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In many textbooks and online answers I have heard people go on about how time dilation accounts for the gap of time experienced between two observers. However, they continue that in reality this is not enough and we must have length contraction as well to truly close the gap.

From an intuitive perspective, can't time just slow down a lot and then this can cover the gap in time all by itself? Why must there also be length contraction to help cover the gap? Can't time just slow down more? I would really like to understand this on both the intuitive and mathematical level.

From an intuitive perspective, can't time just slow down a lot and then this can cover the gap in time all by itself? Why must there also be length contraction to help cover the gap?

Intuitively, the need for length contraction is because the gap you described above depends on the orientation.

For example, when deriving the expression for time dilation, one common approach is to consider what is known as a light clock. In particular, if the relative motion is in the x direction, then typically a light clock oriented along the y direction is analyzed.

What happens if you consider two identical light clocks and orient one in the y direction and the other in the x direction? It turns out that the first can be described by time dilation alone, but using the same approach for the x clock fails. To make both clocks work you need to include length contraction for the x clock.

A less intuitive but conceptually cleaner approach is to derive the Lorentz transform using your favorite approach, and then to view both time dilation and length contraction as features of the Lorentz transform.

First of all, time dilation concerns an observer and an observed particle. If the observed particle has a velocity with respect to the observer, the observation of the aging process of the particle will be subject to time dilation, that means the proper time of the particle is the time before time dilation, and the coordinate time measured by the observer will be the time after time dilation.

So the observer and the observed particle will not agree on the measured time, but in contrast, they will agree on their relative velocity. If A is receding from B, so B is receding with the same velocity from A. Velocity equals distance divided by time, and this is why the observed particle will measure not only a shorter time but also a contracted distance.

That means that length contraction is a direct consequence of time dilation.

I am not sure what you are asking, but my explanation on this site might answer your question Special Relativity: Length Contraction Confusion
If there are more than one object, they will appear closer together as well as shortened. And if you were moving at relativistic speed, objects behind or beside you would appear shortened, while those in front would appear lengthened because the light from the near end has less distance to go, so the near end is seen moving first.

Time, is a term describing temporal motion. Motion through the temporal dimension. Time dilation is a change in the rate at which we time, relative to something or someone else.

Because our experience of timing is subjective, the duration of a period varies, depending on our temporal velocity. A journey of ten light years might subjectively be experienced as a year, therefore the distance traveled feels subjectively shorter.

Think of it as taking a nap on an 8 hour flight. If you sleep for 6 hours, subjectively, your flight is only 2 hour long. The trip feels shorter.

Length contraction is a change in the observers subjective perception of reality. Time dilation is a result of a change in temporal velocity which is inversely proportional to our spatial velocity.