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In classical physics, time and space are absolute and independent, so time and space can be treated separately. However, in modern physics, time and space are relative and dependent: time and space must be treated together. In 4-d s-t frames, we treat time and space independently, then add a constraint to link them together. In teaching, there is a big gap between classical and modern physics. We hope that we are able to find a frame connecting them to make learning simpler. 3-d s-t frame is the best candidate to serve this purpose: time and space are able to be treated dependently by defining the unit of time as T and the unit of space as λ in this frame. Furthermore, the ratio, λ/T, is the velocity, c, of the medium. This paper shows the equivalence between a 4-d s-t frame and a 3-d s-t frame by properly converting coordinates of two frames.

In order to help students to be prepared for modern physics, many professors ask students to forget what they learn in classical physics. If a student ask too many questions based on what they learn in classical physics, it will be difficult for any professor to have progress in teaching modern physics. Even though, Einstein already demonstrated that time and space are related from the thought experiment by emitting photons from the middle of a car of the moving train, we still treat them as independent in a 4-d-s-t frames. In 4-d s-t frame, the representation of 3 dimensions of (x, y, and z) and dimension of time (t) are all perpendicular each other. The only way to depict this is to have three separate graphs of x vs. t, y vs. t, and z vs. t with time which can be negative on those graphs. Later, Minkoski added the same constraint for a pair of inertial frames to make time and space related in the following equation.

We believe that time and spaces are dependent and should be shown in a frame which is applied not only to modern physics but also to classical physics. A 3-d s-t frame serves that purpose. We can draw the time dilation and length contraction between two inertial 3-d s-t frames. We can show that a 4-d s-t frame is the approximation of a 3-d s-t frame, when the velocity of a moving frame is much less than the velocity of the medium. We also show that a 3-d s-t frame is equivalent to a 4-d s-t frame after properly converting coordinates between two frames.

There is a pair of inertial frames [

Because of

which is called time dilation [

The special time for measuring an event happened at the different locations in the 3-d s-t frame O is always longer than the proper time for measuring an event that happens at the same location in the 3-d s-t frame O'. In modern physics, it is called time dilation. We are able use the same graph to derive length contraction formula.

Because of

Because of

The motion of the frame O' is described as the slanted line OA from the frame O and the motion of the frame O is described as the slanted line OB from the frame O' in

Because of

then it can be expressed as

which is called length contraction [

For a pair of inertial 3-d s-time frames, if the velocity of moving frame is much less than the velocity of medium, then time is approximately same for both frames shown in the following graph

From the figure of

From 3-d s-t frames, we can understand why time can be treated as absolute value regardless frame i in classic physics, where u << c, regardless of the frame of reference.

Figures 2(a)-2(d) show the motion of moving train at different percentage velocity of the medium with wave property.

The coordinates of a particle in the 4-d s-t frame are

of the wave,

If the unit of time is sec, then the unit of radius for polar circle is chosen to be the period of the wave-medium, then the unit of x-axis should be chosen as the wavelength of the wave. We construct this new space-time frame to make space and time dependent by using the period and the wavelength of the wave-medium. Furthermore,

the ratio of wavelength to period is equal to the velocity of the wave-medium,

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 [

In the following section, we try to find relations between kinetic terms in a 4-d s-t frame and kinetic terms in a 3-d s-t frame through:

Distance of a particle with a constant velocity

Distance of a particle with a constant acceleration

It shows that all kinetic formulas in a 3-d s-t frame are same as in classical physics.

Newton’s Second Law:

and

Momentum:

and

Kinetic Energy:

and

The other form of Newton’s Second Law:

It shows that the all forms of dynamic formula in a 3-d s-t frame are same forms of dynamic formula in classical physics.

a) In Modern Physics:

For time dilation

For length contraction

It shows that the all forms of kinetic formula in a 3-d s-t frame are same forms of kinetic formula in modern physics.

This paper expounds that a 4-d s-t frame is the approximation of a 3-d s-t frame, when the velocity of a moving frame is much less than the velocity of the medium. While, showing the equivalence between a 4-d s-t frame and a 3-d s-t frame by proper converting coordinates of two frames. In the “Time Dilation and Length Contraction Shown in Three-Dimensional Space-Time Frames”, demonstrates that we are able to visualize time dilations and length contractions through graphs on

We are able to describe the motion of a particle at very small velocity, relative to the velocity of light in a 3-d s-t frame in classical physics. The following graph (see _{m} = 343 m/sec, then the radius of the circle representing 1sec would be

the radius of the circle representing 2 secs would be

etc. In _{m} = 343 m/sec, the velocity of sound which is the medium of the system.

We also able use the velocity of sound

to construct a polar coordinate by letting

P.S.: For all figures except

It might be worth to discuss rewriting textbooks of classical physics and modern physics described in 3-d s-t frames for teaching physics [

We believe that if students study physics using 3-d s-t frames, they will benefit by this approach [

A 4-d s-t frame is independent and is only tied together using an equation constraint. A 3-d s-t frame is dependent and the constraint is embedded into the system [

Tower Chen,Zeon Chen, (2016) A Bridge Connecting Classical Physics and Modern Physics. Journal of Modern Physics,07,1378-1387. doi: 10.4236/jmp.2016.711125