Minkowski Space

Introduction

In general, we deal with three dimensions of space that are very common and related to our daily macroscopic life. However, according to Einstein’s theory of relativity, the time coordinate is also effective in our daily life. This fact was coined and proved by scientist Hermann Minkowski. However, the birth of this concept was from the experiment of Poincare that was introduced in 1905 but then it was in a brief or we can say in raw form. Later, in 1908, the concept was explained by Hermann Minkowski.

He described that Minkowski’s space as an imaginary space that has four dimensions and that satisfies the postulate of the Theory of Relativity. This concept is a mathematical form of the theory of relativity.

What do you mean by Minkowski Space?

When we talk about a space we think about its dimension. We all know very well about the three dimensions of space. Here, the term Minkowski space is a new concept coined by Hermann Minkowski when he was studying the special theory of relativity. He analyzed that the special theory of relativity can be explained in a better manner in a four-dimensional space and explained the concept of spacetime dimension. After the surname of the scientist, the space is titled Minkowski Space.

By definition, we can say that the Minkowski Space is a four-dimension expansion that has three dimensions of space and one dimension of time. It is a concept of mathematical physics which is used for the explanation of Einstein's theory of relativity. When we drive a derivation of Minkowski space we find that the result is equal to a postulate of the theory of relativity. Space-time has four dimensions but it treats the time dimension differently than the other three dimensions; that's why it is consistent with every reference of the frame. The signature of Minkowski Space is denoted as (− + ++). The nature or structure of Minkowski’s space is flat means to say that this space is flat like a page of a book.

Minkowski Space-Time Geometry

Space-time geometry was introduced by a scientist named Hermann Minkowski. He developed this geometry for Maxwell’s equations of electromagnetism. He found that if we take two events the space-time interval between them does not affect on the inertial reference frame.

This geometry was a special postulate of Einstein’s Theory of Relativity. As this space- time geometry has four dimensions that are associated with four coordinates which can be presented as x, y, z, and ct.

In the mathematical form, we can present all the four coordinates, such as

$$\mathrm{(x_{1},x_{2},x_{3},x_{4})}$$

Here, we use the 4^th dimension in the same unit as the three others have. So, the time coordinate is taken as the speed of light in unit time or ct.

Hence, if we measure the differential length of an arc in the plane of space-time we get,

$$\mathrm{\partial\:s^{2}\:=\:\partial\:x^{2}\:\:+\:\partial\:y^{2}\:\:+\:\partial\:z^{2}\:-\:c^{2}\:\partial\:t^{2}}$$

The theory states that space-time is flat, we can prove this following way by using the above equation

$$\mathrm{G_{uv}\:=\:[-\:1\:0\:0\:0\:0\:1\:0\:0\:0\:0\:1\:0\:0\:0\:0\:1]}$$

Interactive Minkowski Diagram

The interactive Minkowski diagram is very important if we want to know the relation between any two bodies which are moving with different velocities. This diagram provides us with the effect of relativity in space-time coordinates.

This diagram has different lines of different colors and shapes. The diagram also has a red dot. Every line and point has its function and value. Let’s have a brief look at these lines and points.

• The black line represents the position axes(𝑥, 𝑡) of the body which is at rest.

• The blue line represents the location of another body that is in motion and the velocity of the second body with respect to the body which is at rest is v. Here, the velocity v is not equal to the speed of light but it is a fraction of light’s speed.

• Here, is the only point denoted by a Red dot, which is a point where something happens at a specific instant in time. But it is important to notice that the red dot is at coordinate(𝑥𝑎𝑡𝑎) from the body at rest and the same point is at coordinate(𝑥′𝑎𝑡′𝑎) from the second body. This difference in measurement can be observed with the help of a dotted line.

• The yellow line denotes the way followed by light.

By using the Minkowski diagram and relativity equation, we can find the path between the origin and the point where the event happens.

Importance of Minkowski Space

Minkowski Space is very useful and important in the terms of the theory of relativity. This Minkowski space is very helpful in representing the Lorentz transformation in the frames.

The Minkowski diagram is also very useful as it uses a single worldline for all sets of coordinates. Mostly we deal with a pair of coordinates so both coordinates have a single worldline.

Conclusion

This tutorial is completely based on the Minkowski space. Here, we learned about the facts that reveal the reason and result of the discovery of Minkowski Space. Moreover, we also came to know how the theory of relativity works and describe the impact of time coordinates on the events that are common for everyone. Also, we came to know about a space having four dimensions, and which is flat.

FAQs

1. What does the Theory of Relativity state?

The theory of relativity state some important statements.

• The laws of physics are the same for every observer who belongs to any inertial frame of reference.

• The speed of light in the vacuum is the same and equal for everyone.

• The speed of the clock is affected by gravity and becomes slow when it is in a gravitational well.

• Light rays are also get affected by the influence of gravity.

• The universe is expanding continuously at a speed greater than the speed of light.

2. Who gives the Theory of Relativity?

The Theory of Relativity was proposed by Albert Einstein.

3. What do you mean by Frame of Reference?

In simple words, we can say that the frame of reference is a set of position coordinates to which we measured the velocity or displacement of any moving or stationary object.

4.What is the difference between an inertial frame of reference and a non-inertial frame of reference?

In an inertial frame of reference, newton’s laws are applicable whereas in a non- inertial frame of reference newton’s laws of motion are not applicable.

5. What is Lorentz transformation?

Lorentz transformation is an equation that defines the relationship between a set of two coordinate frames. Here, both are moving with a constant velocity. Moreover, both frames are relative to one another.