Einstein's theory of special relativity is a scientific theory that focuses on how time, speed, and space interact and how the laws of physics are the same in all inertial frames. The first step to take when studying Einstein's theory of special relativity is to study the two postulates that Einstein put forward for special relativity.
What are Einstein's two postulates about special relativity?
Einstein's theory of special relativity covers movement across colossal distances with a cosmic speed, the speed of light.
Einstein stated that the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is constant regardless of the observer's speed. He backed up his theory with two simple postulates and cautious consideration of how measurements are made.
Einstein's first postulate
Einstein's first postulate is about reference frames. A reference frame is a viewpoint used to determine the movement of an object. According to the first postulate, all velocities are measured relative to some frame of reference.
So what are inertial frames of reference? An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion continues at a constant speed in a straight line unless impacted by an outside force. This may sound familiar, since Newton's first law of motion says the same thing because it is based upon inertial frames of reference. Let's look at some examples.
- If a car is going on a road, its motion is measured relative to the road it is going on.
- If you throw a ball off a large cliff, the motion of the ball is measured relative to your standing point.
The laws of physics are the same and can be stated in much more simple forms in all inertial frames of reference than non-inertial ones.
Imagine you are in the back seat of a car, and the car is driving at a constant speed. The laws of physics appear to function the same way as when you're standing on the surface of the earth. Things get a little trickier when the car is moving.
F, the net force of an object, does not equal the multiplication of mass and acceleration (ma) in many instances like these. Instead, it equals ma plus a fictitious force. Imagine the car is going at a velocity of 10 km/h and you throw a ball inside the car with a velocity of 2 km/h. You will see the ball moving at a velocity of 2km/h while an observer standing on the side of the road will see the ball moving at a velocity of 12km/h.
The famous equation \(E = mc^2\), which is discovered by using the formula for the force in a near light moving frame, is one of the most notable implications of this postulate.
Einstein's second postulate
Einstein's second postulate about the theory of special relativity is about the speed of light and it being a constant independent of the reference frame. Even though in the late 19th century the major tenets of classical physics predicted that light travels at \(c = 3.00 \cdot 10^8 m/s\) in a vacuum, they didn't specify the frame of reference in which light has this speed.
There was a contradiction between this prediction and Newton's laws in which velocities add like simple vectors. If this were true, then the observer moving at a velocity of c would see light as stationary, which went against Maxwell's equations. So Einstein came to the conclusion that an object with mass can't travel at speed c.
As a result of this reasoning, light in a vacuum must always move at c relative to any observer. Maxwell's equations are correct, and Newton's addition of velocities is incorrect in the case of light.
The general outcome of the second postulate is that in a vacuum, the speed of light is constant at \(c = 3.00 \cdot 10^8 m/s\).
The speed of light is slower in matter, as the impact of the index of refraction from the law of refraction shows. Also, this is where special relativity is different from general relativity. Only unaccelerated motion is covered by special relativity, while accelerated motion is covered by general relativity.
What is time dilation, and length contraction?
Can time intervals or the distance travelled be different from one observer to another? Normally we expect the answer to be no, but in some circumstances, the answer may be yes to both of these questions.
The elapsed time of a class is the same for all students (observers). However, at relativistic speeds, the observer's relative velocity and the event being observed impact the elapsed time.
Time dilation
Time dilation is a concept that occurs when one observer moves through space relative to another observer, causing time to flow more slowly. Let's imagine an observer is moving at v, and the proper time is Δt0, which is the time the observer measures at rest relative to the event being observed. This proper time is connected to the time Δt that an observer on Earth measures. You already know that c stands for the speed of light and is a constant that can be determined below. The equation below explains the relationship between them:
\[\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} = \gamma \Delta t_0\]
Where
\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]
Length contraction
When traveling at everyday speeds, observers can and will measure the length of a car journey the same, always independently of the velocity observers are traveling at. But that does not apply under relativistic speeds, which are close to the speed of light.
Under relativistic speeds, the length that an object is moving through is measured to be less than its proper length, this is called length contraction. The proper length (L0) is the length obtained when the distance between two points is measured by an observer who is at rest relative to both points. Take a look at the example below to get a better understanding of the concept:
Let's imagine a spaceship is observed by someone on Earth and travels at a velocity of 0.750c for 9.05µs from the moment it is spotted until it vanishes. It covers the following distance:
\(L_0 = v \Delta t = (0.750) \cdot (3.00 \cdot 10^8 [m/s]) \cdot (9.05 \cdot 10^{-6} [s]) = 2.04 [km]\)
The observer on Earth will observe the proper length, Camacho - StudySmarter Originals
This is relative to Earth. In the spaceship's frame of reference, its lifetime is Δt0:
\(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1-\frac{(0.950 c)^2}{c^2}}} = 1.512\)
In the question, it says 'to an observer on Earth' so 9.05µs is Δt and you need Δt0 to find the length from the spaceship's reference. As we saw before:
\(\Delta t = \gamma \cdot \Delta t_0\)
Let's put the knowns in place, to get \(\Delta t_0 = 5.99 \mu s\)
Now you can find the length relative to the observer inside the ship (L)
\(L = v \Delta t_0 = (0.750) \cdot (3.00 \cdot 10^8 [m/s]) \cdot (5.99 \cdot 10^{-6} [s]) = 1.348 km\)
The observer in the spaceship will observe the length relative, Camacho - StudySmarter Originals
Finally, the distance between when the spaceship appears and when it vanishes, is determined by who is measuring it and how the observer moves relative to it.
While the distances are observed the same by all observers in everyday life, they can be observed differently in relativistic speeds.
What is relativistic energy?
The law of the conservation of energy states that energy has many forms, and each form can be converted to another without being destroyed; the energy in a system remains constant.
Energy is still conserved relativistically if its definition is changed to include the possibility of mass converting to energy. If we define energy to include a relativistic element, Einstein demonstrated that the law of the conservation of energy can be applied relativistically, which led to the concepts of total energy and rest energy.
Total energy
Total energy E can be defined as
\[E = \gamma mc^2\]
m is mass in kg.
c is the speed of light in m / s.
As you may remember we defined \(\gamma\) as:
\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]
You can see that E is related to relativistic momentum. But notice that if the velocity is zero, it will not equal zero but 1. Which will then be called rest energy rest energy.
Rest energy
The rest energy is actually defined as the famous equation below.
\[E_0 = mc^2\]
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, is the rest energy in joules.
This is the correct version of Einstein's most famous equation, which demonstrated for the first time that energy is proportional to an object's mass when it is at rest. When energy is stored in an object, for example, its rest mass increases. This also suggests that energy may be released by destroying mass. Let's look at the example below to understand the concept better.
Calculate the rest energy of a 1 gram mass.
Solution:
Let's apply the equation.
\[E_0 = mc^2\]
In the question, m is given as 1 gram which is equal to \(1 \cdot 10^{-3}\) kg.
\(E_0 = (1 \cdot 10^{-3}) \cdot (3 \cdot 10^8)^2 = 9 \cdot 10^{13} kg \cdot m^2/s^2\)
Let's convert the unit to joules to see how much energy there is. We know that \(1 kg \cdot m^2/s^2 = 1 \space Joule\).
So the result is \(E_0 = 9 \cdot 10^{13} J\)
This is an enormous level of energy. It is about twice the amount of energy released by the Hiroshima atomic bomb.
Einstein's Theory of Special Relativity - Key takeaways
- Einstein's theory of special relativity is a scientific theory that focuses on how time, speed, and space interact and how the laws of physics are the same in all inertial frames.
- Einstein's first postulate upon which he based the theory of special relativity is about reference frames. Einstein's second postulate is about the speed of light.
- Length contraction occurs when the length that an object moving on at relativistic speeds is measured to be less than its proper length.
- Time dilation is a concept that occurs when one observer moves through space relative to another observer, causing time to flow more slowly.
- Relativistic energy states that the energy is conserved relativistically if its definition is changed to include the possibility of mass converting to energy.
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