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You can feel a gravitational field. Lead is much more susceptible to gravitation, and hydrogen much less so, than human tissue. You can also sense the absence of a gravitational field by jumping out of a plane
So it is perfectly acceptable to think of the field as a physical object that is subject to deformation? Is it a 3d +1 physical object?
As Alan says the field is a mathematical abstraction. What you do is place a test mass (1kg) and measure the force vector acting on that mass, that vector is the field strength (unit gravitational force) at that point (you can do the same with a magnetic field, using a test magnet). In this case you create a map of of gravitational force values for a volume of space.That’s all a field is. The value of it is that with gravity we can use a simple formula to calculate the field strength at any point between 2 masses, and hence the force between 2 masses.
Is that the famous Einstein Field Equation I have heard of? (if it is I am glad to learn it is simple )