words: Taking into account the modification of measuring instruments by gravitational
field – in this case the slowing down of clocks – makes it possible to derive
the total energy of a resting particle by use of the equivalence principle. **The contradiction of (2) and (3) is solved.**

What
does formula (6) mean?

(6) _{}

Within
GRT this is interpreted as: ‘time is curved’ or ‘time runs slower within the
gravitational than outside of it’. Lorentz interpretation of GRT (LI of GRT)
says: ‘standard clocks are slowed down inside the gravitational field’ [2].

The picture illustrates this
for measuring rods. Within gravitational fields one can argue - as GRT does:
measuring rods don’t change but space is curved or – as LI of GRT does - space
remains flat but measuring rods contract .

V.
Perlick wrote: „... [Die obige] Argumentation laeuft immer auf dieselbe
Behauptung hinaus:

„Das Problem für die Klassische Allgemeine Relativitätstheorie ist die Entscheidung,
welches Messergebniss die Energie eines im Gravitationsfeld ruhenden
Teilchens wiedergibt und welches umgerechnet werden muss. **Beides zugleich ist falsch.“
Da widerspreche ich entschieden: Beides ist richtig.** Es handelt sich eben
in einem Fall um die Energie in Bezug auf die eine Uhr, im anderen Fall in
Bezug auf die andere Uhr. Beide Uhren sind wohldefiniert und wohlverstanden,
und die Umrechnungsformel ist, wie Sie richtig schreiben, auch wohlbekannt. Von
einer 'Unvollstaendigkeit' der Theorie kann deshalb keine Rede sein. Eine Paradoxie
laege freilich vor, wenn die Situation gegenueber Vertauschung der beiden
Uhren symmetrisch waere. Das ist aber natuerlich nicht der Fall, da ja nur eine
der beiden Uhren eine Standarduhr ist.“

The main argument of V. Perlick, as far as I understand:

Formula (3) is correct since it is proved by standard clocks and formula (2) ist correct as well, since it is the same as formula (3) but using

_{}

and then
convert with formula (8) into the ‘correct’ standard clock time,

_{}and by use
of formula (5) insert _{} instead of _{} getting formula (2).

Everything depends on the clock you are measuring with.

What is
not correct? The measuring result is not the true result if clocks run slow or
fast. Following LI of GRT you can say that *t*-clocks measure the correct
time since standard clocks are slowed down in gravitational fields or following
GRT that instead standard clocks measure correct time and *t*-clocks do
not but you cannot have both of the statements together. Only formula (2) or
(3) is the correct formula.

Classical general theory of relativity knows two
formulas of total energy of a particle resting within gravitational fields
contradicting each other. This contradiction is resolved if one can assume that
measuring instruments become modified by gravitational fields. This is LI of
GRT.

Many
thanks to V. Perlick for our discussion.

**Literature**

[1] Schutz, B. (2003): *Gravity from the ground up.* Cambridge University Press, see p
232ff. Formula (2) is evaluated but with
restricted range of validity.

[2] Brandes, J.; Czerniawski, J. (2010): Spezielle und
Allgemeine Relativitätstheorie für Physiker und Philosophen – Einstein- und
Lorentz-Interpretation, Paradoxien, Raum und Zeit, Experimente. VRI,

4. erweiterte Auflage, 404 Seiten, 100 Abbildungen, ISBN 978-3-930879-08-3 Näheres (Preis,
Inhaltsverzeichnis etc.): http://www.buchhandel.de/
oder http://www.amazon.de/ Suchen mit
9783930879083

[3] Dragon, Norbert (2012): *Geometrie der Relativitätstheorie.* http://www.itp.uni-hannover.de/~dragon. pdf-file on homepage of the
author. There equation (6.24) with *dr/dtau
= 0* and *L=0* results in formula
(2) without restricted range of validity.