Relativity: position/acceleration 4-vectors to Newton's laws

In summary, there are a few ways to approach the problem of relating the given equations for cosh and sinh to the position and acceleration 4-vectors, such as using Taylor series expansions or considering the physical meaning of the first term in the vectors. Keep exploring and experimenting until you find a solution that makes sense to you.
  • #1
Jeff Knapowsk
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Homework Statement


Given the binomial expansions for sinh and cosh, so that the acceleration and position 4-vectors reduce to x=1/2 a t^2 for small t.


Homework Equations



Okay, so we have cosh (x) = 1 + x^2/2 + ... and
sinh(x) = x + x^3/3 ...

We have the position (sigma) 4-vector = (B*sinh(k*tau): B*(cosh(k*tau) -1) and
the acceleration (alpha) 4-vector = (-B*k^2*sinh(k*tau): -B*k^2*(cosh(k*tau))


The Attempt at a Solution



Assume k*tau << 1 and let's ignore higher order terms above x^2.

so, cosh(k*tau) = 1+ 1/2 (k*tau)^2 and sinh(k*tau) = k*tau

So, know I'm somewhat stumped. What are we setting equal to what? What is the small in both of these equations? t? or tau?

I know that the first term in the position 4-vector is c*t. So I could say:

B*k*tau = ct. And the last term in the acceleration 4-vector (alpha) is a. So I could say

-B*k^2*(1 + (k*tau)^2/2) = a. But that gets me nowhere.


Anyone got a thought? What's the direction here?

I read in another thread that we should try to understand the physical meaning of the first term of the acceleration or velocity 4-vectors. It doesn't have a clear explanation. It just sort of "is". Right?
 
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  • #2



Hello, it seems like you're trying to find a way to relate the given equations for cosh and sinh to the position and acceleration 4-vectors. One approach you could take is to use the Taylor series expansion of cosh and sinh to approximate the position and acceleration 4-vectors for small values of t or tau. This would allow you to see how the position and acceleration vectors reduce to x=1/2 a t^2 for small t or tau.

Another approach is to consider the physical meaning of the first term in the acceleration or velocity 4-vectors. In this case, the first term represents the initial velocity or acceleration of the object. By setting this term equal to ct or a, you can see how the equations reduce to x=1/2 a t^2 for small t or tau.

I hope this helps guide your thinking and leads you to a solution. Keep exploring and experimenting with different approaches until you find one that works for you!
 

1. What is the concept of relativity in physics?

Relativity is a theory in physics that explains the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, and the speed of light is constant regardless of the observer's relative motion. The theory of relativity is divided into two parts: special relativity and general relativity.

2. What are position and acceleration 4-vectors in relativity?

In relativity, position and acceleration 4-vectors are mathematical representations of an object's position and acceleration in four-dimensional spacetime. These vectors include both spatial and temporal components, allowing for a more comprehensive understanding of an object's motion in the context of relativity.

3. How does relativity relate to Newton's laws of motion?

Relativity builds upon and expands upon Newton's laws of motion. While Newton's laws are valid for low speeds and weak gravitational fields, relativity provides a more accurate explanation of motion at high speeds and in strong gravitational fields. It also introduces the concept of spacetime curvature, which plays a significant role in general relativity.

4. Can relativity be applied to everyday situations?

Yes, relativity has been extensively tested and verified through various experiments and observations. While it may seem like a complex theory, its principles and equations can be applied to everyday situations, such as GPS technology, which relies on the precise calculations of relativity to function accurately.

5. What are the practical implications of relativity?

The theory of relativity has had numerous practical implications, including advancements in technology, such as GPS, as well as a deeper understanding of the universe and its workings. It has also led to the development of other important theories, such as quantum mechanics, which have further expanded our understanding of the physical world.

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