How Do You Calculate 4-Acceleration in Schwarzschild Coordinates?

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In summary, the discussion involves using Schwarzschild coordinates in the equation of motion, which leads to a coordinate acceleration of zero. The Christoffel symbol is then used to derive the four-acceleration, which results in a minus sign due to a mistake in the sign conventions. Correcting this leads to the correct answer.
  • #1
etotheipi
Homework Statement
Find the 4-acceleration of a particle at rest on the surface of a non-rotating planet of mass ##M##
Relevant Equations
N/A
I'd just like a bit of guidance here, because I'm not sure if what I'm doing is correct. First, the equation of the motion,$$A^{\mu} = \frac{dU^{\mu}}{d\tau} + \Gamma^{\mu}_{\sigma \rho} U^{\sigma}U^{\rho}$$I decided to use the Schwarzschild coordinates ##(t,r,\theta, \phi)##, and in these coordinates I can take ##U^r = U^{\theta} = U^{\phi} = 0##. Since the coordinate acceleration is zero we get$$A^{\mu} = \Gamma^{\mu}_{tt} U^t U^t$$The relevant Christoffel symbol$$\Gamma^{\mu}_{tt} = \frac{1}{2} g^{\mu m} \left( \frac{\partial g_{mt}}{\partial x^t} + \frac{\partial g_{mt}}{\partial x^t} - \frac{\partial g_{tt}}{\partial x^m} \right) = - \frac{1}{2} g^{\mu r} \frac{\partial g_{tt}}{\partial r}$$where I used that the metric is time-independent and that ##g_{tt}## depends only on ##r##, i.e.$$g_{tt} = - \left(1- \frac{2GM}{r} \right) \iff \frac{\partial g_{tt}}{\partial r} = - \frac{2GM}{r^2}$$and so$$A^{\mu} = g^{\mu r} \frac{GM}{r^2}$$So for instance, if ##\mu = r##, we get$$g^{rr} = \left(\frac{2GM}{r} - 1 \right)^{-1} = - \left( 1 + \frac{2GM}{r} \right) + \mathcal{O}(r^{-2})$$and then$$A^{\mu} = - \left( 1 + \frac{2GM}{r} \right) \cdot \frac{GM}{r^2} = - \frac{GM}{r^2} + \mathcal{O}(r^{-3})$$I wondered, why did I get a minus sign? I expected the four-acceleration to have a positive radial component. I wonder if I've mixed up the metric signature somewhere, but don't know where. thanks!
 
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  • #2
Not quite. You seem to have assumed that ##U^t=1##, but ##g_{\mu\nu}U^\mu U^\nu=-1##, so ##U^t=1/\sqrt{|g_{tt}|}## (edit: and not ##U^t=1/\sqrt{g_{tt}}## as I originally wrote) in this case. (Edit: if I finally kept the sign conventions straight, anyway...)
 
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  • #3
Thanks! I also can't seem to get the signs of the metric components right. Starting from where it went wrong,$$A^{\mu} = g^{\mu r} \frac{GM}{r^2} U^t U^t = \frac{g^{\mu r}}{g_{tt}} \frac{GM}{r^2}$$then with$$g^{rr} = (g_{rr})^{-1} = \left(1- \frac{2GM}{r} \right)$$and$$g_{tt} = -\left( 1- \frac{2GM}{r} \right)$$and in that case $$A^{r} = - \frac{GM}{r^2}$$ exactly, but now still with an incorrect sign :cry:
 
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  • #4
I made a mistake in my last post - I said that ##U^t=1/\sqrt{g_{tt}}##, but it's ##1/\sqrt{|g_{tt}|}##, so ##g_{tt}U^tU^t=\mathrm{sgn}(g_{tt})=-1##. When I correct that I get the right answer. My Christoffel symbols agree with yours, so I think you've just made the same mistake.

Sign conventions are an unbelievable pain. I like +--- because four velocities have positive moduli, and that fits in my brain better.
 
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  • #5
Ah, okay, gotcha. Yeah, that's annoying. In the future I'll make more of an effort to make an even number of sign errors.

Also, have a nice new year's eve! 🥃🍾🍻🤮
 

Related to How Do You Calculate 4-Acceleration in Schwarzschild Coordinates?

1. What is 4-acceleration?

4-acceleration is a concept in physics that describes the rate of change of an object's 4-velocity with respect to proper time. It takes into account both the object's change in speed and change in direction.

2. How is 4-acceleration calculated?

4-acceleration can be calculated using the formula aμ = dUμ/dτ, where aμ is the 4-acceleration, Uμ is the 4-velocity, and dτ is the proper time.

3. What are the units of 4-acceleration?

The units of 4-acceleration depend on the units used for 4-velocity and proper time. In the SI system, 4-acceleration has units of meters per second squared (m/s²).

4. How does 4-acceleration relate to Newton's laws of motion?

4-acceleration is a relativistic concept that takes into account the effects of special relativity, such as time dilation and length contraction. It is not directly related to Newton's laws of motion, which were developed for non-relativistic situations.

5. What are some real-world applications of 4-acceleration?

4-acceleration is commonly used in the field of astrophysics to study the motion of objects in space, such as planets and galaxies. It is also used in particle physics to describe the behavior of subatomic particles. Additionally, 4-acceleration has applications in engineering, particularly in the design of spacecraft and other high-speed vehicles.

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