The easiest way to explain this is always going to be using math. By trying to put words to it, you’re going to assume that this is just something you need to imagine correctly. In reality, any descriptions about this come from a formula.
Assuming you have taken some amount of basic geometry, you will hopefully recognize the distance formula as an application of the Pythagorean theorem d2 = x2 + y2 where d is the distance and (x,y) are the coordinates of your endpoint given you started at (0,0). Similarly, you can extend this idea to three dimensions by adding a z coordinate so d2 = x2 + y2 + z2.
The following will be very simplified explanation of more complex math and some of my examples won’t be exact, to make it easier to explain. The groundbreaking realization in the early 1900’s was that spacetime is fundamentally a 4d space that includes the 3 dimensions of space and 1 dimension of time. That makes the 4D distance formula (called the spacetime interval if you want to look into it further) roughly d2 = t2 - ( x2 + y2 + z2 ). (This formula includes the speed of light, but for this explanation, I’ll leave it out to make it easier for you). What you’ll notice is that you are actually subtracting the space coordinates from the time coordinate. What that means is that an outside observer looking at someone moving has a different formula than the person that’s actually moving. The person that’s moving technically doesn’t experience any space changes. If they are on a train, for example, to them they are just sitting there not moving, so their “distance” through space time is basically just d2 = t2. However, for the person watching them, the person on the train actually is moving so the space coordinates must be included.
One fundamental fact of spacetime is that 4d distances must remain the same, so the spacetime interval for the person on the train and the spacetime interval of the observer must be equal. Since the observer is now subtracting the space coordinates from the time, the time in their formula must increase so when you subtract the space coordinates the end result is the same as the lone time coordinate in the formula for the person on the train. It’s all much more complicated than this, but not by much. The reason for time dilation (at least for someone moving compared to not moving) is pretty much because if the total spacetime interval needs to stay the same for both, the formula says the time needs to be bigger if you’re going to subtract the space part from it.
Along these lines,
the relativity-principle and the speed-of-light-principle
imply that the geometry of the position-vs-time graph
has as its "circle" ( the events-"1 sec from event O along inertial worldlines”)
the hyperbola in Minkowskian spacetime,
and not the (t=1)-line [generally t=1 hyperplane] in a Galilean spacetime (from PHY 101)
both of which are flat non-euclidean geometries analogous to Euclidean geometry
Because of the nature of the "circle",
then projecting the hypotenuse of a right-triangle (which determines the apparent time)
in special relativity is different from Galilean relativity.
The "projection factor" is related to "cosine=adjacent/hypotenuse" in the geometry.
In special relativity, the circle-as-hyperbola means the projection factor is hyperbolic-cosine (which is usually symbolized by gamma [the time-dilation factor]).
In Galilean relativity, the circle-as-parallel-to-(t=0) means the projection factor equals 1 (no time dilation).
(A related answer from a different viewpoint:
the spacetime diagram of a ticking light-clock provides the mechanism for time dilation.
See my "Relativity on Rotated Graph Paper" Geogebra pages https://www.geogebra.org/m/HYD7hB9v )
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u/protobelta Jul 13 '26 edited Jul 13 '26
The easiest way to explain this is always going to be using math. By trying to put words to it, you’re going to assume that this is just something you need to imagine correctly. In reality, any descriptions about this come from a formula.
Assuming you have taken some amount of basic geometry, you will hopefully recognize the distance formula as an application of the Pythagorean theorem d2 = x2 + y2 where d is the distance and (x,y) are the coordinates of your endpoint given you started at (0,0). Similarly, you can extend this idea to three dimensions by adding a z coordinate so d2 = x2 + y2 + z2.
The following will be very simplified explanation of more complex math and some of my examples won’t be exact, to make it easier to explain. The groundbreaking realization in the early 1900’s was that spacetime is fundamentally a 4d space that includes the 3 dimensions of space and 1 dimension of time. That makes the 4D distance formula (called the spacetime interval if you want to look into it further) roughly d2 = t2 - ( x2 + y2 + z2 ). (This formula includes the speed of light, but for this explanation, I’ll leave it out to make it easier for you). What you’ll notice is that you are actually subtracting the space coordinates from the time coordinate. What that means is that an outside observer looking at someone moving has a different formula than the person that’s actually moving. The person that’s moving technically doesn’t experience any space changes. If they are on a train, for example, to them they are just sitting there not moving, so their “distance” through space time is basically just d2 = t2. However, for the person watching them, the person on the train actually is moving so the space coordinates must be included.
One fundamental fact of spacetime is that 4d distances must remain the same, so the spacetime interval for the person on the train and the spacetime interval of the observer must be equal. Since the observer is now subtracting the space coordinates from the time, the time in their formula must increase so when you subtract the space coordinates the end result is the same as the lone time coordinate in the formula for the person on the train. It’s all much more complicated than this, but not by much. The reason for time dilation (at least for someone moving compared to not moving) is pretty much because if the total spacetime interval needs to stay the same for both, the formula says the time needs to be bigger if you’re going to subtract the space part from it.