Special Relativity

The basic theory is based on the discovery that the speed of light is the same for observers at all frames of reference.

The inferences are that space and time are not absolute, but rather they contract according to the relative motion of two bodies. In other words, length contraction and time dilation (contraction of space and time).

The discovery was made through the Michelson-Morley experiment, which attempted to measure the relative motion of earth with the universal frame of reference, which they termed the lumineferous aether.

Now, classical relativity states that if a vehicle moves with a velocity \(v_1\), and a body moving along it moves with a velocity \(v_2\), the velocity of the body from the frame of reference of the vehicle would simply be \(v_2\), while the velocity from frame of reference of the ground would be \((v_1+v_2)\).

But when it comes to light, the velocity of light from the frame of reference of the vehicle is \(c\), but the velocity of the body from the frame of reference of the ground was also \(c\), rather than \(v_1+c\).

Now with this in mind, if you draw the space vs time diagram for the body and the light ray, in classical relativity, from the frame of reference of the vehicle, the vehicle will remain at a constant position along time, and light will move along the line that moves at \(c\), while the ground moves along the line that moves at \(-v_1\), and from the frame of reference of the ground, the vehicle will move along the line that moves at \(v_1\) and light will move along the line that moves at \(v_1+c\) and the ground will remain at a constant position. This transformation of frames is called the Galilean transformation, since it needs just a simple rotation of the world lines to change the frame of reference. Also, since the rotation of the lines is the result of the simple shifting of the position line to make the reference frame appear stationary, all objects remain in the same line of time.

But under special relativity, if you draw the space vs time diagram for the body and the light ray, from the frame of reference of the vehicle, the diagram remains the same, with the ground moving at the line that moves at \(-v_1\) and light moving along the line that moves at \(c\), while the vehicle remains at a constant position along time. However since the light moves along the line that moves at \(c\) in all frames of reference, this remains true for the frame of reference of the ground too, while the ground remains at a constant position along time, while the vehicle moves along the line that moves at \(v_1\).

Since the velocity of the vehicle with respect to light remains the same in this diagram too, to the only way to account for that is to consider the line that the vehicle moves along as its time line, and assume that time varies for it with changing speed to keep the same timings, and since the velocity of light remains constant, the space line for the vehicle also has to tilt to the same degree that the time line has shifted, so that light remains at the velocity.

[Also, since the world line for the vehicle has now become its time line from the perspective of the observer, when reference frames are switched to from a velocity closer to the vehicle, the world line of the vehicle rotates closer to stationary axis, and the time dilation decreases appropriately, leading to a lowering of the temporal - I forgot why the Minkowski spacetime is hyperbolic]

Doubts

• Why is the line shifting frames of references hyperbolic, rather than the lines resizing to match the flat line?
• What is the non-mathematical intuition for spacetime compression as opposed to expansion, or remaining the same?
• Is Doppler effect due to spacetime contraction? When the relative velocity of the radiator increases, the frequency of radiation increases, rather than the velocity of light, and the wavelength decreases.
• Similar, does all matter occupying space an time follow the principles of Fourier transformation? m = hf, all mass has frequency and also time, which are inverse to one another.

Idea

If everything is made of light, then, just like resistors, everything is simply taking a longer route, and that's the cause of relativity.