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In Newton’s classical physics, space and time are treated as absolute quantities. Space and time are treated as independent quantities and can be discussed sepa-rately. With his theory of relativity, Einstein proved that space and time are de-pendent and must be treated inseparably. Minkowski adopted a four-dimensional space-time frame and indirectly revealed the dependency of space and time by adding a constraint for an event interval. Since space and time are inseparable, a three-dimensional space-time frame can be constructed by embedding time into space to directly show the interdependency of space and time. The formula for time dilation, length contraction, and the Lorenz transformation can be derived from graphs utilizing this new frame. The proposed three-dimensional space-time frame is an alternate frame that can be used to describe motions of objects, and it may improve teaching and learning Special Relativity and provide additional insights into space and time.

In order to describe the position of a static object, Descartes constructed three axes perpendicular to one another in the space using

In order to describe the position of a moving object, Galileo constructed an time axis which is perpendicular to three axes in the space, which using

In order to describe the position of a moving object in Special Relativity, Minkowski constructed a time axis, ct, which is perpendicular to three axes in the space simultaneously. First, he treated space and time independently, then added a constraint:

In order to describe the position of a moving object in Special Relativity, we construct polar coordinate for time on

Theory of one Big Bang creating the universe is based on 4-d s-t frame. There are many unsolved paradoxes in this theory: Hubble’s constant should be a fixed value, but having wide range; There are two different methods to measure the distance of a quasar, but results are very different; In order to raise up the density of the universe keeping present status, there is need of dark matter; In order to explain the observation of acceleration of the universe, there is need of dark energy. The paper of “The Shell Model of the Universe: a universe generated from multiple big bangs” [

Any particle’s motion in space can be described by choosing a 3-d s-t frame with the proper velocity of a medium [

In order to describe the motion of micro quanta, there is uncertainty relation between its momentum and its position. When the motion of a macro object or a micro quantum is observed, the only difference between a macro object and a micro quantum is that one is visible and the other is invisible while interacting with measurement equipment. There are two uncertain measurements related to this measuring: the probability of hitting different spots which is inversely proportional to mass and velocity, and the probability of hitting either the front or the rear of the wave of the photon wave which is proportional to the wave length. The matter wavelength can be explained as the probability of uncertainty in measuring a quantum with the unit of length. The second beam of photon may hit a different spot from the first one because of the rotation of the particle. To verify the assumptions made previously, the Heisenberg uncertainty relationship can be derived by multiplying these two independent probabilities (matter wavelength of an object and light wavelength of measuring medium) [

For quantum entanglement, there is a medium affecting each of a pair of particles with a velocity much faster than light, and it might be with an infinity velocity. It is against the main assumption of Special Relativity: the velocity of light is the upper limit of particle in the universe. If we locate a particle on the platform and the other particle at any distance from the platform, then the medium can be treated as the moving train. If the moving train travels with the velocity of light, the observers on the train will reach the other particle at any distance from the platform with zero second. It means that the medium will affect both particles instantly and the distance between both particles is also zero meter measured by observers. It can apply to any force between two objects including gravitational force, as long as the medium between two objects traveling with velocity of light. The proposed 3-d s-t frame shows the advantage of 3-d s-t frame [

The motion of any particle in space can be decomposed into its x, y, and z directions. In order to describe the motion of an object in 3-dimensional space along the locations of x-axis, y-axis, and z-axis, we can construct a new space-time frame. Spheres with different radius representing different outgoing time, polar coordinates will be formed from circles of intersections between spheres and x-y plane, y-z plane, and z-x plane [

Its component along the x-axis can be described as a function of time, which is represented by the time circles created from the intersections between the x-y plane and the concentric time spheres. If the velocity of an appropriate medium is

intersection between the z-x plane and the concentric time spheres The point with the properties,

If messages are relayed by sound of V_{m} = 350 m/sec then the radius of the sphere representing one second is equivalent to (V_{m})(1 sec) = 350 m; the radius of the sphere representing two seconds is equivalent to (V_{m})(2 sec) = 700 m; …; and the radius of the sphere representing n seconds is equivalent to (V_{m})(n sec) = n(350) m.

If the message is transmitted by light of V_{m} ~3(10^{8}) m/sec, then the radius of the sphere representing one second is equivalent to (V_{m})(1 sec) = 3(10^{8}) m; the radius of the sphere representing two second is equivalent to (V_{m})(2 sec) = 6(10^{8}) m; …; and the radius of the sphere representing n seconds is equivalent to (V_{m})(n sec) = 3n(10^{8}) m. Since the velocity of light is the limiting velocity, all possible motions of a particle can be described using this 3-d s-t frame.

In cosmology, the expansion velocity of the universe is very high, as the recession velocities of some galaxies away from the earth are nearly 90% of the velocity of light [_{m})(1 year) = 9.46(10^{15}) m = 1ly; the radius of the sphere representing two years is equivalent to (V_{m})(2 year) = 1.89(10^{16}) m = 2ly; …; and the radius of the sphere representing n years is equivalent to (V_{m})(n year) = 9.46n(10^{15}) m = nly.

In high energy physics, if a particle’s velocity approaches the velocity of light, the interval of 1 sec would be too large to meaningfully describe its motion. The units can be scaled down by choosing the period

The proposed new coordinate frame can also be used to describe the motion of the object moving along the x-axis in various trajectories and at different speeds using the methods described above. In

A 3-d s-t frame, created by embedding time into space directly, reveals the dependency of space and time. Although the space coordinates are bi-directional, time cannot be given a negative value thus, because it only has one outgoing direction in this 3-d s-t frame.

Before describing time dilation and length contraction, we will first define some terms. If two frames have a constant relative velocity between them, two frames are called a pair of inertial frames which are inertial to each other [

In

using a sensor attached to the front of the train, i.e. the origin O’ of the moving frame S’. The length of the rod as measured by an observer in the stationary frame S, is defined as proper length,

At the same time the sensor touches the left end of the rod, the observer in the moving frame S’ sends a pulse of light towards the ceiling of the car, where a mirror is placed. To the observer in the moving frame S’, the light travels vertically up towards the ceiling and is then reflected vertically down. The ceiling height of the boxcar is adjustable, such that the pulse of light reaches the ceiling at the same time that the sensor reaches the right end of the rod. In

From

From the previous discussion, we know that

When an observer on the train moves to the right with velocity

The two equations,

and

are derived from

The event where the sensor attached to the observer on the train moves from the left of the rod to the right of the rod can be described by two the different observers. To the observer at the origin of the moving frame S’, the event occurred at the same location, and the duration of the event is called the proper time,

Since

In the next discussion, we designate the occurrence of an event at the coordinate x on the x-axis of a stationary frame S, by laying a rod on the x-axis from 0 to x. The proper length

equal to

In order to check that the time

on the clock on the wall at x'. When the light reaches the wall at x', the time

both walls on the moving frame S’. In the moving frame S’, if

then the time

on the wall at x'. When the light reaches the wall at x', the time

be theoretically recorded on the clock on the wall at 0 because it takes extra time

recorded on the clock on the wall at x'. To observers in the stationary frame S, if

then the time _{ }on the clock on the wall at x' is synchronized with the time

in the moving frame S’. This means that the proper time is adjusted by

the coordinate x' when the proper time is

by

two coordinates is

Combining all relationships between coordinates of the stationary frame S and the moving frame S’ forms the following Lorentz transformation:

In order to derive the reverse Lorentz transformation, we can construct a stationary frame S’ on the moving train, and a moving frame S on the platform which is moving to the left with respect to the train with the constant velocity

in the moving frame S is equal to

In order to check that the time

on the clock on the wall at x. When the light reaches the wall at x, the time

both walls on the moving frame S. In the moving frame S, if

then the time

x'. When the light reaches the wall at x, the time

the wall at x. To observers in the stationary frame S’, if

then the time _{ }on the clock on the wall at x is synchronized with the time

in the moving frame S. This means that the proper time is adjusted by

the coordinate x when the proper time is

by

two coordinates is

Combining all relationships between coordinates of the moving frame S and the stationary frame S’ forms the following reverse Lorentz transformation:

Length contraction and time dilation between two inertial frames was discussed in the section IV. In the following example, we particularly select a blue light as the median to transmit message with wavelength

as the unit of length and period

as the unit of time to construct 3-d s-t frames [

For a rod of length

by length contraction.

In order to draw length into graph, we can change the unit of length from m to

and

These values with new units satisfies the formula of length contraction

It takes regular time

for the origin O’ of the moving frame S’ to pass from the left to the right ends of the rod measured by observers on the stationary frame S, thus the proper time measured from

observers on the moving frame S’ is

formula. We can calculate the proper time

In order to draw time into graph, we change the unit of time from sec to T. Because

then

and

then

These values with new units also satisfy the length contraction equation

From

then

and

then

This example shows that the actual value of time dilation and the actual value of length contraction can be measured simultaneously in this 3-d s-t frame by selecting

In classical physics, time and space are treated independently. Einstein demonstrated the inseparability of time and space. The realistic difference between time and space is the single direction of time and the two directions of space. In the proposed 3-d s-t frame, time is represented by spheres of different radii with the origin of the space axes as their center and time can only have a single direction.

In Special Relativity, two 3-d s-t inertial frames can be constructed by choosing light as a medium for transmitting messages. The geometric meaning of time dilation of an event occurring at the same location in the moving frame for an observer in the stationary frame and length contraction of a rod lying still in the stationary frame for an observer in the moving frame can be clearly illustrated in this 3-d s-t. The Lorenz transformation can also be derived from graphs of time dilation and length contraction. The universe generated from multiple big bangs based on a 3-d s-t frame solves the problems which are unsolved by the universe generated from Big Bang. Time contraction and length contraction on the moving train helps us explain quantum entanglement. These demonstrate the value of 3-d s-t frames.

We would like to thank Elizabeth Chen, Min Chou, Zen-Fu Chow, Angel Garciel, Li-Shing Hsu, Bo Liu, Yaijei Sun, Chu-Tak Tseng, Chun-Zin Wu, Lin Wu, Wan-Zin Zhaw, Yousuo Zou for their support, encouragement, suggestions.

Chen, T. and Chen, Z. (2016) Special Relativity in Three- Dimensional Space-Time Frames. International Journal of Astronomy and Astrophysics, 6, 410-424. http://dx.doi.org/10.4236/ijaa.2016.64033