Why does time have to be a complex (Minkowski metric)?

In summary, special relativity requires working with a four-dimensional space where time is represented as a complex variable. This leads to the Minkowski metric, with a negative time component and positive space components. The use of complex numbers is a mathematical convention, with no specific physical meaning. The Lorentz transformation is the key physical concept behind this representation. The Bondi k-calculus is a useful tool for understanding the origin of this signature.
  • #1
feryopo
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I am studying special relativity, and I found that you have to work with a four dimensional space, where time is a complex variable. If you do so, you end up with the Minkowski metric, were the time component is negative and space components are positive (or vice versa). My questions are, why do we have to work with a complex time component, what is the physical meaning of time being complex and mathematically how can you deduct the Minkowski metric?
Thank you very much in advance.
 
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  • #2
Welcome to PF;
You will learn to discard this notation and use an object called the metric tensor instead.
Minkovski coordinates uses a complex time coordinate because of work done by Poincare, that the 4D euclidean metric had to subtract time-squared, where the usual pythagorean approach would suggest adding time squared. This makes lorentz transformaton look like ordinary rotations as opposed to something weird and mysterious.

There is no special physical significance to the time component being represented as a complex number - the convention is exploiting the properties of complex numbers to shortcut some maths. The physics is the Lorentz transformation.
 
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  • #4
##x_4 = ict ## was the invention of Hermann Minkowski in 1907 (published in 1908). I wonder why people still teach it nowadays.
 
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  • #5
dextercioby said:
##x_4 = ict ## was the invention of Hermann Minkowski in 1907 (published in 1908). I wonder why people still teach it nowadays.
The Law of Relativistic Inertia? o0)
 
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Related to Why does time have to be a complex (Minkowski metric)?

1. Why is time considered a "complex" concept in physics?

Time is considered complex in physics because it is not a tangible entity that can be measured or observed directly. It is a fundamental concept that is necessary for understanding the behavior of matter and energy in the universe, but it is not fully understood or explained by any one theory or equation.

2. What is the Minkowski metric and how does it relate to time?

The Minkowski metric is a mathematical tool used to describe the geometry of spacetime in Einstein's theory of relativity. It is a four-dimensional metric that combines space and time into a single entity, allowing for the measurement of distances and intervals in spacetime. In this metric, time is treated as a fourth dimension, and is represented as a complex number to account for its unique properties in relation to space.

3. How does the concept of time change in different reference frames?

According to Einstein's theory of relativity, the passage of time is relative and can vary depending on the observer's reference frame. This means that time can appear to pass at different rates for different observers, depending on their relative motion and gravitational fields. This concept is known as time dilation and is an essential component of understanding the complex nature of time.

4. What other theories attempt to explain the concept of time?

Aside from the Minkowski metric in relativity, there are several other theories that attempt to explain the nature of time. These include the theory of quantum mechanics, which suggests that time is not continuous but is instead made up of discrete moments. Another theory, known as the "block universe" theory, proposes that time is an illusion and that all events, past, present, and future, exist simultaneously.

5. How does the concept of time impact our understanding of the universe?

The concept of time is essential in understanding the behavior and evolution of the universe. It allows us to measure the duration of events, track changes, and make predictions about the future. Our understanding of time has also led to significant advancements in technology and the development of theories such as relativity and quantum mechanics, which have revolutionized our understanding of the universe and its origins.

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