We mentioned in our previous blog post about the Alcubierre drive that negative mass-energy behaves very differently in general relativity versus our own model.
The Einstein field equations are linear in the Ricci curvature. If we flip the sign of the stress energy tensor T on the right side, then the sign of the Ricci curvature is flipped.
Thus, if positive mass causes positive curvature, then general relativity predicts that negative mass flips the sign of the Ricci curvature of the metric. Time close to negative mass would run faster than in the surrounding asymptotic Minkowski space. Radial distances around negative mass would be contracted. This would open doors for faster-than-light travel. By faster than light we mean that one could move from one location in the asymptotic Minkowski space to another location faster than would be allowed by the Minkowski metric, if the Minkowski metric would hold everywhere in the space.
light
x • ---------------- O --------------- • y
donut of
negative mass
An example: we want to send a signal from x to y faster than light in the Minkowski space. The solution is to put a donut of negative mass between x and y. A Minkowski observer sees light to move surprisingly fast through the donut. It is like a gravitational lens, but instead of slowing down light, it makes light to go faster.
In our own Minkowski & newtonian model an interaction can only increase the inertia of a test mass, which means that clocks will run slower, and optionally, some distances become longer. Thus, in our model the donut has the opposite effect: it slows down light.
As far as we know, there does not exist negative energy. However, there does exist negative pressure.
For a massive rod, negative pressure acts like negative mass
Suppose that we have a very rigid rod. A gravitational wave arrives, normal to the rod, and causes spatial distances along the rod to get longer for a while.
s
m • -->
===================== rod
----------------------
^ gravitational
| waves
----------------------
We assume that the wave was born from a movement of positive masses, for example, from a binary black hole.
We assume that the gravitational attraction of the rod is stronger than repusion generated by negative pressure in the rod. This is the weak energy condition.
s
m • -->
======= ======= rod
M1 M2 positive mass
- - - - - - - - - - - - negative pressure
In the diagram we have removed the central part of the rod to concentrate our attention on the ends of the rod.
If there is no negative pressure, then the masses M1 and M2 cause extra inertia to the test mass m, because if we move m to the right, then energy flows from the field of M2 to the field of M1.
However, if there is a little of negative pressure, then energy flows also to the opposite direction. We claim that the opposite energy flow decreases the extra inertia of the test mass m if we move it a short distance s.
What does the rod do to the wave?
The stretching of the metric on the upper side of the rod is decreased by the negative pressure. That means that the wave has problems going "through" the rod. Some of the wave is reflected back.
The quadrupole moment of the rod, relative to the center of the rod, is, in a sense, reduced by negative pressure. The negative pressure in the rod acts as a source of a new gravitational wave and causes some destructive interference to the passing gravitational wave.
The behavior for a massive rod is similar in general relativity and our model.
What if we would have a gravitational wave generated by a movement of negative mass? The negative pressure could strengthen the wave? That cannot happen. We have to think about this.
Gravitational waves generated by pressure: does pressure possess a charge?
In our Minkowski & newtonian model gravity is an ordinary force whose charge is mass-energy. Accelerating mass-energy generates waves just like an accelerating electric charge.
But in general relativity, changing pressure generates gravitational waves, too. How do we explain those waves in our model? Does pressure possess a charge?
In our model, the attractive or repulsive force on a test mass is an indirect consequence of the gravity of the test mass. The Schwarzschild metric around the test mass interacts with the force that causes the pressure. It is a different interaction than direct gravitational attraction, or is it?
Should we assign a charge to pressure?
Another option is to claim that any long-distance interaction creates "waves" if an object of that interaction is in an accelerated motion or periodic motion.
Then we do not need to specify if pressure has a charge. It is enough to claim that "waves" will carry that interaction over large distances.
In this blog we have tossed the idea that a wave carries a "copy" of the transmitting system over a large distance, and puts it close to the receiving system. The copy is distorted, though, because it does not contain longitudinal fields or static fields, and the copy is "weaker" than the original transmitting system.
Thus, it may be that we do not need to specify if pressure has some kind of a charge.
Another option is the following: the interaction with pressure is relayed from the test mass through gravity, just like gravitational attraction. We could claim that any interaction which is relayed through gravity can cause gravitational waves. That is, the force on the test mass is defined as the gravity field. We assume that the test mass does not possess any other charges but mass.
What about more complicated interactions? Suppose that the test mass also possesses an electric charge, and the hamiltonian of our system contains some complex formula of gravity and the electric field. If we move the system, does it generate some kind of "hybrid" waves?
Maybe the "copy" model is the way to define the waves generated by very complicated interactions?
We need to think about this in detail. Can we always treat pressure as if it would generate a gravity field? That would be simple, but do any problems arise?
Negative pressure in general relativity generates waves which have the opposite phase to our Minkowski & newtonian model?
The weak energy condition prevents us from creating a static negative gravity field in general relativity. But it does not prevent us from creating a gravitational wave which does have a negative gravity field. For that, we need negative pressure.
rod
pull <---- ================== ----> pull
mass M
We periodically pull the ends of the rod, which generates periodic negative pressure inside the rod. The rod also contains the mass M, which ensures that the sum of gravity forces is always attractive, despite the negative pressure.
However, the gravitational wave, which is created, ignores the static field of the mass M. The wave is born from the negative gravity of the negative pressure.
We assume approximate linearity: static fields do not produce a wave. The wave is born from the dynamic field.
Let us ignore the mass M and concentrate on the negative pressure. In the diagram we have cut the rod to two pieces, to emphasize the ends of the rod.
s
• -->
m test mass
======= ======= rod
- - - - - - - - - - - - - - negative pressure
In the Minkowski & newtonian model, the negative pressure produces repulsive fields. If we move the test mass horizontally a short distance s, that ships energy between the fields. The inertia is higher, and horizontal distances are longer.
In general relativity, negative pressure generates a wave which makes horizontal distances shorter? We have to check this. Probably no one has calculated what kind of waves pressure generates in general relativity.
Conclusions
Lots of open problems remain.
If we have a gravitational wave generated by negative pressure, how does the wave react to a rigid wall?
Can we consistently say that an indirect interaction of the field of a test mass and pressure causes "gravity"? Does that "gravity" generate gravitational waves just like masses do?
Pressure generates "gravity". What is the analogous effect in electromagnetics. Can pressure generate an electric field?
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