Before talking about relativisic kinematics, let's briefly run through Einstein's theories of Special and General Relavitity. Einstein's theory of special relativity was derived from the following two postulates:

- Postulate 1: The laws of physics are the same in all inertial reference frames.
- Postulate 2: The speed of light in a vacuum is equal to the value c, independent of the motion of the source

To summarize, the laws of physics do not change regardless of what your perspective is. Nothing can go faster than the speed of light in the universe. Everyone *always* measures the speed of light to be the speed of light regardless of how fast or slow one is moving.

However, this theory of relativity does not take into account acceleration. It took Einstein another 10 years to develop his General Theory of Relativity.

General Relativity tells us:

- Space and time are interwoven into a 4-dimensional fabric known as space-time. This fabric can be bent and stretched which causes changes in how we perceive time.
- Massive objects like black holes cause distortions in space-time.

*Keep these ideas in mind*

Relativistic kinematics sounds like a complicated term that the average person would be intimidated by. However, it basically means how the laws of physics operate at incredibly fast speeds. But what is considered a fast speed? You might have heard before that the speed of light is the speed limit of the universe. This means that no object that has mass can reach or exceed that speed (**Sidenote**: This law only says that anything *inside* space cannot exceed this speed. Space itself is actually expanding at an accelerated rate that is faster than the speed of light!). What can reach the speed of light? Only particles that have a mass of zero can reach the speed of light, therefore only photons, the particles of light, are able to reach this speed. Before we go any further, we need to define a term known as the Lorentz factor which plays a key role in understanding relativistic kinematics. The Lorentz factor tells us how time, length, and relativistic mass change for an object in motion.

$\beta = \frac{v}{c}$ tells you the percentage of light speed that the object of interest is moving at.

Time can also be dilated by massive objects. If you have seen the movie *Interstellar* then you have witnessed the effects of general relativity. Being close to a massive object like a black hole (which is millions to billions times the mass of the sun) causes space-time to be stretched massively! Because of this, time happens a lot slower than it would if you were on Earth. The more massive the object, the more space-time is stretched. The more space-time is stretched, the longer it is going to take for time to occur for you.

Fun Fact: Since sea level is closer to the center of the Earth where gravity is the strongest, time is technically happening slower than it is on a mountatin. The time difference is infinitesimal, but still not the same!

Check out this great visual below of the effect mass has on space-time!

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In [7]:
```from IPython.display import Image
from IPython.core.display import HTML
Image(url= "http://sci.esa.int/science-e-media/img/72/ESA_LISA-Pathfinder_spacetime_curvature_above_orig.jpg")

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As objects approach the speed of light, they contract in the direction of motion! In other words, when a stationary observer measures the length of a rod moving at near the speed of light, they see it as smaller than it is in its actual reference frame. In the reference frame of the rod, the rod is its actual length and time runs normally. It's only when we observe moving reference frames that we notice these differences. The equation to calculate length contraction is:

$$L=\frac{1}{\gamma}L_p = \sqrt{1-\frac{v^2}{c^2}} L_p$$where $L_p$ is the proper length, or the length of the object at rest in a reference frame. It makes sense mathematically that the length is smaller in a moving reference frame because we are dividing by gamma, which is always greater than 1 unless v=c. The effect speed has on length contraction is shown with the baseball figure below.

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In [8]:
```from IPython.display import Image
from IPython.core.display import HTML
Image(url= "http://www.patana.ac.th/secondary/science/anrophysics/relativity_option/images/length_cont1.JPG")

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Out[8]:
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Many of you have probably heard of Albert Einstein's famous equation: $E=mc^2$ It turns out that this isn't always true. $E=mc^2$ is only true for objects at rest! The equation for objects in motion is $E=\gamma m c^2$ where the Lorentz factor comes into play again. When we multiply $mc^2$ by the Lorentz factor, the mass increases. Therefore, as objects approach the speed of light, they become more massive.

Our Newtonian understanding of momentum is also incorrect. In general physics classes, we learn that $p=mv$, however the relativistic momentum changes at near light speeds. Therefore our new formula for momentum is: $$p =\gamma mv$$

In particle physics, we often don't know the masses of particles, but know the energies and momenta in the x, y, and z directions. CERN can reach energies of up to 13 TeV! At particle accelerators like CERN, we are able to use machines that can accelerate particles to near the speed of light and then collide them. We then study the byproducts of these collisions. Since some particles have shorter lifetimes than others, it is possible that some decay before they reach our detectors. We then have to trace our decay products back to something called the displaced vertex (where the original particle was in space right before it decayed). We are able to calculate the invariant mass (mass of particles at rest in their reference frame) using the following formula:

$$E^2=m^2c^4+p^2c^2$$This equation allows us to calculate the masses of particles that arise from these collisions.

This equation also proves that although photons have no mass, they still have a momentum! The energy for photons is defined as: $$E = pc$$ We can manipulate the formula and solve for the rest mass as follows:

$$m = \sqrt{E^2 - (p^2_x + p^2_y + p_z^2)}$$Now we can apply our knowledge of time dilation! Since these particles are moving at near the speed of light, time is happening slower for them relative to us. Essentially, particles will have a longer lifetime when they travel at higher speeds. Because of this, they are able to travel greater distances! We can utilize time dilation to observe some particles before they decay and therefore learn more about them.

In some cases, particles can move faster than the speed of light in different materials! When this happens, a cone of light is given off in the form of Cherenkov radiation. The angle at which this light is given off is directly related to the velocity. Knowing the velocity combined with the momentum, we can measure the mass.

- The speed of light is the same for all observers.
- Massive objects bend the fabric of space-time.
- Crazy things begin to happen when you approach the speed of light!
- Time happens slower
- Objects contract in length
- Mass increases

- We can measure masses of particles given energies and momenta