Why Are the Space-Like Components of the 4-Acceleration Non-Zero in the IRF?

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In summary, the conversation discusses the components of 4-acceleration in the instantaneous reference frame (IRF) and how they are affected by the values of 4-velocity. The conversation also provides an example of a ball thrown in the air to illustrate how the 4-acceleration can still have a non-zero magnitude even when the 4-velocity is constant. Additionally, it mentions that the 4-acceleration is always orthogonal to the 4-velocity.
  • #1
Lorna
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Homework Statement



I read in my textbook that "in the instantanuous reference frame (IRF) [tex][u^{'\gamma}]=(c,\overline{0}) [/tex] imply that the components of the 4-acceleration in the IRF are [tex] [a^{'\gamma}]=(0,\overline{a'}) [/tex]"

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I don't know why they got a' for the space-like components of the 4-acceleration. Isn't the accelaration = du/d[tex]\tau[/tex] if so then using [tex][u^{'\gamma}=(c,\overline{0}) [/tex] we get zero for both the time-like and space-like components of the 4-acceleration.
 
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  • #2
Think back to first-year calculus, and suppose [itex]f\left(x\right) = x^3[/itex]. Then, [itex]f\left(2\right) = 8[/itex], and [itex]8[/itex] is a constant, so

[tex]\frac{d}{dx}8 = 0,[/tex]

but

[tex]\frac{df}{dx}\left(2) = 12 \ne 0.[/tex]

Just because the 4-velocity has a particular value (which could have zero spatial part) in a particular frame at a particular instant doesn't mean that the 4-velocity is constant. If the 4-velocity is not constant, then the 4-acceleration doesn't have to be zero.

Throw a ball in the air. When the ball reaches its greatest height, its (spatial) velocity is zero, but its (spatial) acceleration still has magnitude [itex]g[/itex]
 
  • #3
In addition, note that the 4-acceleration of a particle is orthogonal to its 4-velocity. (Why?)
 
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  • #4
That example was very helpful. thanks a lot.

Thanks robphy too.
 

FAQ: Why Are the Space-Like Components of the 4-Acceleration Non-Zero in the IRF?

What is the definition of the 4-acceleration?

The 4-acceleration is a four-dimensional vector quantity that describes the rate of change of a particle's 4-velocity with respect to proper time.

What are the components of the 4-acceleration?

The components of the 4-acceleration are the time component (the rate of change of the time component of the 4-velocity), and the spatial components (the rates of change of the spatial components of the 4-velocity).

How is the 4-acceleration related to the 4-velocity?

The 4-acceleration is the derivative of the 4-velocity with respect to proper time. This means that the 4-acceleration describes the change in the direction and magnitude of the 4-velocity as the particle moves through space and time.

What is the significance of the 4-acceleration in relativity?

The 4-acceleration is a crucial quantity in relativity because it is what allows us to understand how objects move through space and time. It is a fundamental part of Einstein's theory of general relativity and is essential for understanding the curvature of spacetime.

How can the 4-acceleration be measured or calculated?

The 4-acceleration can be calculated using the equations of motion in special or general relativity. It can also be measured using specialized instruments, such as accelerometers, which measure the acceleration of an object in different directions.

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