Fundamental Physics by Ramamurti Shankar at Yale_5_Relativity

Einstein’s famous relativity theory contains two parts – special relativity and general relativity. Let’s start from special relativity.

Physicists always like to study from simplifying the complex. So here is the coordinate transformation with “me” x and “you” s.

However this classical Newtonian transformation is defied by physicists’ attempt to measure the speed of light over and over again. It baffled them immensely until Einstein came up with the following two postulations: all inertia observers are equivalent with respect to natural laws; the velocity of light is independent of the state of motion of observer.

Based on this new ground, if we look at the following simple case, me is described in x and t, while you is described in x’ and t’, and you have a speed of u relative to me moving from the origin to the right side on x-axis.

So if at time the very beginning, a light pulse is emitted and detected at the distance x for me, then x = c*t, while for you, it’s x’ = c*t’. With these conditions, next, Einstein just applied simple math by multiplying x and x’ on the left side, equated the product to the right side:

This is the final space-time equation Einstein deduced:

The latter two equations are significant. We live in a world of low velocity, meaning u/c <<< 1, so these parts can be negligible, and one can see from the math above, it’s reduced back to Newton world, or Galileo transformation: X’ = x – u*t; t = t’. From this equation, we can also see that no velocity can exceed that of the light, or it would be meaningless. It essentially give the relationship between two coordinates with respect to the same event.

This set of equations fundamentally explained the observation in physical world by fundamentally change our perception of space and time. The two are tangled together and time can be changed per space…

So if you turn the coordinate system to a theta of 45 degrees or pei/4, the same location defined by x’ and y’ can be computed.

And the Einstein’s space-time equation exactly follows the same logic, even the format looks alike:

Implication of the Lorentz transformation of space-time equations are plenty, quite mind boggling but need contemplation. Assume there are two events, event one, a gun is fired, even number two, the bullet hit the wall. Here come x1, x2 delta x, delta t, x1′, x2′, delta x’, delta t’, concluding the below result: your speed of the bullet w is equal to my speed minus your speed relative to me(u), divided by (1-uv/c^2).

Conversely, my speed v in your eye is

Your speed is jacked up by the denominator, while my speed v is diminished by the denominator of (1+uw/c^2). This explains one of the most baffling questions – if there is probability to get a speed greater than that of the light? What if we put one rocket into another, so on and so forth to augment the speed until it’s greater than 3*10^8/second?

Seems right in old good world, let’s assume there is light pulse instead of bullet being fired, then w = c – u from your perspective. However, plugging in Einstein’s equation, w has to be modified by the denominator (1 – uv/c^2) = (1 – u/c), so you get (c-u)/(1-u/c) = c.

The second conclusion we can derive from this equation is that a simultaneously occurring events will be different from high speed moving observer. It’s simply because if delta t is zero for me, it’s definitely not zero for you. Shocking! However, the caveat is that the distance/space has to be considered. This phenomena satisfies only when the simultaneously occurring events take place in two different locations. It can’t be at the same place too. Collision of cars, clap of two hands, these are good example of the later form. That is truly occurring same for both me and you. (weird, why Einstein’s separate equation of delta doesn’t account for this significant condition?)

From Einstein’s theory, there is this most famous claim about how a clock runs – it is not running at the same rate if two identical clocks are placed into my and your hands, premised on you and me are relatively running with different speed, or put in previous setting all along, we are in different coordinating system. The clock of yours – running with the speed of u, not omit-table compared to the speed of light – ticks slower than mine.

Why is that?

if I am the prime observer holding a clock and measure the time from Tick to Tack as 1 second, so delta x = 0 because I hold the clock still; then alternate to use your framework, delta t’ = (delta t – u*delta x/c^2)/square root of 1- u^2/c^2, if u is 3/4 of c, then the delta t’ is approximately 16/7,double the time of mine.

Paradoxically, if we switch the perspective, you set your delta t, and compare that to mine with the same equation, you would find your my time is elongated than yours too. Can both be correct?

An intuitive explanation be seen from below time measuring technique using light, given we all agree that speed of light is constant. hence, the distance for both relative observers are extended.

What if the clocks we have are not light clocks as above, say, an extreme case, biological clock, does it abide by the same rule, or run at same speed as light clock? The answer is yes. Hence, you can say the aging can be delayed in different frameworks.

This leads to another famous paradox – twin paradox. Using the above case, we know both can look down the other party as slower one, in biological case such as this twins, who is the younger one then? There is a clear answer – the one went out to the space is younger. The symmetrical characteristic doesn’t exist because the twin who went off, has to be landed back and experiencing acceleration and deceleration. Imagine there are triplets, two goes to opposite direction, one stays on earth, the two coming back will both be younger than the one staying on earth. microscopic level experiments have been conducted and proved this theory is correct.

Time distortion also leads to another perspective – length contraction. What can deduce is that the length at rest appears to be longer, while in the moving fashion, it will be shorter, opposite to the time distortion.

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