Geodesics in 4D Spacetime: An Overview

[Apashanka]

The geodesic for 2-D, 3-D are straight lines.
For a 4-D spacetime (x¹,x²,x³,t) what would be it's geodesic.??
The tangent vector components are $V^0=\frac{∂t}{∂λ} , V^i=\frac{∂x^i}{∂λ},i=1,2,3$ & $(\nabla_V V)^\mu=(V^\nu \nabla_\nu V)^\mu=0,(\nu,\mu=0,1,2,3)$

 

[Orodruin]

Geodesics are generalisations of straight lines to curved spaces. In any flat space they are going to be straight lines regardless of dimensionality.

 

[Apashanka]

Geodesics are generalisations of straight lines to curved spaces. In any flat space they are going to be straight lines regardless of dimensionality.

Sir one thing to clarify ,the curve traced by a particle in the 4-D space time in it's rest frame is a geodesic??
In rest frame $x^\mu$={$\tau,constt$}

 

[Dale]

the curve traced by a particle in the 4-D space time in it's rest frame is a geodesic??

Only if the particle is inertial. The worldlines of non-inertial particles are not geodesic. Also, the frame doesn’t matter

 

[PeroK]

The geodesic for 2-D, 3-D are straight lines.
For a 4-D spacetime (x¹,x²,x³,t) what would be it's geodesic.??
The tangent vector components are $V^0=\frac{∂t}{∂λ} , V^i=\frac{∂x^i}{∂λ},i=1,2,3$ & $(\nabla_V V)^\mu=(V^\nu \nabla_\nu V)^\mu=0,(\nu,\mu=0,1,2,3)$

It's not clear from your posts that you are studying GR in any systematic way. Your questions show a lack of knowledge of basic concepts such as spacetime, coordinate systems, reference frames and the relationship between the physics you are trying to learn and the mathematics that underpins it.

In terms of geodesics, you should be developing your knowldege in a systematic way. You may approach the subject in many ways, but not all at the same time. For example, you might meet geodesics as solutions to the equations of motion for a given spacetime geometry. This could be developed from the Lagrangian principle of maximal proper time. Then, you might meet geodescics again in the light of the covariant derivative and more generally in terms of differential geometry and manifolds.

But, you can't dive in from all directions at once.

I'd also say that unless you have a rock-solid understanding of Special Relativity and other prerequisites - such as Lagrangian mechanics - you are largely wasting your time, as any knowledge you gain with your haphazard approach will be fragile and transient.

You must find yourself a reputable textbook and work though it systematically, taking time out as necessary to study the physics and mathematics that are prerequisites to the study of GR.

 

[Apashanka]

Only if the particle is inertial. The worldlines of non-inertial particles are not geodesic. Also, the frame doesn’t matter

That means the world lines of all inertial frames are geodesic irrespective of frame

 

[Orodruin]

That means the world lines of all inertial frames are geodesic irrespective of frame

There is no such thing as the world line of an inertial frame. I am sorry, but as has been pointed out already in this thread and others, you seem not to have the proper foundation to stand on to learn relativity at this moment in time. I would suggest that you work through the basics before attempting relativity.

 

[Dale]

That means the world lines of all inertial frames are geodesic irrespective of frame

Frames don’t have worldlines. However, the integral curves of the tetrad of an inertial frame are all geodesics. Also, particles at rest in an inertial frame have geodesic worldlines

 

[PeterDonis]

the 4-D space time

Which 4-D spacetime? You keep on using the term "the 4-D spacetime" as though there were only one. There are an infinite number of them. You need to specify which one.

 

[pervect]

The geodesic for 2-D, 3-D are straight lines.
For a 4-D spacetime (x¹,x²,x³,t) what would be it's geodesic.??
The tangent vector components are $V^0=\frac{∂t}{∂λ} , V^i=\frac{∂x^i}{∂λ},i=1,2,3$ & $(\nabla_V V)^\mu=(V^\nu \nabla_\nu V)^\mu=0,(\nu,\mu=0,1,2,3)$

Step 1. Write down the geodesic equation - equation 56 in Carroll's treatment. Look it up.

What's the metric of the 4-d spacetime in question?
From the metric, compute the Christoffel symbols $\Gamma^i{}_{jk}$

Can you solve the resulting equations?