Has the time "dimension" no spatial extension?

In summary: Te_j=0##?In summary, the GR works with the metric and does not depend on the embedding. The intrinsic curvature does not change if one composition f with a diffeomorphism. However, to find the f one needs to know the metric of the 5th dimension.
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It's just think that if we measure 1 second with a clock we should be able to "see" a 300'000km long piece of something in space or not ?

Or does the time extension only has to be understood as a set of numbers indicating timelaps, so that there is no "geometry" of time ?
 
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None of the above. It is just a four dimensional space time geometry. There is no physical fabric on which a clock could leave a 300,000 km brown streak.
 
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So the fourth dimension were in some sense hidden to the eye ?

Rephrasing the question were like this, maybe more precise :

Is it possible to have a metric coefficient for time depending only on r but in external geometry the manifold coordinate for time depends also on let say the polar angle ?

I know that GR works only with the metric and hence shall be independent of the embedding. I think this is based on Gauß theorem of intrinsic curvature, but if I remember well this says the intrinsic curvature does not depend on reparametring the manifold, but somewhere the shape of the manifold has to be given.

Like we give a manifold via ##f:\mathbb{R}^2\rightarrow\mathbb{R}^3## then the curvature does not change if we compose f with a diffeomorphism ##g:\mathbb{R}^2\rightarrow\mathbb{R}^2##.

But it seems to me that the intrinsic curvature is but depending on f of course, but not g ?

Since for example symmetry arguments would depend on f too, one needs to find f.

I was asking myself which work should be done, like :

Find ##M1\in M_{5\times 5} s.t. \exists P\in M_{4\times 5} s.t. P^T M1 P=(g_{\mu\nu})## where the latter is the GR given metric and then find ##f:\mathbb{R}^4\rightarrow\mathbb{R}^5## s.t ##M1=\langle e_i|e_j\rangle## with as usual ##e_i=D_i f##

But then since this 5th dimension is unknown shall it be considered only mathematically a euclidean background metric hence ##e_i^Te_j## or an asymptotic vanishing dimension with ##\lim_{\epsilon\rightarrow 0}e_i^T\cdot diag(1,1,1,-1,\epsilon)e_j##
 
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Related to Has the time "dimension" no spatial extension?

1. What is the "time dimension" and does it have spatial extension?

The "time dimension" refers to the concept of time as a measurable and continuous entity. It is often described as the fourth dimension in space-time. While time does not have physical spatial extension like length, width, and height, it is still considered a dimension because it is a necessary component in understanding the physical world.

2. How is the concept of time different from the other dimensions?

The concept of time is different from the other dimensions because it is not directly observable or measurable like length, width, and height. It is also not affected by gravity or motion in the same way as the other dimensions. However, time is still a fundamental aspect of our universe and is essential for understanding the laws of physics.

3. Can time be considered a physical dimension?

Time is often considered a physical dimension because it is a crucial component in understanding the physical world. However, it is not a physical dimension in the same sense as the other dimensions, as it cannot be measured or observed in the same way. Time is often described as a "spacetime interval" that combines both temporal and spatial components.

4. How does the concept of time relate to the theory of relativity?

The theory of relativity, specifically Einstein's theory of special relativity, states that time and space are intertwined and are relative to the observer's frame of reference. This means that time can be perceived differently by different observers depending on their relative speeds and positions. The theory of relativity also explains how time can be affected by gravity and motion.

5. Is time travel possible if time has no spatial extension?

While time travel is a popular concept in science fiction, it is not currently possible according to our current understanding of physics. Theories such as the theory of relativity suggest that time travel could be possible under certain conditions, but it is still a topic of ongoing research and debate. However, even if time has no spatial extension, it does not necessarily mean that time travel is impossible.

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