Can we build a four dimensional compass?

[SleepDeprived]

The reason I ask is that when we use the surface of the Earth as an analogy, if we travel in one direction then eventually we will come back to where we started. But to do that we need a compass or some similar instruments to keep track of our path to make sure we are always in a "straight" line otherwise we may come back at some other place. Obviously the flat land people cannot use our GPS since they cannot have access to the satellites since the satellite would be somewhere above the flat land at some other dimension and that would be cheating.
So like the people in the flat land, we are trapped in a three dimension space in a supposedly four dimensional space time universe, and if the theory is correct and if we travel in a "straight line" we will eventually come back to where we started (assuming the universe is close and finite). But a "straight line" in a four dimensional space time is not the same "straight line" like on the surface of the earth. I mean when our spaceship takes off from earth, it will take off in a curved trajectory but even after that our spaceship will be affected by gravity from other celestial bodies the path will not be straight in a three dimensional sense. But you get my point. So the only way to know if our spaceship travels in a four dimensional straight line is the need of having a four dimensional compass to keep track of our path, but to visualize such a thing is a little bit foggy for my mind.
The closest thing I can visualize is shining a light beam into space and wait for it to come back again assuming the universe is close and finite and not expanding faster than the speed of light. We will inevitably use the flat Earth as an analogy, so when I look at the WMAP picture, I assume that any point in the universe, any observe will get the exactly the same picture of the cosmic radiation, just as any person on the Earth flat land will see the same picture.
Someone had said that if you can look far enough, you will eventually see the back of your head. I am still having a hard time able to visualize it though.

 

[jedishrfu]

There's been some developments to map many X-ray pulsars to be used to pinpoint your location in space.

https://en.m.wikipedia.org/wiki/X-ray_pulsar-based_navigation

 

[Orodruin]

we are trapped in a three dimension space in a supposedly four dimensional space time universe,

No we are not.

So the only way to know if our spaceship travels in a four dimensional straight line is the need of having a four dimensional compass to keep track of our path, but to visualize such a thing is a little bit foggy for my mind.

No, the way of doing this is not a compass, it is something that checks if a line curves or not. In space-time, this is an accelerometer.

We will inevitably use the flat Earth as an analogy, so when I look at the WMAP picture, I assume that any point in the universe, any observe will get the exactly the same picture of the cosmic radiation, just as any person on the Earth flat land will see the same picture

I do not see how this is relevant. It is also not true.

Someone had said that if you can look far enough, you will eventually see the back of your head. I am still having a hard time able to visualize it though.

"Someone said" is not a valid reference.

But to do that we need a compass or some similar

No, a compass will not lead you on a great circle on the sphere. What you need is an instrument that can check if your path is straight.

 

[SleepDeprived]

There's been some developments to map many X-ray pulsars to be used to pinpoint your location in space.

https://en.m.wikipedia.org/wiki/X-ray_pulsar-based_navigation

So is this your final answer? :-)

 

[jedishrfu]

So is this your final answer? :-)

Only if this is your final question. :-)