Not quite.
First of all, what is space-time?
It's like the fabric of the universe. It's what we live in, and it's made of four dimensions: three for space (up-down, left-right, forward-backward) and one for time.
Now, according to Einstein's theory of general relativity, massive objects warp this fabric.
They make it bend and curve around them, like a bowling ball on a trampoline. And this bending and curving is what we feel as gravity.
The more massive the object, the more it warps space-time. For example, the Sun warps space-time way more than the Earth, and a black hole warps it even more than the Sun.
But here's the kicker: not all objects warp space-time in the same way.
They can warp it differently depending on their shape, their spin, their speed and their charge. For instance, a spinning object drags space-time around with it, like a spoon stirring honey.
This is called frame-dragging. A moving object stretches space-time behind it and squeezes it in front of it, like a rubber band. This is called length contraction.
And a charged object twists space-time around it, like a corkscrew.
This is called gravitomagnetism.
Verdict:
No, not all objects in space warp space-time in the same direction. They can warp it in different directions depending on their properties. And this can have some crazy effects on other objects and light that pass near them.
For example, a black hole can warp space-time so much that it can make light from behind it appear in front of it.
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