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In this section, we will explore *highly novel ideas* of
direct connections, potential synergies, and construction a
*complex-time interpretation* of the universe, without a
fundamental time variable. Specifically, we will examine spacekime in
the context of observability, Wheeler-DeWitt equation, Boltzmann theory,
entropy, quantum gravity, and artificial intelligence applicaitons.

In special relativity, as an object moves faster, its mass appears to
increase from the perspective of an observer in a different frame of
*reference frame.* This effect becomes significant as the
object’s speed approaches the speed of light. The increase in mass due
to speed can be derived from the principles of special relativity.

**Motivating Data: Observations and Experimental
Evidence**

**Particle Accelerators**: When particles such as electrons or protons are accelerated to speeds close to the speed of light in particle accelerators like the Large Hadron Collider, their behavior confirms the relativistic mass increase. The energy required to continue accelerating these particles increases dramatically as their speed approaches the speed of light, indicating an increase in their relativistic mass.**Muons in Cosmic Rays**: Muons are particles that decay relatively quickly when at rest. However, muons generated by cosmic rays traveling near the speed of light reach the Earth’s surface more often than expected. This is explained by time dilation and the increase in their relativistic mass, which prolongs their lifespan from the perspective of an observer on Earth.

To derive the relativistic mass increase with velocity, we start with
the *total energy-momentum* relation in special relativity:

\[E^2 = (pc)^2 + (mc^2)^2\ ,\]

where \(E\) is the *total
energy*, \(p\) is the
*momentum*, \(m\) is the
*rest mass*, and \(c=300,000\
km/s\) is the speed of light. The *momentum* \(p\) in relativity is \(p = \gamma mv\). The total energy \(E\) can also be expressed in terms of the
velocity \(v\) of the object relative
to the observer as \(E = \gamma mc^2\),
where \(\gamma\) (the *Lorentz
factor*) is \[\gamma = \frac{1}{\sqrt{1 -
\frac{v^2}{c^2}}}\ .\]

In the context of special relativity, the mass of an object as
measured by an observer moving relative to it (often called
*relativistic mass*)

\[m_{\text{rel}} = \gamma m = \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}}\ .\]

This equation shows that as the speed \(v\) of the object increases, its relativistic mass \(m_{\text{rel}}\) increases as well. Specifically, as velocity increases, \(v\to c\), the denominator \(\sqrt{1 - \frac{v^2}{c^2}}\to 0\), causing \(m_{\text{rel}}\to\infty\).

The increase in mass due to speed is given by the difference between
the *relativistic mass* and the *rest mass*:

\[\Delta m = m_{\text{rel}} - m = \left(\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1\right) m\]

This \(\Delta m\) represents the
*additional mass* that appears due to the object’s velocity. This
relativistic mass increase has been confirmed through experiments and
observations, particularly in high-energy physics (e.g., LHC), where
particles are accelerated to relativistic speeds. This relationship is
fundamental to understanding the behavior of objects in motion at speeds
close to the speed of light, influencing everything from particle
physics to cosmological models.

Spacetime is a four-dimensional manifold \(\mathcal{M}\) where events are described by coordinates \((x^0, x^1, x^2, x^3)\), typically represented as \((t, x, y, z)\), where \(t\) is the time coordinate and \((x, y, z)\) are the spatial coordinates. The geometry of spacetime is determined by the metric tensor \(g_{\mu\nu}\), which encodes the distances and angles in the manifold. The line element (or interval) between two infinitesimally close events is given by \(ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu\), where \(\mu, \nu = 0, 1, 2, 3\) and Einstein summation convention is used.

A **spacetime event** is a point in spacetime,
representing a specific location and time. Mathematically, an event
\(E\) can be defined as a 4-tuple \(E = (t, x, y, z)\). In general relativity,
events are points on the spacetime manifold \(\mathcal{M}\). The set of all possible
events constitutes the entire spacetime.

**Observability** in spacetime refers to the ability to
detect or measure certain events or quantities in the spacetime
continuum. It depends on the availability of observers and the causal
structure of spacetime. For an observer \(O\) with a worldline parameterized by
proper time \(\tau\), the observability
of an event \(E = (t, x, y, z)\) is
determined by the light cones emanating from \(E\)

**Future Light Cone:**The set of all possible future events that can be influenced by \(E\). These are events \(E'\) such that \(ds^2 \geq 0\) and \(t' > t\).**Past Light Cone:**The set of all possible past events that could influence \(E\). These are events \(E'\) such that \(ds^2 \geq 0\) and \(t' < t\).

The **causal structure** defines the relationships
between events and whether one event can influence another. For two
events \(E_1\) and \(E_2\)

- \(E_1\) and \(E_2\) are
**timelike separated**if \(ds^2 > 0\). This means a causal relationship is possible, where one event can influence the other. - \(E_1\) and \(E_2\) are
**lightlike (null) separated**if \(ds^2 = 0\). The events can be connected by a light signal, making them potentially observable to each other. - \(E_1\) and \(E_2\) are
**spacelike separated**if \(ds^2 < 0\). There is no causal connection between the events, and they are not mutually observable in any reference frame.

The observability of a spacetime event from a particular observer’s perspective is governed by the causal structure of spacetime and the worldline of the observer. If an ob