Help with integration conventions in "Spacetime and Geometry" by Sean Carroll

In summary, the conversation discusses the use of a formula (1.132) in section 1.10 of the book "Spacetime and Geometry" by Sean Carroll. The formula involves the Lagrange density, action, and vector potential. The integral in the formula can be written in different ways, with one option being ##d^4x## instead of ##dx\, dy\, dz\, dt##. The reason for this is to save space and not emphasize a particular order for the iterated integrals. However, there are certain properties that must be satisfied for the integral over a region to be equivalent to iterated integration over intervals.
  • #1
George Keeling
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I am reading Spacetime and Geometry by Sean Carroll. In section 1.10 on classical field theory, he uses this formula (1.132)
shortintegral.png

The curly L is a Lagrange density. S is an action, Φ is a vector potential.

Could the integral also be written as follows?
longintegral.png
 

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  • #2
Yes, assuming you mean ##dx\, dy\, dz\, dt##. One reason for using ##d^4x## instead is to save space, and to not emphasize any particular order for the iterated integrals. (There is also a little matter of the fact that the integral over a region is only equivalent to iterated integration over a series of intervals when the region and function being integrated satisfy certain properties, such as Fubini's theorem, but this is not really a big deal most of the time.)
 
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  • #3
I did indeed mean dx dy dz dt. Thank you, I liked your answer!
 
  • #5
That's enough thanks!
 

1. What is the definition of integration conventions in "Spacetime and Geometry"?

Integration conventions refer to the mathematical rules and techniques used to integrate various equations and functions in the context of spacetime and geometry. These conventions are important in understanding the physical laws and principles governing the behavior of objects in space and time.

2. Why are integration conventions important in "Spacetime and Geometry"?

Integration conventions are important because they allow us to solve complex equations and understand the behavior of objects in spacetime. By using these conventions, we can also make predictions and calculations about the motion and interactions of particles and systems in the universe.

3. What are some common integration conventions used in "Spacetime and Geometry"?

Some common integration conventions used in "Spacetime and Geometry" include the use of differential equations, the use of vector calculus, and the use of complex numbers and tensors. These conventions allow us to solve equations in curved spacetime and understand the effects of gravity on the behavior of particles.

4. How do integration conventions relate to general relativity?

Integration conventions are an essential part of understanding general relativity, as they allow us to solve the equations that describe the curvature of spacetime and the behavior of objects in it. By using these conventions, we can also make predictions about the gravitational effects of massive objects and the expansion of the universe.

5. Are there any challenges or controversies surrounding integration conventions in "Spacetime and Geometry"?

There may be some challenges and controversies surrounding integration conventions in "Spacetime and Geometry" due to the complexity of the subject matter and the ongoing research in this field. Some physicists may have different opinions or approaches to solving certain equations or using certain conventions, leading to debates and discussions within the scientific community.

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