Gravitational Fields

Einstein's widely accepted portrayal of gravity defines it as a field, a concept that denotes a value in space assigned to each coordinate position. In everyday language, we often simplify such concepts by grouping them, referring to temperature as "room" or "wall" temperature, for example. However, in precise terms, a field possesses varying values at every infinitesimal coordinate, as illustrated in Figure 1 for a two-dimensional field.

The labels in Figure 1 are intentionally left out to signify that they represent positions in space, which can be expressed in various coordinate systems, including number line, polar, cylindrical, homogeneous, and Cartesian coordinates. Einstein utilized such a system to articulate gravity's presence across the x, y, z coordinates, which he termed spacetime. His work culminated in the formulation of what we now recognize as Einstein’s Field Equations, typically depicted in tensor form as follows:

In this equation, R is Ricci's tensor, R is its trace, g is the spacetime metric, T represents the matter energy–momentum tensor, including the cosmological constant, G is Newton's universal gravitational constant, and c indicates the speed of light in a vacuum. It's noteworthy that these equations are termed "field equations" in the plural form, suggesting they can be decomposed into multiple equations—specifically ten. One method of decomposing this equation is through the Taylor series expansion, introduced by British mathematician Brook Taylor in 1715.

Einstein’s field equations are often regarded as complex and computationally demanding. Notably, Einstein himself doubted whether an exact solution could ever be achieved, ultimately producing only approximate solutions. The principal source of error in these equations stems from the Taylor series expansion method, which allows errors to accumulate based on the number of terms included in the expansion. It is fair to state that his efforts were constrained by the limited computational resources available during his era.

Despite these challenges, numerous attempts have been made to solve the equations, with the first exact solution attributed to Karl Schwarzschild, a German physicist and World War I veteran, in 1915—the same year Einstein published his field equations. The solutions derived from these equations are known as metrics, with the Schwarzschild solution also referred to as the Schwarzschild metric. This solution defines what is known as the Schwarzschild radius r, which delineates the event horizon of a non-rotating black hole. The mathematical representation is as follows:

Over time, additional notable solutions to Einstein's field equations have emerged, including the Reissner–Nordström, Kerr, and Kerr-Newman metrics. Each of these corresponds to specific types of black holes. To sum up this section, while referring to gravity as a field does not address the question of its origin, it does indicate where gravity exists in our universe—within the field itself.

Gravity as a Curvature in Space

Another perspective on gravity is to view it as a curvature within the spacetime matrix caused by massive objects. This notion is widely understood, although it can lead to confusion as it does not clearly explain the source of gravity, particularly as demonstrated by Isaac Newton, who questioned what causes attraction between two objects of differing masses at the same level.

In essence, combining these two models reveals that while Newtonian gravity is perceived as a pulling force between masses, Einstein’s framework quantifies gravity through the "depression" or vortex created by the mass in the spacetime matrix. Consequently, smaller masses tend to move toward this created vortex.

Spacetime-f: The Extension of Spacetime

This section aims to demonstrate that for every coordinate x, y, z, t in Einstein's gravitational field description, an additional frequency term exists that displays wave-like properties.

It is widely accepted that our universe comprises pure frequencies. However, this notion has not transitioned from philosophical speculation to mathematical rigor, particularly concerning extensive objects. For example, notable methods exist to estimate the wave properties of elementary particles, including de Broglie and Compton wavelengths. Compton's approach was groundbreaking, enabling precise determination of particles' rest mass, which would otherwise be challenging to measure. According to Compton scattering, a particle's wavelength equals that of a photon whose energy matches its mass, expressed as follows:

Here, h represents Planck's constant, and c indicates the speed of light in a vacuum. However, it's important to note that this equation applies only to individual, homogeneous particles, not composite ones. For instance, calculating a wavelength for Earth would yield trivial and less intuitive results.

This limitation compels us to seek an equivalent of the Compton formula for larger objects that would allow us to associate any mass with wave properties, such as frequency or wavelength. The solution lies in applying an exact solution to Einstein’s field equations. For any object, regardless of its size, its Compton wavelength equivalent can be expressed as follows:

Where r is the Schwarzschild radius. The wave numbers for this wavelength can also be obtained by taking the reciprocal k = 1/?. A range of nearby planetary bodies has been presented in Table 1 using the same methodology.

The primary advantage of transforming massive objects into their equivalent single-particle Compton wavelengths is that their corresponding radio frequencies can be derived from the classical electromagnetic wave relation f = c/?. This discovery unveils a new avenue for estimating Newtonian gravitational potential g as follows:

The underlying rationale for this method's applicability to all celestial bodies—planets, moons, neutron stars, and even black holes—lies in its reduction to the familiar framework of Newtonian gravity. This reduction can be achieved through the following steps:

Where M is the mass of the object in question. An additional benefit of this methodology is its ability to characterize celestial bodies, ranging from planets to black holes, including their detection. This can be accomplished through any radio frequency mapping (radar) technique, enabling the calculation of gravitational potential g using the previous equation, with radii determined by the following method:

In this case, the minimum value for r in the earlier equation is r?, indicating that any object with a radius equal to or less than its Schwarzschild radius is categorized as a black hole. For such cases, the equation simplifies to ? = 2r?. This suggests that the properties of black holes—including mass, energy density, and Schwarzschild radius—can be estimated once the relevant radio frequencies are acquired. For Schwarzschild black holes, gravitational radii and masses can be estimated as follows:

It is known that stars like our sun gradually lose energy—referred to as nucleosynthetic power—albeit in minuscule amounts. The eventual outcome is the formation of a black hole at the core. If we were to model this phenomenon using our methodology, we would allow the wavelength to recede by an amount ?? = ? - r at any given moment. The resulting metric solution can be expressed as follows:

This implies that an object will not become a black hole as long as the difference ?? between its radius and the corresponding wavelength exceeds r?. A significant challenge in obtaining the wave frequencies associated with gravitational fields is the potential lack of accurate detection tools, as these frequencies fall within the ultra-low frequency (ULF) range of the electromagnetic spectrum.

The frequencies we seek lie at the far right of the electromagnetic spectrum, implying they are long wavelengths and low energy, measuring in the range of nanoHertz. For Earth, this frequency is approximately 32.71 nanoHertz, with Jupiter having the highest at 77.05 and Pluto the lowest at 2.335 nanoHertz.

As a side note, the universal gravitational constant is not as universal as traditionally believed. It is rather a constant relative to the specific planet for which calculations are performed, meaning the constant 6.6743E-11 applies only to Earth. In the strictest terms, it functions as a variable for each massive object, proportional to its extent r, mass M, and the associated frequencies discussed in this article.

This observation underscores the reasoning behind why many attempts to apply general relativistic calculations to quantum mechanics often falter, as in these contexts, G can approach values like 10E-54.

Conclusion

There are strong indications that every coordinate in a gravitational field, as described by Albert Einstein through spacetime, can be associated with wave properties. When utilized effectively, this association allows for a more comprehensive characterization of the gravitational field. Notably, massive objects in such a field tend to emit a wavelength that is significantly larger than their own radii.

These wavelengths correspond to radio frequencies that, if accurately mapped, can provide insights into various properties linked to the gravitational fields surrounding these objects. This methodology has been termed spacetime-f, as it introduces a frequency response variable f quantifiable for each coordinate position x, y, z, and t. Consequently, mass is no longer the sole means of detecting spacetime curvatures, as these radio frequencies also present viable candidates for exploration in space, provided they fall within our electromagnetic spectrum.

A successful comprehension and implementation of these methods could pave the way for technological advancements in sound navigation, detection ranging (sonar, sodar, radar), and potentially lead to innovations in anti-gravitation technologies.

References

¹ Miller, Mark. “Accuracy requirements for the calculation of gravitational waveforms from coalescing compact binaries in numerical relativity.” Physical Review D 71.10 (2005): 104016.

² https://science.nasa.gov/science-news/science-at-nasa/2005/16nov_gpb

³ https://www.nasa.gov/multimedia/guidelines/index.html