Dimensional Physics
Everything consists of space-time.
Everything consists of space-time.
ART is based on only 3 principles
We have already derived the principle of relativity and the speed of light in the chapter on relativity. The equivalence principle is still missing for GR. There are two of these. The weak and the strong equivalence principle. We will deal with both separately. The strong one is sufficient as it contains the weak one. Hence the names chosen. The separate derivation is interesting for the logical structure. The astonishing result of the derivation is that spacetime itself is a potential field. This becomes very important again in cosmology, in a different form. Here, the vectorial potential field of spacetime is identical to the potential field of gravity. All other potential fields in physics function according to the same principle. In QFT in different spacetime configurations.
In this chapter, we will also clarify what a force is in the classical description of physics. This will help us to understand gravity. Einstein’s ingenious idea of a force as a geometric representation in spacetime is not always immediately understandable. We then recognize more easily why we can use such different descriptions for an identical phenomenon.
Let’s start with a weak principle and then we can improve it. The weak equivalence principle is already included in the good old mechanics of Newton. In classical mechanics, however, it was unclear why this is so. Here, the principle is often referred to as the equality of inertial and gravitational mass.
What Einstein’s E\space =\space mc^2 is for Newton F\space =\space ma. The two most famous formulas in the world. Force equals mass times acceleration. Newton’s second law. The mass m in the formula is the inertial mass. Inertial because it does not change its state of motion when no acceleration acts on it. No acceleration, no force, and thus no change => inertia. Since mass is the only object in the formula, this inertia must be associated with the mass. So far, so good.
So why is there a first axiom? Well, do you know it off by heart? I’ll help you: “A body at rest in a force-free environment remains at rest or moves in a straight line and uniformly”. We already had that in the second axiom. No acceleration, no change. Why is this statement in two separate axioms? To make sense of this, we have to read the first axiom differently. We turn the statement around: If no forces act on a body, then what the body does is rest or a straight-line and uniform motion.
The first axiom is a measurement specification. We can measure what a straight and uniform motion is. In a spacetime with spacetime curvature, “straight” is not so easy to determine. This makes a popular statement about gravity questionable. A body in a gravitational field falls force-free in a straight line to the center of gravity. We will see that this statement should be treated with caution. Here we will learn about the difference between the potential field and the force.
Newton’s next famous formula is the formula for gravitational force
F\space =\space \cfrac{G\space *\space M_{heavy}\space *\space m_{heavy}}{r^2}
The capital M shall be the earth and the small m a test mass. The mass here is the heavy mass. That which the scale indicates. We put this formula together differently.
F\space =\space \cfrac{G\space *\space M_{heavy}}{r^2}\space *\space m_{heavy}
The first term with the fraction is, according to the units of measurement, an acceleration. For the earth as M, the well-known small g for the acceleration due to gravity comes out here. This gives:
m_{inert}\space *\space a\space =\space g\space *\space m_{heavy}
If anything is to fit together here, then we must be able to cancel out the different m or g and a. This leads us to the following statements:
The identity of inertial and heavy mass was a mystery to Newton. You can see that it must be so, but there was no reason for it. This identity has been very carefully examined in 2025. There can only be a deviation after the 14th place behind the decimal point. One of the best-examined values ever.
In the DP, the approach is completely different. Each mass is a spacetime density. There is no characteristic for a distinction. All known characteristics for a distinction lie in the QFT and not in the ART. Thus, these characteristics must not produce any difference when a “force” is exerted via gravity. We do not have to justify equality, it is necessarily given by the approach. We turn the tables. We don’t even have the option of describing a difference.
If a difference is ever detected, no matter how far behind the decimal point, the DP is falsified.
Somehow there must be a connection between force and gravitation as a geometric figure. The strong equivalence principle refers to an acceleration. In classical mechanics, this always produces a force. The solution is already contained in Newton’s axioms. First and second axiom: A force is a change.
In DP, we can understand the classical force as a change in the density of spacetime. Without an interaction, a density of spacetime remains what it is. It can change through an interaction. That’s very simple. But we have a big problem, especially with gravity. What is exchanged in an interaction? The long-sought graviton as the exchange particle of quantum gravity? No, definitely not!
In GR, there is only a geometric mapping as spacetime curvature for gravity. All mass-energy equivalents are collected in the energy-momentum tensor. In the Einstein tensor, we have no spacetime density as exchange particles. However, we still need a change in a space-time density. This is precisely where the strength of the DP lies. We have a curvature or a density, that’s all there is. It cannot be a density. There is only one possibility left. The space-time curvature must cause a change in the space-time density without an exchange particle.
Ultimately, we have to come up with the strong equivalence principle. There, gravity must be indistinguishable from an acceleration in its effect on a mass. Thus, space-time curvature must produce a change in space-time density that corresponds to an acceleration. One could also have concluded the DP if one wanted to fully explain the concept of a potential and here the gravitational potential. Unfortunately, people were already satisfied with the exact calculation. The why was no longer interesting.
For us, force is a change in the density of spacetime. Since the density of spacetime is also a state of motion, it should come as no surprise that acceleration is associated with force. Changing a state of motion requires acceleration. This clarifies the concept of force. Let’s move on and finally take a look at the strong equivalence principle.
In the strong equivalence principle, the effect of gravity cannot be distinguished from acceleration. These do not have to be identical, we just must not be able to distinguish the effect.
We saw the first approach in the weak equivalence principle. There, a and g had to be identical. Einstein then came up with the idea that a motion in a curved space must correspond exactly to this acceleration. As we can see from the word “effect”, it was already clear to him that this is realized with different phenomena.
Figures 29 and 30 show a closed “box”. On earth or in a spaceship with acceleration
We are once again traveling with the locked box from Galileo. With the SR, it was without an external effect. Here it is gravity and the rocket with acceleration. In both boxes, we cannot determine with any experiment whether it is gravity or acceleration. The effect is identical.
Since a deformation of space-time was not very useful at the beginning of the GR, the old analogy with acceleration was used. In order to obtain an effect like acceleration, the test object m must “fall” into the center of gravity in curved space-time. I believe that this analogy has slowed down the search for the why-question. The moon falls to Earth. Since space-time is curved, the moon falls on its orbit around the Earth. This can also be calculated very well. Everyone can understand this and everyone is satisfied.
Not us! This analogy explains nothing. According to the calculation in the GR, the moon moves on a geodesic around the Earth. This term describes the direction of motion without the influence of a force. In spacetime without gravity, this is a straight line. With gravity, it is the almost circular orbit around the Earth. Force-free, that reminds us of Newton’s first axiom. In a flat spacetime, it is straight and uniform. In a curved spacetime, it is always following the curvature. But that is exactly the measurement specification that says the moon is not subject to any interaction. No acceleration and therefore no change. Where should an effect as acceleration come from? The first axiom and the second axiom mutually exclude each other when it comes to acceleration. In the GR, however, it is assumed that both can be present at the same time. The force-free moon (since on geodetics) falls (and thus accelerates) around the earth. No, it doesn’t work that way.
Let’s calm down a bit and continue. No interaction from the outside and yet we still need a change. This change remains constant over billions of years, using the moon as an example. This question has never been solved. So let’s do it now.
The first idea we can have is that the value of the space-time density changes in a space-time curvature. Then we have no interaction from the outside and yet a changed value. That sounds very much like the solution we are looking for. In the space-time curvature, the length increases and the length of the space-time density remains the same. Then, in relation to the space-time density, the density increases. Thus, the space-time density receives a perpetual change = acceleration from the environment. Yes, but we have space-time. With the time dimension, it is exactly the opposite and everything balances out again.
Don’t be sad, it’s a good thing. We need energy conservation. The space-time curvature does not change the space-time density for its area. You remember the constant surface area. Thus, the ratio of a space-time density to the surrounding space-time with space-time curvature does not change either.
Ultimately, we have no interaction from the outside. Thus, the ratio of the spacetime densities of the environment and the object cannot change. However, we only have spacetime curvature and spacetime density, so where can it come from?
Attention! To simplify matters, I only explained the facts with length in the YouTube channel. This is not correct. There is no black hole in a black hole. Here, too, I’m afraid I have to say “sorry.” At the time, this wasn’t 100% thought out.
The only thing left now is the shifts between space dimension and time dimension in the curvature of space-time. Let’s take a closer look.
A space-time density moves towards the earth at 1 m/s. Far away from the earth, this is a straight-line and uniform motion. Since we have no interaction from the outside, the speed must remain the same. However, in the curvature of space-time, space and time change their definition. The meter becomes longer and the second becomes slower. But this only happens for the surrounding space-time and not for the space-time density. The speed must remain at 1 m/s. So the space-time density must become faster. It now has to cover a longer distance in less time. The space-time density is accelerated simply because of the change in the dimensions of space and time.
This somewhat strange acceleration is exactly what we need:
The strong equivalence principle arises from the counter-rotating deformations of the space and time components in a spacetime curvature. The spacetime density is not changed. Here we see again how important it is that this deformation is a change in definition and not just a point of view. The equivalence principle only works if the definition is changed.
A change in the components can also have the opposite effect. This happens when acceleration no longer allows the speed to be increased. We have to consider the special case of the speed of light. Here we have two possibilities:
To see this, let’s look at the following picture:
Figure 31 shows “very exaggeratedly” how a beam of light is “extended” around the sun.
A photon traveling at the speed of light passes very close to the sun. As the photon travels towards the sun, it undergoes a blue shift. When it travels away from the sun, it undergoes a red shift. There is no change in frequency.
However, the photon must follow the curvature of space. This results in a longer path for the photon. Then it must simply fly through the longer path at the speed of light and everything is perfect. This was also the idea until Mr. Shapiro, for light in the mathematics of GR, discovered a deviation. Light signals must show a lower speed when flying past a mass. The effect has been experimentally confirmed to about 4 decimal places.
Even at the risk of you being sick of it. Here, too, we see, as with the equivalence principle, that the change in the space-time metric must necessarily be a change in the definition of geometry. If this curvature were just a longer distance, then this effect would not occur.
The photon has the maximum speed. In the space-time curvature, by definition, the path becomes longer and the time shorter. It is not possible to accelerate. The photon becomes slower for an observer in this environment. Locally, the photon retains the speed of light, as we discussed in the SR.
The final act for this chapter should be the gravitational potential. From my point of view, the term potential is one of the least understood but most frequently used terms in physics for calculations. If it doesn’t have to be 100% exact, then we always calculate with the potential and not directly with the curvature of space-time when dealing with a problem involving gravity. Otherwise it’s much too complicated. The trajectories of almost all the bodies that we have shot into space and will shoot into space in the future were calculated in this way.
If we ask a physicist what a potential is, the answer is almost always something along the lines of: the potential is the ability to convert potential energy into kinetic energy. Ok, where does this ability come from, is it in the body? Everyone agrees that this ability is in the potential and not in the body. The identical body outside of a potential experiences no acceleration. What then is this ability? In most cases, rest then sets in. The answer often comes: a property of the potential. We are back to square one.
For almost all potentials, it is important whether the test body participates at all in the interaction of the potential. A neutral neutrino is completely unaffected by an electric potential. In the case of gravity, we have the peculiarity that absolutely everything that we can identify as an object participates in the gravitational potential. This makes sense, since in the DP everything is a spacetime density in curved spacetime and must therefore participate. In the other interactions, the geometry in QFT indicates whether an interaction is allowed to take place.
We have another peculiarity compared to other potentials. In the case of an electric potential, the strength of the test body’s charge plays just as important a role as the strength of the potential itself. Not in the case of gravity. It doesn’t matter how much spacetime density the test body has. It is not about the value of the space-time density itself. It is about the deviation from the length dimension to the time dimension. This is always zero for the space-time density, since the space and time dimensions deform identically. The deviation comes only and exclusively from the environment with space-time curvature.
If we want to get out of the gravitational potential again, then we have to compete against this acceleration. We need a rocket. The acceleration is not just an apparent effect. A rocket must have a decent power output to successfully work against this acceleration. This time, we use the rocket to apply a force by interaction, and this will actually increase the space-time density. The state of motion of the space-time density (rocket) and thus the energy itself must be increased by acceleration to escape velocity.
The acceleration converts the kinetic energy into potential energy. This is the classic statement about a gravitational potential. In fact, nothing is converted into potential energy. The rocket must actually generate the acceleration against gravity through an interaction. The rocket comes out of the gravitational potential and then has a higher state of motion outside the potential..
Calculating with the potential is very simple. Energy conservation ensues because the mutual accelerations must cancel each other out. The energy of the rocket has increased in real terms. We simply assign a negative energy to the potential. However, the object, our rocket, has actually increased its space-time density when leaving the potential. In the calculation, all this is lumped together. With a negative energy in the potential, we get energy conservation and the calculations are very simple.
What about the special case of light? If we look at a photon in terms of waves, it’s a bit easier. The photon doesn’t necessarily have to slow down, it can do something else. If the energy of a photon is determined by its wavelength and acceleration corresponds to an increase in energy, then the photon can increase its energy at the same speed with a shorter wavelength. This is the blue shift. If the photon wants to get out of the gravitational potential, it works the other way around and we get the red shift. It has to use part of its existing space-time density to counteract the acceleration of gravity. However, this only works into or out of the potential. There is no Shapiro delay here because the acceleration can be mapped into the wavelength.
This explains the origin of the equivalence principle. To summarize again in a few sentences:
With this little explanation, it should now be clear why Lagrange and Hamilton work so well. This all comes from the conservation of energy. A spacetime density can only change and does not simply disappear or multiply.
With this knowledge, we can turn to cosmology. The development of our universe.