# 1905 paper Does the Inertia of a Body Depend Upon its Energy-Content?

• edpell
In summary, Einstein used a Taylor expansion to derive E=mc2 which is good if v<<c but if v is not <<c then we need more terms. If we keep the first three terms then mc2 is divided by (1 + 3/4(v/c)^2 + 15/8(v/c)^4)have I done the Taylor series correctly? Do we all agree? This number of terms would be good as long as (v/c)^4 <<1 say up to v=0.4c. Is this correct?Your expansion is wrong. The second term is (1/2)(v/c)^2 which lead to (1/2
edpell
1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?

In his 1905 paper Einstein uses a Taylor expansion and keeps only the first term (an approximation) to derive E=mc^2 which is good if v<<c but if v is not <<c then we need more terms. If we keep the first three terms then mc^2 is divided by
(1 + 3/4(v/c)^2 + 15/8(v/c)^4)
have I done the Taylor series correctly? Do we all agree? This number of terms would be good as long as (v/c)^4 <<1 say up to v=0.4c. Is this correct?

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Your expansion is wrong. The second term is (1/2)(v/c)^2 which lead to (1/2)mv^2.
Also, generally in an expansion like that if the second term isn't already small, there's not much point in continuing the expansion.

clem the Taylor expansion is
1 + (1/2)(v/c)^2 + (3/8)(v/c)^4 + (15/8)(v/c)^6 + ...
the 1 is canceled by the -1 in the previous equation
a factor of (1/2)(v/c)^2 is factored out so that Einstein gets
K0-K1 = (1/2)(E/c^2)v^2 by taking only the first term he then equates m=E/c^2
or E=mc^2
if we factor out (1/2)(v/c)^2 the Taylor expansion minus one is to four terms
1+3/4(v/c)^2+(15/8)(v/c)^4 which must appear below the mc^2 yes? no?

You mean you are finding corrections to (1/2)mv^2? That is an archaic, clumsy way to proceed, only of interest to historians.

In his paper Einstein says "Neglecting magnitudes of fourth and higher orders we may place..."
He seems to say that it is an approximation. I never knew that E = m$$c^2$$ is an approximation that only hold if $$(v/c)^2$$ <<1.

edpell said:
In his 1905 paper Einstein uses a Taylor expansion and keeps only the first term (an approximation) to derive E=mc^2 which is good if v<<c but if v is not <<c then we need more terms. If we keep the first three terms then mc^2 is divided by
(1 + 3/4(v/c)^2 + 15/8(v/c)^4)
have I done the Taylor series correctly? Do we all agree? This number of terms would be good as long as (v/c)^4 <<1 say up to v=0.4c. Is this correct?

Einstein is not an easy read! One thing to keep in mind in reading Einstein is his humble respect for Newton. So he is careful not to say something like "I'm right, and Newton was wrong". You have to carefully read between the lines.

E=mc2 is the accurate result. Einstein is claiming that the Newtonian result, E = E0 + (1/2)mv2 is an approximation and true when v<<c.

He claims that the expansion is accurate and the Newtonian result, E = E0 + (1/2)mv2 is an approximation.

E0 doesn't really show-up in Newtonian energy because measurements of energy are always measurements of energy differences. For instance, when we calculate the energy in a battery we don't include the energy that is the mass of the battery. Since E0 is the rest mass energy, it doesn't usually change except when elementry particles change into other elementry particles. E0 drops-out of energy differerence equations except for these cases. So without atomic reactions it shows-up on both sides of an equation, and cancels:

$$E_0 + E_a = E_0 + E_b$$

$$E_a = E_b$$

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So the politics of physics is getting in the way of a clear statement of the physics of physics (in Einstein's paper)? Of course he ended up with lifetime tenure at the Institute for Advanced Studies so I would have to say he made the smart choice.

## 1. What is the significance of the 1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?"

The 1905 paper, also known as Einstein's mass-energy equivalence paper, is a groundbreaking work in the field of physics. It introduced the famous equation E=mc², which states that energy (E) and mass (m) are equivalent and interchangeable. This paper laid the foundation for the theory of special relativity and had a profound impact on our understanding of the universe.

## 2. Who wrote the 1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?"

The paper was written by Albert Einstein, a German theoretical physicist who is widely regarded as one of the most influential scientists of the 20th century. His work on the theory of relativity and the concept of mass-energy equivalence revolutionized our understanding of space, time, and the relationship between matter and energy.

## 3. What is the main idea of the 1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?"

The main idea of the paper is that mass and energy are two forms of the same thing. It proposes that a body's inertia, or resistance to change in motion, is directly related to its energy content. This revolutionary concept paved the way for further developments in physics and had a significant impact on our understanding of the universe.

## 4. How did the 1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?" impact the scientific community?

The paper had a profound impact on the scientific community and sparked a revolution in our understanding of the physical world. It challenged long-held beliefs and paved the way for further discoveries in the field of physics. Today, the concept of mass-energy equivalence is widely accepted and is a fundamental principle in modern physics.

## 5. What are some real-world applications of the ideas presented in the 1905 paper "Does the Inertia of a Body Depend Upon its Energy-Content?"

The paper's ideas have numerous real-world applications, including nuclear power, nuclear weapons, and medical imaging technologies such as PET scans. The famous equation E=mc² has also been used to develop technologies like nuclear fusion and fission, which have the potential to provide clean and abundant energy. Additionally, the paper's concepts have been applied in space exploration and have led to important discoveries in astrophysics and cosmology.

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