How can I expand this expression in powers of 1/c²?

In summary, the conversation is about the Taylor series expansion presented in James B. Hartle's book on gravity and the equation for total energy per unit rest mass. The equation in question is not a Taylor series expansion, but rather a multiplication of terms.
  • #1
Joao Victor
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Homework Statement
I was studying James B. Hartle's book - Gravity: an introduction to Einstein's General Relativity -, and in one section he expanded an expression in powers of 1/c², but I couldn't follow what he did. I do know this is related to a Taylor Series Expansion, and I do know how to construct the taylor expansion for a function f(x) [or even for a function of several variables], but I have no idea on how to proceed with this kind of expansion presented on the book.
Relevant Equations
e = \frac {mc^2+E} {mc^2} = 1 + \frac {2E} {mc^2} + ...
As I said before, I really have no idea on how to proceed.
 
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  • #2
You almost have it! [tex]\frac{mc^2+ E}{mc^2}= 1+ \frac{E}{mc^2}= 1+ \frac{E}{m}\frac{1}{c^2}[/tex]. That is the 'power series" [tex]a_0+ a_1\frac{1}{c^2}+ a_2\left(\frac{1}{c^2}\right)^2+ \cdot\cdot\cdot[/tex] with [tex]a_0= 1[/tex], [tex]a_1= \frac{E}{m}[/tex], and all other coefficients equal to 0.
 
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  • #3
Joao Victor said:
Problem Statement: I was studying James B. Hartle's book - Gravity: an introduction to Einstein's General Relativity -, and in one section he expanded an expression in powers of 1/c², but I couldn't follow what he did. I do know this is related to a Taylor Series Expansion, and I do know how to construct the taylor expansion for a function f(x) [or even for a function of several variables], but I have no idea on how to proceed with this kind of expansion presented on the book.
Relevant Equations: e = \frac {mc^2+E} {mc^2} = 1 + \frac {2E} {mc^2} + ...

As I said before, I really have no idea on how to proceed.
The total energy per unit rest mass, ##e##, is defined as
$$e = \frac {mc^2+E_\text{Newt}} {mc^2},$$ but in equation (9.53), the lefthand side is ##e^2##. That's why the two appears in the second term. It's not a Taylor series expansion. Just multiply it out.
 

Related to How can I expand this expression in powers of 1/c²?

1. How do I expand an expression in powers of 1/c²?

The first step in expanding an expression in powers of 1/c² is to rewrite the expression using the rules of algebra and simplify it as much as possible. Then, identify any terms in the expression that contain a c² in the denominator. These terms will become the base for the expansion. Finally, use the binomial theorem or other methods to expand the expression into a series of terms with increasing powers of 1/c².

2. Why would I want to expand an expression in powers of 1/c²?

Expanding an expression in powers of 1/c² can be useful in situations where c is a very large or very small number. This can make calculations and approximations easier, especially in physics and engineering problems where c is often a constant value.

3. Can I expand any expression in powers of 1/c²?

In theory, any expression can be expanded in powers of 1/c². However, the usefulness of the expansion may depend on the complexity of the expression and the desired level of accuracy. In some cases, it may be more practical to use other methods of approximation or simplification.

4. What is the significance of 1/c² in the expansion process?

The value of 1/c² is chosen as the base for the expansion because it is a common factor in many physical equations, particularly those involving the speed of light (c). By expanding an expression in powers of 1/c², the terms can be rearranged and simplified to reveal important relationships and approximations within the equation.

5. Are there any limitations to expanding an expression in powers of 1/c²?

One limitation is that the expansion may only be accurate for a certain range of values for c. For example, if c is a very small number, the expansion may only be valid for values of c that are close to zero. Additionally, the expansion may become increasingly complex as the number of terms increases, making it difficult to use in practical calculations.

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