Why are forces weaker in a gravitational potential well?
Why do the forces grow weaker for an outside observer? Because if the outside observer sends in a light pulse 1 joule of energy to another observer inside the well, the receiver thinks he gets 1 + E joule. Gravitational acceleration adds to the energy. We then use the equivalence principle here: 1 + E joule should do the same work for the observer inside the potential well as 1 + E joule does for the observer outside.
From the point of view of the outside observer, the 1 joule he sent was able to do work for 1 + E joule. He thinks forces are weaker in the well.
Another way to show the weakening of forces is this: suppose that we have two electrons in a low gravitational potential on a neutron star. The repulsive force is a consequence of the increase in the electric field energy if we push the electrons closer to each other. But in a low gravitational potential, the field energy increases less when we move the electrons a distance ds. In this we used the assumptions that:
1) forces are a result of field energy;
2) field energy is worth less for an outside observer, if the field resides in a gravitational potential well.
Inertial mass for horizontal movement
Our discussion of the inertial mass of the electron revealed that it is by no means simple to determine the inertial mass of an object under a potential.
m -------------------->
\O
|
/\
=======================
x y
<------------------- |V|
We employ once again our model of a man standing on a surface. In this case, it is a surface of a planet and the force is gravitation.
The man lets the test mass m approach from far away, and fills an energy store at x with the energy |V| that the test mass gets in the approach. Then the man moves the test mass to y and uses an energy store |V| there to push the test mass far away again. The inertial mass of the test mass might be
m + |V|,
but we need to think more carefully how the field energy of the gravitational field moves when we move the test mass.
The field is weaker in a wide area between the test mass and the center of the planet. Everywhere else, the field is stronger.
The inertia in an electric field versus gravitational
The electric field has a positive energy density of E^2, where E is the field strength. The energy density of the gravitational field is negative. Suppose that we add a small field dE to a strong electric field E. The energy density change is
(E + dE)^2 - E^2
= 2 dE E + dE^2.
For the gravity field G, the change is
G^2 - (G + dG)^2
= -2 dG G - dG^2.
Let us then compare the fields of an electron close to a large positively charged sphere, and a test mass m close to a planet, such that their field strengths have equal numerical absolute values. The direction of the field of the electron is opposite to the field of the test mass. The change in the energy density then has the same sign in both cases far away from the electron or test mass, and is approximately equal
Close to the electron, the field has positive energy, while it has a negative energy close to the test mass.
Speed of a horizontal clock in gravitational potential
Suppose that the gravitational potential on the surface of a neutron star is -0.01 m for an object of mass m. We have set c = 1.
Let observer A be far away and observer B stand on the surface of the star.
A thinks that all forces between objects which are close to B have weakened by 1 %.
A
B 1 kg 1 kg
==================
neutron star
A and B want to compare the speed of their time. B puts two 1 kg weights (which were weighed far away and exported to the neutron star) horizontally at the distance of 1 meter from each other. Both weights will carry 1 coulomb of negative electric charge. B measures the end velocity of the weights as they push each other far. A uses a telescope and measures the velocity, too.
The electric force is 1 % weaker on the surface as measured by A. The kinetic energy will be 1 % less. Furthermore, the inertial mass is 1 % larger as measured by A, if our hypothesis from a previous section is valid.
E_kin = 1/2 m_inertial v^2
<=>
v^2 = 2 E_kin / m_inertial.
The velocity that A sees from his telescope is 1 % less than A would see if he himself would perform the same experiment in space. A concludes that the time of B has slowed down by 1 %. That 1 % is the redshift of light from B to A, according to general relativity.
What did we assume?
What did we assume?
1. We used the fact from general relativity that horizontal distances look the same for A and B.
2. We used the equivalence principle to deduce that forces are weaker in the potential well, or alternatively that field energy is worth less in a gravitational potential well.
3. We used a hypothesis of gravitational field energy distribution to determine the inertial mass in the potential well.
We did not need to assume that the geometry of spacetime is anything else than the flat Minkowski.
The Schwarzschild geometry and vertical distances
In the Schwartzschild geometry, if the time measured by an observer A on a neutron star has diminished by a factor b, then vertical distances measured by him have increased by a factor 1 / b. In this, we compare his measurements to the global Schwarzschild coordinates.
A spacetime element in the global coordinates is squeezed when A measures it, but the spacetime volume measured by A agrees with global coordinates.
Can we explain with the gravitational potential why the ruler of A is shortened when it is vertical, but not shortened when it is horizontal?
When we move the test mass horizontally, there is energy flow within the field but not to/from the test mass.
If we move the test mass vertically, there is energy flow also between the test mass and the field. The inertial mass for the test mass might be larger in the vertical direction.
Let the observer A throw the test mass horizontally so that it hits a wall at a 45 degree angle, and bounces vertically up. If the inertial mass of the test mass is bigger in the vertical direction, it will fly slower.
Any object, an electron in an atom or anything, will move slower to the vertical direction. That might be the reason why the ruler of A is shorter in the vertical direction.
We need to estimate the energy flow in a vertical movement.
Let us have a neutron star of radius 1. The potential V of the test mass m is
V = -k / r,
for some k.
V
<---- m ---->
==========
neutron star
In horizontal movement, if we move the test mass a distance 0.01, the negative potential energy V, in a sense, seems to cause positive energy |V| to flow a distance 0.01. The total energy displacement is
m * 0.01 + |V| * 0.01.
The inertial mass is m + |V| for horizontal movement.
^
|
m V
|
v
==========
neutron star
If we move the test mass vertically +0.01, we move|V| the same distance. Our test mass has stolen its extra inertial mass |V| from the field of the neutron star. When the test mass moves up 0.01, it returns 0.01 |V| of the inertial mass to the field of the neutron star. Maybe the energy displacement for that 0.01 |V| is the radius (= 1) of the neutron star? The total energy displacement may be
m * 0.01 + |V| * 0.01 + 0.01 |V| * 1.
We see that the inertial mass in vertical movement is m + 2 |V|.
The movement of electrons in an atom is 1 % slower in the vertical direction. That squeezes the ruler 1 % vertically.
If the gravitational potential |V| is much less than m, our model explains the Schwarzschild geometry from the potential of a newtonian gravity force.
The behavior close to the horizon of a black hole requires a separate analysis. Since time has slowed very much there, forces are very weak, and the inertial mass is very large.
When we move the test mass horizontally, there is energy flow within the field but not to/from the test mass.
If we move the test mass vertically, there is energy flow also between the test mass and the field. The inertial mass for the test mass might be larger in the vertical direction.
Let the observer A throw the test mass horizontally so that it hits a wall at a 45 degree angle, and bounces vertically up. If the inertial mass of the test mass is bigger in the vertical direction, it will fly slower.
Any object, an electron in an atom or anything, will move slower to the vertical direction. That might be the reason why the ruler of A is shorter in the vertical direction.
We need to estimate the energy flow in a vertical movement.
Let us have a neutron star of radius 1. The potential V of the test mass m is
V = -k / r,
for some k.
V
<---- m ---->
==========
neutron star
In horizontal movement, if we move the test mass a distance 0.01, the negative potential energy V, in a sense, seems to cause positive energy |V| to flow a distance 0.01. The total energy displacement is
m * 0.01 + |V| * 0.01.
The inertial mass is m + |V| for horizontal movement.
^
|
m V
|
v
==========
neutron star
If we move the test mass vertically +0.01, we move|V| the same distance. Our test mass has stolen its extra inertial mass |V| from the field of the neutron star. When the test mass moves up 0.01, it returns 0.01 |V| of the inertial mass to the field of the neutron star. Maybe the energy displacement for that 0.01 |V| is the radius (= 1) of the neutron star? The total energy displacement may be
m * 0.01 + |V| * 0.01 + 0.01 |V| * 1.
We see that the inertial mass in vertical movement is m + 2 |V|.
The movement of electrons in an atom is 1 % slower in the vertical direction. That squeezes the ruler 1 % vertically.
If the gravitational potential |V| is much less than m, our model explains the Schwarzschild geometry from the potential of a newtonian gravity force.
The behavior close to the horizon of a black hole requires a separate analysis. Since time has slowed very much there, forces are very weak, and the inertial mass is very large.
No comments:
Post a Comment