I guess you don’t understand what is meant by spacetime warp. If you agree that time moves differently at different elevations, you are conceding spacetime warp, whether you realize it or not. Spacetime warp is about how the relationship between space and time changes. If time changes due to elevation, then so does space because the two are inextricably linked. Space expands as time contracts and vice versa. If time changes, so does space. The faster you go in space, the slower you go in time. Everything moves at c in spacetime. If velocity is increased in space, then velocity through time has to decrease.

In the diagram below, three objects are moving through space at different elevations. If space time were flat, space and time would “line up” and you could connect the objects with the straight red lines. Instead, connecting the objects with the green line shows how the relationship between space and time changes from the higher elevation and the lower. That relationship is clearly

* warped*. Space expands and time contracts at lower elevations and space contracts and time expands at higher elevations. If that’s not the definition of warped, I don’t know what is.

If time moves faster at the top of an accelerating rocket ship it is because the space inside the rocket is warped

* due to acceleration*. This goes back to the point I made before about the equivalence principle and what happens to the worldline of an inertial object when you accelerate the frame of reference. It changes from straight to curved. I am sure you have seen or read about how light falls in an accelerating elevator. If you shine a light across an elevator at rest, the beam will shine straight across. When you accelerate the elevator upwards, the beam curves down. But there is no reason for it to curve down unless the space in the accelerating elevator is warped. There is no force on it to change its path through spacetime, so it must be the spacetime itself that is warped. Same concept applies to an accelerating rocket. As it accelerates, the spacetime inside becomes warped, just like the spacetime in the elevator as it accelerates.

Really, and why should it cause time to slow down at lower altitudes rather than cause time to speed up at lower altitudes?

I told you why. Lower altitude means more curvature. More curvature means more space and more space means less time.

The speed of light is always constant, and the distance is fixed, so time equals distance divided by speed. Now consider light traveling from point C to point D but in a curved line. The speed is constant, but the distance is longer than before. This means that the numerator is bigger than the previous equation. With a bigger numerator, this means that the time it takes for the light to travel the exact same distance is longer. Therefore time has just been warped.

Here is another way of looking at it. The lines are traveling at the same speed but the straight line is higher in elevation and not as effected by the warp. Therefore, it travels farther in the same amount of time.